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diff --git a/buch/papers/dreieck/main.tex b/buch/papers/dreieck/main.tex
index 75ba410..d7bc769 100644
--- a/buch/papers/dreieck/main.tex
+++ b/buch/papers/dreieck/main.tex
@@ -3,19 +3,21 @@
 %
 % (c) 2020 Hochschule Rapperswil
 %
-\chapter{Dreieckstest und Beta-Funktion\label{chapter:dreieck}}
-\lhead{Dreieckstest und Beta-Funktion}
+\chapter{$\int P(t) e^{-t^2} \,dt$ in geschlossener Form?
+\label{chapter:dreieck}}
+\lhead{Integrierbarkeit in geschlossener Form}
 \begin{refsection}
 \chapterauthor{Andreas Müller}
 
 \noindent
-Mit dem Dreieckstest kann man feststellen, wie gut ein Geruchs-
-oder Geschmackstester verschiedene Gerüche oder Geschmäcker
-unterscheiden kann.
-Seine wahrscheinlichkeitstheoretische Erklärung benötigt die Beta-Funktion,
-man kann die Beta-Funktion als durchaus als die mathematische Grundlage
-der Weindegustation
-bezeichnen.
+Der Risch-Algorithmus erlaubt, eine definitive Antwort darauf zu geben,
+\index{Risch-Algorithmus}%
+\index{elementare Stammfunktion}%
+ob eine elementare Funktion eine Stammfunktion in geschlossener Form hat.
+Der Algorithmus ist jedoch ziemlich kompliziert.
+In diesem Kapitel soll ein spezieller Fall mit Hilfe der Theorie der
+orthogonale Polynome, speziell der Hermite-Polynome, behandelt werden,
+wie er in der Arbeit \cite{dreieck:polint} untersucht wurde.
 
 \input{papers/dreieck/teil0.tex}
 \input{papers/dreieck/teil1.tex}
diff --git a/buch/papers/dreieck/references.bib b/buch/papers/dreieck/references.bib
index d2bbe08..47bd865 100644
--- a/buch/papers/dreieck/references.bib
+++ b/buch/papers/dreieck/references.bib
@@ -4,32 +4,12 @@
 % (c) 2020 Autor, Hochschule Rapperswil
 %
 
-@online{dreieck:bibtex,
-	title = {BibTeX},
-	url = {https://de.wikipedia.org/wiki/BibTeX},
-	date = {2020-02-06},
-	year = {2020},
-	month = {2},
-	day = {6}
+@article{dreieck:polint,
+	author = { George Stoica },
+	title = { Polynomials and Integration in Finite Terms },
+	journal = { Amer. Math. Monthly },
+	volume = 129,
+	year = 2022,
+	number = 1,
+	pages = {80--81}
 }
-
-@book{dreieck:numerical-analysis,
-	title = {Numerical Analysis},
-	author = {David Kincaid and Ward Cheney},
-	publisher = {American Mathematical Society},
-	year = {2002},
-	isbn = {978-8-8218-4788-6},
-	inseries = {Pure and applied undegraduate texts},
-	volume = {2}
-}
-
-@article{dreieck:mendezmueller,
-        author = { Tabea Méndez and Andreas Müller },
-        title = { Noncommutative harmonic analysis and image registration },
-        journal = { Appl. Comput. Harmon. Anal.},
-        year = 2019,
-        volume = 47,
-        pages = {607--627},
-        url = {https://doi.org/10.1016/j.acha.2017.11.004}
-}
-
diff --git a/buch/papers/dreieck/teil0.tex b/buch/papers/dreieck/teil0.tex
index bcf2cf8..65eff7a 100644
--- a/buch/papers/dreieck/teil0.tex
+++ b/buch/papers/dreieck/teil0.tex
@@ -3,7 +3,48 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Testprinzip\label{dreieck:section:testprinzip}}
-\rhead{Testprinzip}
+\section{Problemstellung\label{dreieck:section:problemstellung}}
+\rhead{Problemstellung}
+Es ist bekannt, dass das Fehlerintegral
+\[
+\frac{1}{\sqrt{2\pi}\sigma} \int_{-\infty}^x e^{-\frac{t^2}{2\sigma}}\,dt
+\]
+nicht in geschlossener Form dargestellt werden kann.
+Mit der in Kapitel~\ref{buch:chapter:integral} skizzierten Theorie von
+Liouville und dem Risch-Algorithmus kann dies strengt gezeigt werden.
 
+Andererseits gibt es durchaus Integranden, die $e^{-t^2}$ enthalten,
+für die eine Stammfunktion in geschlossener Form gefunden werden kann.
+Zum Beispiel folgt aus der Ableitung
+\[
+\frac{d}{dt} e^{-t^2}
+=
+-2te^{-t^2}
+\]
+die Stammfunktion
+\[
+\int te^{-t^2}\,dt
+=
+-\frac12 e^{-t^2}.
+\]
+Leitet man $e^{-t^2}$ zweimal ab, erhält man
+\[
+\frac{d^2}{dt^2} e^{-t^2}
+=
+(4t^2-2) e^{-t^2}
+\qquad\Rightarrow\qquad
+\int (t^2-{\textstyle\frac12}) e^{-t^2}\,dt
+=
+{\textstyle\frac14}
+e^{-t^2}.
+\]
+Es gibt also eine viele weitere Polynome $P(t)$, für die der Integrand
+$P(t)e^{-t^2}$ eine Stammfunktion in geschlossener Form hat.
+Damit stellt sich jetzt das folgende allgemeine Problem.
+
+\begin{problem}
+\label{dreieck:problem}
+Für welche Polynome $P(t)$ hat der Integrand $P(t)e^{-t^2}$
+eine elementare Stammfunktion?
+\end{problem}
 
diff --git a/buch/papers/dreieck/teil1.tex b/buch/papers/dreieck/teil1.tex
index 4abe2e1..45c1a23 100644
--- a/buch/papers/dreieck/teil1.tex
+++ b/buch/papers/dreieck/teil1.tex
@@ -3,9 +3,92 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Ordnungsstatistik und Beta-Funktion
-\label{dreieck:section:ordnungsstatistik}}
-\rhead{}
+\section{Hermite-Polynome
+\label{dreieck:section:hermite-polynome}}
+\rhead{Hermite-Polyome}
+In Abschnitt~\ref{dreieck:section:problemstellung} hat sich schon angedeutet,
+dass die Polynome, die man durch Ableiten von $e^{-t^2}$ erhalten
+kann, bezüglich des gestellten Problems besondere Eigenschaften
+haben.
+Zunächst halten wir fest, dass die Ableitung einer Funktion der Form
+$P(t)e^{-t^2}$ mit einem Polynom $P(t)$ 
+\begin{equation}
+\frac{d}{dt} P(t)e^{-t^2}
+=
+P'(t)e^{-t^2} -2tP(t)e^{-t^2}
+=
+(P'(t)-2tP(t)) e^{-t^2}
+\label{dreieck:eqn:ableitung}
+\end{equation}
+ist.
+Insbesondere hat die Ableitung wieder die Form $Q(t)e^{-t^2}$
+mit einem Polynome $Q(t)$, welches man auch als
+\[
+Q(t)
+=
+e^{t^2}\frac{d}{dt}P(t)e^{-t^2}
+\]
+erhalten kann.
 
+Die Polynome, die man aus der Funktion $H_0(t)=e^{-t^2}$ durch
+Ableiten erhalten kann, wurden bereits in
+Abschnitt~\ref{buch:orthogonalitaet:section:rodrigues}
+bis auf ein Vorzeichen hergeleitet, sie heissen die Hermite-Polynome 
+\index{Hermite-Polynome}%
+und es gilt
+\[
+H_n(t) 
+=
+(-1)^n
+e^{t^2} \frac{d^n}{dt^n} e^{-t^2}.
+\]
+Das Vorzeichen dient dazu sicherzustellen, dass der Leitkoeffizient
+immer $1$ ist.
+Das Polynom $H_n(t)$ hat den Grad $n$.
+
+In Abschnitt wurde auch gezeigt, dass die Polynome $H_n(t)$
+bezüglich des Skalarproduktes
+\[
+\langle f,g\rangle_{w}
+=
+\int_{-\infty}^\infty f(t)g(t)e^{-t^2}\,dt,
+\qquad
+w(t)=e^{-t^2},
+\]
+orthogonal sind.
+Ausserdem folgt aus \eqref{dreieck:eqn:ableitung}
+die Rekursionsbeziehung
+\begin{equation}
+H_{n}(t)
+=
+2tH_{n-1}(t)
+-
+H_{n-1}'(t)
+\label{dreieck:eqn:rekursion}
+\end{equation}
+für $n>0$.
+
+Im Hinblick auf die Problemstellung ist jetzt die Frage interessant,
+ob die Integranden $H_n(t)e^{-t^2}$ eine Stammfunktion in geschlossener
+Form haben.
+Mit Hilfe der Rekursionsbeziehung~\eqref{dreieck:eqn:rekursion}
+kann man für $n>0$ unmittelbar verifizieren, dass
+\begin{align*}
+\int H_n(t)e^{-t^2}\,dt
+&=
+\int \bigl( 2tH_{n-1}(t) - H'_{n-1}(t)\bigr)e^{-t^2}\,dt
+\\
+&=
+-\int \bigl( \exp'(-t^2) H_{n-1}(t) + H'_{n-1}(t)\bigr)e^{-t^2}\,dt
+\\
+&=
+-\int \bigl( e^{-t^2}H_{n-1}(t)\bigr)' \,dt
+=
+-e^{-t^2}H_{n-1}(t)
+\end{align*}
+ist.
+Für $n>0$ hat also $H_n(t)e^{-t^2}$ eine elementare Stammfunktion.
+Die Hermite-Polynome sind also Lösungen für das
+Problem~\ref{dreieck:problem}.
 
 
diff --git a/buch/papers/dreieck/teil2.tex b/buch/papers/dreieck/teil2.tex
index 83ea3cb..8e89f6a 100644
--- a/buch/papers/dreieck/teil2.tex
+++ b/buch/papers/dreieck/teil2.tex
@@ -3,7 +3,113 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Wahrscheinlichkeiten im Dreieckstest
-\label{dreieck:section:wahrscheinlichkeiten}}
-\rhead{Wahrscheinlichkeiten}
+\section{Beliebige Polynome
+\label{dreieck:section:beliebig}}
+\rhead{Beliebige Polynome}
+Im Abschnitt~\ref{dreieck:section:hermite-polynome} wurden die
+Hermite-Polynome $H_n(t)$ mit $n>0$ als Lösungen des gestellten
+Problems erkannt.
+Eine Linearkombination von solchen Polynomen hat natürlich
+ebenfalls eine elementare Stammfunktion.
+Das Problem kann daher neu formuliert werden:
+
+\begin{problem}
+\label{dreieck:problem2}
+Welche Polynome $P(t)$ lassen sich aus den Hermite-Polynomen
+$H_n(t)$ mit $n>0$ linear kombinieren?
+\end{problem}
+
+Sei also
+\[
+P(t) = p_0 + p_1t + \ldots + p_{n-1}t^{n-1} + p_nt^n
+\]
+ein beliebiges Polynom vom Grad $n$.
+Eine elementare Stammfunktion von $P(t)e^{-t^2}$ existiert sicher,
+wenn sich $P(t)$ aus den Funktionen $H_n(t)$ mit $n>0$ linear
+kombinieren lässt.
+Gesucht ist also zunächst eine Darstellung von $P(t)$ als Linearkombination
+von Hermite-Polynomen.
+
+\begin{lemma}
+Jedes Polynome $P(t)$ vom Grad $n$ lässt sich auf eindeutige Art und
+Weise als Linearkombination
+\begin{equation}
+P(t) = a_0H_0(t) + a_1H_1(t) + \ldots + a_nH_n(t)
+=
+\sum_{k=0}^n a_nH_n(t)
+\label{dreieck:lemma}
+\end{equation}
+von Hermite-Polynomen schreiben.
+\end{lemma}
+
+\begin{proof}[Beweis]
+Zunächst halten wir fest, dass aus der
+Rekursionsformel~\eqref{dreieck:eqn:rekursion}
+folgt, dass der Leitkoeffizient bei jedem Rekursionsschnitt
+mit $2$ multipliziert wird.
+Der Leitkoeffizient von $H_n(t)$ ist also $2^n$.
+
+Wir führen den Beweis mit vollständiger Induktion.
+Für $n=0$ ist $P(t)=p_0 = p_0 H_0(t)$ als Linearkombination von
+Hermite-Polynomen darstellbar, dies ist die Induktionsverankerung.
+
+Wir nehmen jetzt im Sinne der Induktionsannahme an,
+dass sich ein Polynom vom Grad $n-1$ als
+Linearkombination der Polynome $H_0(t),\dots,H_{n-1}(t)$ schreiben
+lässt und untersuchen ein Polynom $P(t)$ vom Grad $n$.
+Da der Leitkoeffizient des Polynoms $H_n(t)$ ist $2^n$, ist zerlegen
+wir
+\[
+P(t)
+=
+\underbrace{\biggl(P(t) - \frac{p_n}{2^n} H_n(t)\biggr)}_{\displaystyle = Q(t)}
++
+\frac{p_n}{2^n} H_n(t).
+\]
+Das Polynom $Q(t)$ hat Grad $n-1$, besitzt also nach Induktionsannahme
+eine Darstellung
+\[
+Q(t) = a_0H_0(t)+a_1H_1(t)+\ldots+a_{n-1}H_{n-1}(t)
+\]
+als Linearkombination der Polynome $H_0(t),\dots,H_{n-1}(t)$.
+Somit ist
+\[
+P(t)
+= a_0H_0(t)+a_1H_1(t)+\ldots+a_{n-1}H_{n-1}(t) +
+\frac{p_n}{2^n} H_n(t)
+\]
+eine Darstellung von $P(t)$ als Linearkombination der Polynome
+$H_0(t),\dots,H_n(t)$.
+Damit ist der Induktionsschritt vollzogen und das Lemma für alle
+$n$ bewiesen.
+\end{proof}
+
+\begin{satz}
+\label{dreieck:satz1}
+Die Funktion $P(t)e^{-t^2}$ hat genau dann eine elementare Stammfunktion,
+wenn in der Darstellung~\eqref{dreieck:lemma}
+von $P(t)$ als Linearkombination von Hermite-Polynomen $a_0=0$ gilt.
+\end{satz}
+
+\begin{proof}[Beweis]
+Es ist
+\begin{align*}
+\int P(t)e^{-t^2}\,dt
+&=
+a_0\int e^{-t^2}\,dt
++
+\int
+\sum_{k=1} a_kH_k(t)\,dt
+\\
+&=
+a_0
+\frac{\sqrt{\pi}}2
+\operatorname{erf}(t)
++
+\sum_{k=1} a_k\int H_k(t)\,dt.
+\end{align*}
+Da die Integrale in der Summe alle elementar darstellbar sind,
+ist das Integral genau dann elementar, wenn $a_0=0$ ist.
+\end{proof}
+
 
diff --git a/buch/papers/dreieck/teil3.tex b/buch/papers/dreieck/teil3.tex
index e2dfd6b..c0c046a 100644
--- a/buch/papers/dreieck/teil3.tex
+++ b/buch/papers/dreieck/teil3.tex
@@ -3,8 +3,75 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Erweiterungen
-\label{dreieck:section:erweiterungen}}
-\rhead{Erweiterungen}
+\section{Integralbedingung
+\label{dreieck:section:integralbedingung}}
+\rhead{Lösung}
+Die Tatsache, dass die Hermite-Polynome orthogonal sind, erlaubt, das
+Kriterium von Satz~\ref{dreieck:satz1} in einer besonders attraktiven
+Integralform zu formulieren.
+
+Aus den Polynomen $H_n(t)$ lassen sich durch Normierung die
+\index{orthogonale Polynome}%
+\index{Polynome, orthogonale}%
+orthonormierten Polynome
+\[
+\tilde{H}_n(t)
+=
+\frac{1}{\| H_n\|_w} H_n(t)
+\qquad\text{mit}\quad
+\|H_n\|_w^2
+=
+\int_{-\infty}^\infty H_n(t)e^{-t^2}\,dt
+\]
+bilden.
+Da diese Polynome eine orthonormierte Basis des Vektorraums der Polynome
+bilden, kann die gesuchte Zerlegung eines Polynoms $P(t)$ auch mit
+Hilfe des Skalarproduktes gefunden werden:
+\begin{align*}
+P(t)
+&=
+\sum_{k=1}^n
+\langle \tilde{H}_k, P\rangle_w
+\tilde{H}_k(t)
+=
+\sum_{k=1}^n
+\biggl\langle \frac{H_k}{\|H_k\|_w}, P\biggr\rangle_w
+\frac{H_k(t)}{\|H_k\|_w}
+=
+\sum_{k=1}^n
+\underbrace{
+\frac{ \langle H_k, P\rangle_w }{\|H_k\|_w^2}
+}_{\displaystyle =a_k}
+H_k(t).
+\end{align*}
+Die Darstellung von $P(t)$ als Linearkombination von Hermite-Polynomen
+hat somit die Koeffizienten
+\[
+a_k = \frac{\langle H_k,P\rangle_w}{\|H_k\|_w^2}.
+\]
+Aus dem Kriterium $a_0=0$ dafür, dass eine elementare Stammfunktion
+von $P(t)e^{-t^2}$ existiert, wird daher die Bedingung, dass
+$\langle H_0,P\rangle_w=0$ ist.
+Da $H_0(t)=1$ ist, folgt als Bedingung
+\[
+a_0
+=
+\langle H_0,P\rangle_w
+=
+\int_{-\infty}^\infty P(t) e^{-t^2}\,dt
+=
+0.
+\]
+
+\begin{satz}
+Ein Integrand der Form $P(t)e^{-t^2}$ mit einem Polynom $P(t)$
+hat genau dann eine elementare Stammfunktion, wenn
+\[
+\int_{-\infty}^\infty P(t)e^{-t^2}\,dt = 0
+\]
+ist.
+\end{satz}
+
+
 
 
diff --git a/buch/papers/ellfilter/Makefile.inc b/buch/papers/ellfilter/Makefile.inc
index 8f20278..97e4089 100644
--- a/buch/papers/ellfilter/Makefile.inc
+++ b/buch/papers/ellfilter/Makefile.inc
@@ -3,12 +3,11 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-dependencies-ellfilter =						\
-	papers/ellfilter/packages.tex					\
-	papers/ellfilter/main.tex					\
-	papers/ellfilter/references.bib				\
-	papers/ellfilter/teil0.tex					\
-	papers/ellfilter/teil1.tex					\
-	papers/ellfilter/teil2.tex					\
-	papers/ellfilter/teil3.tex
-
+dependencies-ellfilter =                        \
+    papers/ellfilter/packages.tex               \
+    papers/ellfilter/main.tex                   \
+    papers/ellfilter/references.bib             \
+    papers/ellfilter/einleitung.tex             \
+    papers/ellfilter/tschebyscheff.tex          \
+    papers/ellfilter/jacobi.tex                 \
+    papers/ellfilter/elliptic.tex
diff --git a/buch/papers/ellfilter/einleitung.tex b/buch/papers/ellfilter/einleitung.tex
new file mode 100644
index 0000000..37fd89f
--- /dev/null
+++ b/buch/papers/ellfilter/einleitung.tex
@@ -0,0 +1,56 @@
+\section{Einleitung}
+
+% Lineare filter
+
+% Filter, Signalverarbeitung
+
+
+Der womöglich wichtigste Filtertyp ist das Tiefpassfilter.
+Dieses soll im Durchlassbereich unter der Grenzfrequenz $\Omega_p$ unverstärkt durchlassen und alle anderen Frequenzen vollständig auslöschen.
+
+% Bei der Implementierung von Filtern
+
+In der Elektrotechnik führen Schaltungen mit linearen Bauelementen wie Kondensatoren, Spulen und Widerständen immer zu linearen zeitinvarianten Systemen (LTI-System von englich \textit{time-invariant system}).
+Die Übertragungsfunktion im Frequenzbereich $|H(\Omega)|$ eines solchen Systems ist dabei immer eine rationale Funktion, also eine Division von zwei Polynomen.
+Die Polynome habe dabei immer reelle oder komplex-konjugierte Nullstellen.
+
+
+\begin{equation} \label{ellfilter:eq:h_omega}
+    | H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p}
+\end{equation}
+
+$\Omega = 2 \pi f$ ist die analoge Frequenz
+
+
+% Linear filter
+Damit das Filter implementierbar und stabil ist, muss $H(\Omega)^2$ eine rationale Funktion sein, deren Nullstellen und Pole auf der linken Halbebene liegen.
+
+$N \in \mathbb{N} $ gibt dabei die Ordnung des Filters vor, also die maximale Anzahl Pole oder Nullstellen.
+
+Damit ein Filter die Passband Kondition erfüllt muss $|F_N(w)| \leq 1 \forall |w| \leq 1$ und für $|w| \geq 1$ sollte die Funktion möglichst schnell divergieren.
+Eine einfaches Polynom, dass das erfüllt, erhalten wir wenn $F_N(w) = w^N$.
+Tatsächlich erhalten wir damit das Butterworth Filter, wie in Abbildung \ref{ellfilter:fig:butterworth} ersichtlich.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/python/F_N_butterworth.pgf}
+    \caption{$F_N$ für Butterworth filter. Der grüne Bereich definiert die erlaubten Werte für alle $F_N$-Funktionen.}
+    \label{ellfilter:fig:butterworth}
+\end{figure}
+
+wenn $F_N(w)$ eine rationale Funktion ist, ist auch $H(\Omega)$ eine rationale Funktion und daher ein lineares Filter. %proof?
+
+\begin{align}
+    F_N(w) & =
+    \begin{cases}
+        w^N                            & \text{Butterworth} \\
+        T_N(w)                         & \text{Tschebyscheff, Typ 1}  \\
+        [k_1 T_N (k^{-1} w^{-1})]^{-1} & \text{Tschebyscheff, Typ 2}  \\
+        R_N(w, \xi)                    & \text{Elliptisch (Cauer)}    \\
+    \end{cases}
+\end{align}
+
+Mit der Ausnahme vom Butterworth filter sind alle Filter nach speziellen Funktionen benannt.
+Alle diese Filter sind optimal für unterschiedliche Anwendungsgebiete.
+Das Butterworth-Filter, zum Beispiel, ist maximal flach im Durchlassbereich.
+Das Tschebyscheff-1 Filter sind maximal steil für eine definierte Welligkeit im Durchlassbereich, währendem es im Sperrbereich monoton abfallend ist.
+Es scheint so als sind gewisse Eigenschaften dieser speziellen Funktionen verantwortlich für die Optimalität dieser Filter.
diff --git a/buch/papers/ellfilter/elliptic.tex b/buch/papers/ellfilter/elliptic.tex
new file mode 100644
index 0000000..88bfbfe
--- /dev/null
+++ b/buch/papers/ellfilter/elliptic.tex
@@ -0,0 +1,92 @@
+\section{Elliptische rationale Funktionen}
+
+Kommen wir nun zum eigentlichen Teil dieses Papers, den elliptischen rationalen Funktionen
+\begin{align}
+    R_N(\xi, w) &= \cd \left(N~f_1(\xi)~\cd^{-1}(w, 1/\xi), f_2(\xi)\right) \\
+                &= \cd \left(N~\frac{K_1}{K}~\cd^{-1}(w, k), k_1)\right) , \quad k= 1/\xi, k_1 = 1/f(\xi) \\
+                &= \cd \left(N~K_1~z , k_1 \right), \quad w= \cd(z K, k)
+\end{align}
+
+
+sieht ähnlich aus wie die trigonometrische Darstellung der Tschebyschef-Polynome \eqref{ellfilter:eq:chebychef_polynomials}
+Anstelle vom Kosinus kommt hier die $\cd$-Funktion zum Einsatz.
+Die Ordnungszahl $N$ kommt auch als Faktor for.
+Zusätzlich werden noch zwei verschiedene elliptische Module $k$ und $k_1$ gebraucht.
+
+
+
+Sinus entspricht $\sn$
+
+Damit die Nullstellen an ähnlichen Positionen zu liegen kommen wie bei den Tschebyscheff-Polynomen, muss die $\cd$-Funktion gewählt werden.
+
+Die $\cd^{-1}(w, k)$-Funktion ist um $K$ verschoben zur $\sn^{-1}(w, k)$-Funktion, wie ersichtlich in Abbildung \ref{ellfilter:fig:cd}.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/cd.tikz.tex}
+    \caption{
+        $z$-Ebene der Funktion $z = \sn^{-1}(w, k)$.
+        Die Funktion ist in der realen Achse $4K$-periodisch und in der imaginären Achse $2jK^\prime$-periodisch.
+    }
+    \label{ellfilter:fig:cd}
+\end{figure}
+Auffallend ist, dass sich alle Nullstellen und Polstellen um $K$ verschoben haben.
+
+Durch das Konzept vom fundamentalen Rechteck, siehe Abbildung \ref{ellfilter:fig:fundamental_rectangle} können für alle inversen Jaccobi elliptischen Funktionen die Positionen der Null- und Polstellen anhand eines Diagramms ermittelt werden.
+Der erste Buchstabe bestimmt die Position der Nullstelle und der zweite Buchstabe die Polstelle.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/fundamental_rectangle.tikz.tex}
+    \caption{
+        Fundamentales Rechteck der inversen Jaccobi elliptischen Funktionen.
+    }
+    \label{ellfilter:fig:fundamental_rectangle}
+\end{figure}
+
+Auffallend an der $w = \sn(z, k)$-Funktion ist, dass sich $w$ auf der reellen Achse wie der Kosinus immer zwischen $-1$ und $1$ bewegt, während bei $\mathrm{Im(z) = K^\prime}$ die Werte zwischen $\pm 1/k$ und $\pm \infty$ verlaufen.
+Die Funktion hat also Equirippel-Verhalten um $w=0$ und um $w=\pm \infty$.
+Falls es möglich ist diese Werte abzufahren im Sti der Tschebyscheff-Polynome, kann ein Filter gebaut werden, dass Equirippel-Verhalten im Durchlass- und Sperrbereich aufweist.
+
+
+
+Analog zu Abbildung \ref{ellfilter:fig:arccos2} können wir auch bei den elliptisch rationalen Funktionen die komplexe $z$-Ebene betrachten, wie ersichtlich in Abbildung \ref{ellfilter:fig:cd2}, um die besser zu verstehen.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/cd2.tikz.tex}
+    \caption{
+        $z_1$-Ebene der elliptischen rationalen Funktionen.
+        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen passiert.
+    }
+    \label{ellfilter:fig:cd2}
+\end{figure}
+% Da die $\cd^{-1}$-Funktion 
+
+
+
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/python/F_N_elliptic.pgf}
+    \caption{$F_N$ für ein elliptischs filter.}
+    \label{ellfilter:fig:elliptic}
+\end{figure}
+
+\subsection{Degree Equation}
+
+Der $\cd^{-1}$ Term muss so verzogen werden, dass die umgebene $\cd$-Funktion die Nullstellen und Pole trifft.
+Dies trifft ein wenn die Degree Equation erfüllt ist.
+
+\begin{equation}
+    N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
+\end{equation}
+
+
+Leider ist das lösen dieser Gleichung nicht trivial.
+Die Rechnung wird in \ref{ellfilter:bib:orfanidis} im Detail angeschaut.
+
+
+\subsection{Polynome?}
+
+Bei den Tschebyscheff-Polynomen haben wir gesehen, dass die Trigonometrische Formel zu einfachen Polynomen umgewandelt werden kann.
+Im gegensatz zum $\cos^{-1}$ hat der $\cd^{-1}$ nicht nur Nullstellen sondern auch Pole.
+Somit entstehen bei den elliptischen rationalen Funktionen, wie es der name auch deutet, rationale Funktionen, also ein Bruch von zwei Polynomen.
+
+Da Transformationen einer rationalen Funktionen mit Grundrechenarten, wie es in \eqref{ellfilter:eq:h_omega} der Fall ist, immer noch rationale Funktionen ergeben, stellt dies kein Problem für die Implementierung dar.
diff --git a/buch/papers/ellfilter/jacobi.tex b/buch/papers/ellfilter/jacobi.tex
new file mode 100644
index 0000000..6a208fa
--- /dev/null
+++ b/buch/papers/ellfilter/jacobi.tex
@@ -0,0 +1,189 @@
+\section{Jacobische elliptische Funktionen}
+
+%TODO $z$ or $u$ for parameter?
+
+Für das elliptische Filter wird statt der, für das Tschebyscheff-Filter benutzen Kreis-Trigonometrie die elliptischen Funktionen gebraucht.
+Der Begriff elliptische Funktion wird für sehr viele Funktionen gebraucht, daher ist es hier wichtig zu erwähnen, dass es ausschliesslich um die Jacobischen elliptischen Funktionen geht.
+
+Im Wesentlichen erweitern die Jacobi elliptischen Funktionen die trigonometrische Funktionen für Ellipsen.
+Zum Beispiel gibt es analog zum Sinus den elliptischen $\sn(z, k)$.
+Im Gegensatz zum den trigonometrischen Funktionen haben die elliptischen Funktionen zwei parameter.
+Zum einen gibt es den \textit{elliptische Modul} $k$, der die Exzentrizität der Ellipse parametrisiert.
+Zum andern das Winkelargument $z$.
+Im Kreis ist der Radius für alle Winkel konstant, bei Ellipsen ändert sich das.
+Dies hat zur Folge, dass bei einer Ellipse die Kreisbodenstrecke nicht linear zum Winkel verläuft.
+Darum kann hier nicht der gewohnte Winkel verwendet werden.
+Das Winkelargument $z$ kann durch das elliptische Integral erster Art
+\begin{equation}
+    z
+    =
+    F(\phi, k)
+    =
+    \int_{0}^{\phi}
+    \frac{
+        d\theta
+    }{
+        \sqrt{
+            1-k^2 \sin^2 \theta
+        }
+    }
+    =
+    \int_{0}^{\phi}
+    \frac{
+        dt
+    }{
+        \sqrt{
+            (1-t^2)(1-k^2 t^2)
+        }
+    } %TODO which is right? are both functions from phi?
+\end{equation}
+mit dem Winkel $\phi$ in Verbindung liegt.
+
+Dabei wird das vollständige und unvollständige Elliptische integral unterschieden.
+Beim vollständigen Integral
+\begin{equation}
+    K(k)
+    =
+    \int_{0}^{\pi / 2}
+    \frac{
+        d\theta
+    }{
+        \sqrt{
+            1-k^2 \sin^2 \theta
+        }
+    }
+\end{equation}
+wird über ein viertel Ellipsenbogen integriert also bis $\phi=\pi/2$ und liefert das Winkelargument für eine Vierteldrehung.
+Die Zahl wird oft auch abgekürzt mit $K = K(k)$ und ist für das elliptische Filter sehr relevant.
+Alle elliptishen Funktionen sind somit $4K$-periodisch.
+
+Neben dem $\sn$ gibt es zwei weitere basis-elliptische Funktionen $\cn$ und $\dn$.
+Dazu kommen noch weitere abgeleitete Funktionen, die durch Quotienten und Kehrwerte dieser Funktionen zustande kommen.
+Insgesamt sind es die zwölf Funktionen
+\begin{equation*}
+    \sn \quad
+    \ns \quad
+    \scelliptic \quad
+    \sd \quad
+    \cn \quad
+    \nc \quad
+    \cs \quad
+    \cd \quad
+    \dn \quad
+    \nd \quad
+    \ds \quad
+    \dc.
+\end{equation*}
+
+Die Jacobischen elliptischen Funktionen können mit der inversen Funktion des kompletten elliptischen Integrals erster Art
+\begin{equation}
+    \phi = F^{-1}(z, k)
+\end{equation}
+definiert werden. Dabei ist zu beachten dass nur das $z$ Argument der Funktion invertiert wird, also
+\begin{equation}
+    z = F(\phi, k)
+    \Leftrightarrow
+    \phi = F^{-1}(z, k).
+\end{equation}
+Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem Elliptischen integral dargestellt werden:
+\begin{equation}
+    \sin(\phi)
+    =
+    \sin \left( F^{-1}(z, k) \right)
+    =
+    \sn(z, k)
+    =
+    w
+\end{equation}
+
+\begin{equation}
+    \phi
+    =
+     F^{-1}(z, k)
+     =
+     \sin^{-1} \big( \sn (z, k ) \big)
+     =
+    \sin^{-1} ( w )
+\end{equation}
+
+\begin{equation}
+    F(\phi, k)
+    =
+    z
+    =
+    F( \sin^{-1} \big( \sn (z, k ) \big) , k)
+    =
+    F( \sin^{-1} ( w ), k)
+\end{equation}
+
+\begin{equation}
+    \sn^{-1}(w, k)
+    =
+    F(\phi, k),
+    \quad
+    \phi = \sin^{-1}(w)
+\end{equation}
+
+\begin{align}
+    \sn^{-1}(w, k)
+        & =
+    \int_{0}^{\phi}
+    \frac{
+        d\theta
+    }{
+        \sqrt{
+            1-k^2 \sin^2 \theta
+        }
+    },
+    \quad
+    \phi = \sin^{-1}(w)
+    \\
+        & =
+    \int_{0}^{w}
+    \frac{
+        dt
+    }{
+        \sqrt{
+            (1-t^2)(1-k^2 t^2)
+        }
+    }
+\end{align}
+
+Beim $\cos^{-1}(x)$ haben wir gesehen, dass die analytische Fortsetzung bei $x < -1$ und $x > 1$ rechtwinklig in die Komplexen zahlen wandert.
+Wenn man das gleiche mit $\sn^{-1}(w, k)$ macht, erkennt man zwei interessante Stellen.
+Die erste ist die gleiche wie beim $\cos^{-1}(x)$ nämlich bei $t = \pm 1$.
+Der erste Term unter der Wurzel wird dann negativ, während der zweite noch positiv ist, da $k \leq 1$.
+\begin{equation}
+    \frac{
+        1
+    }{
+        \sqrt{
+            (1-t^2)(1-k^2 t^2)
+        }
+    }
+    \in \mathbb{R}
+    \quad \forall \quad
+    -1 \leq t \leq 1
+\end{equation}
+Die zweite stelle passiert wenn beide Faktoren unter der Wurzel negativ werden, was bei $t = 1/k$ der Fall ist.
+
+
+
+
+Funktion in relle und komplexe Richtung periodisch
+
+In der reellen Richtung ist sie $4K(k)$-periodisch und in der imaginären Richtung $4K^\prime(k)$-periodisch.
+
+
+
+%TODO sn^{-1} grafik
+
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/sn.tikz.tex}
+    \caption{
+        $z$-Ebene der Funktion $z = \sn^{-1}(w, k)$.
+        Die Funktion ist in der realen Achse $4K$-periodisch und in der imaginären Achse $2jK^\prime$-periodisch.
+    }
+    % \label{ellfilter:fig:cd2}
+\end{figure}
diff --git a/buch/papers/ellfilter/main.tex b/buch/papers/ellfilter/main.tex
index 26aaec1..c58dfa7 100644
--- a/buch/papers/ellfilter/main.tex
+++ b/buch/papers/ellfilter/main.tex
@@ -8,29 +8,10 @@
 \begin{refsection}
 \chapterauthor{Nicolas Tobler}
 
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht.
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile.
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt
-anzuwenden.
-\item
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
-
-\input{papers/ellfilter/teil0.tex}
-\input{papers/ellfilter/teil1.tex}
-\input{papers/ellfilter/teil2.tex}
-\input{papers/ellfilter/teil3.tex}
+\input{papers/ellfilter/einleitung.tex}
+\input{papers/ellfilter/tschebyscheff.tex}
+\input{papers/ellfilter/jacobi.tex}
+\input{papers/ellfilter/elliptic.tex}
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/ellfilter/packages.tex b/buch/papers/ellfilter/packages.tex
index c94db34..9a550e2 100644
--- a/buch/papers/ellfilter/packages.tex
+++ b/buch/papers/ellfilter/packages.tex
@@ -8,3 +8,20 @@
 % following example
 %\usepackage{packagename}
 
+% \usepackage[dvipsnames]{xcolor}
+
+\usetikzlibrary{trees,shapes,decorations}
+
+\DeclareMathOperator{\sn}{\mathrm{sn}}
+\DeclareMathOperator{\ns}{\mathrm{ns}}
+\DeclareMathOperator{\scelliptic}{\mathrm{sc}}
+\DeclareMathOperator{\sd}{\mathrm{sd}}
+\DeclareMathOperator{\cn}{\mathrm{cn}}
+\DeclareMathOperator{\nc}{\mathrm{nc}}
+\DeclareMathOperator{\cs}{\mathrm{cs}}
+\DeclareMathOperator{\cd}{\mathrm{cd}}
+\DeclareMathOperator{\dn}{\mathrm{dn}}
+\DeclareMathOperator{\nd}{\mathrm{nd}}
+\DeclareMathOperator{\ds}{\mathrm{ds}}
+\DeclareMathOperator{\dc}{\mathrm{dc}}
+
diff --git a/buch/papers/ellfilter/presentation/presentation.tex b/buch/papers/ellfilter/presentation/presentation.tex
new file mode 100644
index 0000000..7fdb864
--- /dev/null
+++ b/buch/papers/ellfilter/presentation/presentation.tex
@@ -0,0 +1,413 @@
+\documentclass[ngerman, aspectratio=169, xcolor={rgb}]{beamer}
+
+% style
+\mode<presentation>{
+	\usetheme{Frankfurt}
+}
+%packages
+\usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+\usepackage[english]{babel}
+\usepackage{graphicx}
+\usepackage{array}
+
+\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\usepackage{ragged2e}
+
+\usepackage{bm} % bold math
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{mathtools}
+\usepackage{amsmath}
+\usepackage{multirow} % multi row in tables
+\usepackage{booktabs} %toprule midrule bottomrue in tables
+\usepackage{scrextend}
+\usepackage{textgreek}
+\usepackage[rgb]{xcolor}
+
+\usepackage{ marvosym } % \Lightning
+
+\usepackage{multimedia} % embedded videos
+
+\usepackage{tikz}
+\usepackage{pgf}
+\usepackage{pgfplots}
+
+\usepackage{algorithmic}
+
+%citations
+\usepackage[style=verbose,backend=biber]{biblatex}
+\addbibresource{references.bib}
+
+
+%math font
+\usefonttheme[onlymath]{serif}
+
+%Beamer Template modifications
+%\definecolor{mainColor}{HTML}{0065A3} % HSR blue
+\definecolor{mainColor}{HTML}{D72864} % OST pink
+\definecolor{invColor}{HTML}{28d79b} % OST pink
+\definecolor{dgreen}{HTML}{38ad36} % Dark green
+
+%\definecolor{mainColor}{HTML}{000000} % HSR blue
+\setbeamercolor{palette primary}{bg=white,fg=mainColor}
+\setbeamercolor{palette secondary}{bg=orange,fg=mainColor}
+\setbeamercolor{palette tertiary}{bg=yellow,fg=red}
+\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic
+\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points)
+\setbeamercolor{section in toc}{fg=black} % TOC sections
+\setbeamertemplate{section in toc}[sections numbered]
+\setbeamertemplate{subsection in toc}{%
+	\hspace{1.2em}{$\bullet$}~\inserttocsubsection\par}
+
+\setbeamertemplate{itemize items}[circle]
+\setbeamertemplate{description item}[circle]
+\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true]
+\beamertemplatenavigationsymbolsempty
+
+\setbeamercolor{footline}{fg=gray}
+\setbeamertemplate{footline}{%
+	\hfill\usebeamertemplate***{navigation symbols}
+	\hspace{0.5cm}
+	\insertframenumber{}\hspace{0.2cm}\vspace{0.2cm}
+}
+
+\usepackage{caption}
+\captionsetup{labelformat=empty}
+
+%Title Page
+\title{Elliptische Filter}
+\subtitle{Eine Anwendung der Jaccobi elliptischen Funktionen}
+\author{Nicolas Tobler}
+% \institute{OST Ostschweizer Fachhochschule}
+% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}}
+\date{\today}
+
+\input{../packages.tex}
+
+\newcommand*{\QED}{\hfill\ensuremath{\blacksquare}}%
+
+\newcommand*{\HL}{\textcolor{mainColor}}
+\newcommand*{\RD}{\textcolor{red}}
+\newcommand*{\BL}{\textcolor{blue}}
+\newcommand*{\GN}{\textcolor{dgreen}}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+
+\makeatletter
+\newcount\my@repeat@count
+\newcommand{\myrepeat}[2]{%
+	\begingroup
+	\my@repeat@count=\z@
+	\@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}%
+	\endgroup
+}
+\makeatother
+
+\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings}
+
+\begin{document}
+
+	\begin{frame}
+		\titlepage
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Content}
+		\tableofcontents
+	\end{frame}
+
+	\section{Linear Filter}
+
+	\begin{frame}
+		\frametitle{Lineare Filter}
+
+
+		\begin{equation}
+			| H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p}
+		\end{equation}
+
+		\pause
+
+		\begin{equation}
+			F_N(w) = w^N
+		\end{equation}
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Beispiel: Butterworth Filter}
+
+		\begin{equation}
+			F_N(w) = w^N
+		\end{equation}
+
+		\begin{center}
+			\input{../python/F_N_butterworth.pgf}
+		\end{center}
+
+	\end{frame}
+
+
+	\begin{frame}
+		\frametitle{Arten von linearen filtern}
+
+		\begin{align*}
+			F_N(w) & =
+			\begin{cases}
+				w^N                            & \text{Butterworth} \\
+				T_N(w)                         & \text{Tschebyscheff, Typ 1}  \\
+				[k_1 T_N (k^{-1} w^{-1})]^{-1} & \text{Tschebyscheff, Typ 2}  \\
+				R_N(w,\xi)                      & \text{Elliptisch (Cauer)}    \\
+			\end{cases}
+		\end{align*}
+
+	\end{frame}
+
+	\section{Tschebycheff Filter}
+
+	\begin{frame}
+		\frametitle{Tschebyscheff-Polynome}
+
+
+		\begin{columns}
+			\begin{column}[T]{0.35\textwidth}
+
+				\begin{align*}
+					T_{0}(x)&=1\\
+					T_{1}(x)&=x\\
+					T_{2}(x)&=2x^{2}-1\\
+					T_{3}(x)&=4x^{3}-3x\\
+					T_{n+1}(x)&=2x~T_{n}(x)-T_{n-1}(x)
+				\end{align*}
+
+			\end{column}
+			\begin{column}[T]{0.65\textwidth}
+
+				\begin{center}
+					\resizebox{\textwidth}{!}{
+					\input{../python/F_N_chebychev2.pgf}
+					}
+				\end{center}
+
+			\end{column}
+		\end{columns}
+
+
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Tschebyscheff-Filter}
+
+		\begin{equation*}
+			| H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 T_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p}
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.9}{
+				\input{../python/F_N_chebychev.pgf}
+			}
+		\end{center}
+
+	\end{frame}
+
+
+	\begin{frame}
+		\frametitle{Tschebyscheff-Filter}
+
+		Darstellung mit trigonometrischen Funktionen:
+
+		\begin{align} \label{ellfilter:eq:chebychef_polynomials}
+			T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\
+				&= \cos \left(N~z \right), \quad w= \cos(z)
+		\end{align}
+
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Tschebyscheff-Filter}
+
+		\begin{equation*}
+			z = \cos^{-1}(w)
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.85}{
+				\input{../tikz/arccos.tikz.tex}
+			}
+		\end{center}
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Tschebyscheff-Filter}
+
+		\begin{equation*}
+			z_1 = N~\cos^{-1}(w)
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.85}{
+				\input{../tikz/arccos2.tikz.tex}
+			}
+		\end{center}
+
+	\end{frame}
+
+
+	\section{Jaccobi elliptische Funktionen}
+
+	\begin{frame}
+		\frametitle{Jaccobi elliptische Funktionen}
+
+
+		\begin{equation}
+			z
+			=
+			F(\phi, k)
+			=
+			\int_{0}^{\phi}
+			\frac{
+				d\theta
+			}{
+				\sqrt{
+					1-k^2 \sin^2 \theta
+				}
+			}
+			=
+			\int_{0}^{\phi}
+			\frac{
+				dt
+			}{
+				\sqrt{
+					(1-t^2)(1-k^2 t^2)
+				}
+			}
+		\end{equation}
+
+		\begin{equation}
+			K(k)
+			=
+			\int_{0}^{\pi / 2}
+			\frac{
+				d\theta
+			}{
+				\sqrt{
+					1-k^2 \sin^2 \theta
+				}
+			}
+		\end{equation}
+
+
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Jaccobi elliptische Funktionen}
+
+		\begin{equation*}
+			z = \sn^{-1}(w, k)
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.7}{
+				\input{../tikz/sn.tikz.tex}
+			}
+		\end{center}
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Fundamentales Rechteck}
+
+		Nullstelle beim ersten Buchstabe, Polstelle beim zweiten Buchstabe
+
+		\begin{center}
+			\scalebox{0.8}{
+				\input{../tikz/fundamental_rectangle.tikz.tex}
+			}
+		\end{center}
+
+	\end{frame}
+
+
+	\begin{frame}
+		\frametitle{Jaccobi elliptische Funktionen}
+
+		\begin{equation*}
+			z = \cd^{-1}(w, k)
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.7}{
+				\input{../tikz/cd.tikz.tex}
+
+			}
+		\end{center}
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Periodizität in realer und imaginärer Richtung}
+
+		\begin{center}
+			\input{../python/k.pgf}
+		\end{center}
+
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Elliptisches Filter}
+
+		\begin{equation*}
+			z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k)
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.8}{
+				\input{../tikz/cd2.tikz.tex}
+			}
+		\end{center}
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Elliptisches Filter}
+
+		\begin{columns}
+
+			\begin{column}[T]{0.5\textwidth}
+
+				\begin{center}
+					\resizebox{\textwidth}{!}{
+					\input{../python/F_N_elliptic.pgf}
+					}
+				\end{center}
+
+			\end{column}
+			\begin{column}[T]{0.5\textwidth}
+
+				\begin{center}
+					\resizebox{\textwidth}{!}{
+					\input{../python/elliptic.pgf}
+					}
+				\end{center}
+
+			\end{column}
+		\end{columns}
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Gradgleichung}
+
+		\begin{equation}
+			N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
+		\end{equation}
+
+	\end{frame}
+
+	\end{document}
diff --git a/buch/papers/ellfilter/python/F_N_butterworth.pgf b/buch/papers/ellfilter/python/F_N_butterworth.pgf
new file mode 100644
index 0000000..857e363
--- /dev/null
+++ b/buch/papers/ellfilter/python/F_N_butterworth.pgf
@@ -0,0 +1,1083 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%%   \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%%   \usepackage{pgf}
+%%
+%% Also ensure that all the required font packages are loaded; for instance,
+%% the lmodern package is sometimes necessary when using math font.
+%%   \usepackage{lmodern}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%%   \usepackage{import}
+%%
+%% and then include the figures with
+%%   \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
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new file mode 100644
index 0000000..72d5834
--- /dev/null
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new file mode 100644
index 0000000..43ebb91
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diff --git a/buch/papers/ellfilter/python/F_N_elliptic.pgf b/buch/papers/ellfilter/python/F_N_elliptic.pgf
new file mode 100644
index 0000000..03084c6
--- /dev/null
+++ b/buch/papers/ellfilter/python/F_N_elliptic.pgf
@@ -0,0 +1,847 @@
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diff --git a/buch/papers/ellfilter/python/chebychef.py b/buch/papers/ellfilter/python/chebychef.py
new file mode 100644
index 0000000..254ad4b
--- /dev/null
+++ b/buch/papers/ellfilter/python/chebychef.py
@@ -0,0 +1,66 @@
+# %%
+
+import matplotlib.pyplot as plt
+import scipy.signal
+import numpy as np
+
+
+order = 5
+passband_ripple_db = 1
+omega_c = 1000
+
+a, b = scipy.signal.cheby1(
+    order,
+    passband_ripple_db,
+    omega_c,
+    btype='low',
+    analog=True,
+    output='ba',
+    fs=None,
+)
+
+w, mag, phase = scipy.signal.bode((a, b), w=np.linspace(0,2000,256))
+f, axs = plt.subplots(2,1, sharex=True)
+axs[0].plot(w, 10**(mag/20))
+axs[0].set_ylabel("$|H(\omega)| /$ db")
+axs[0].grid(True, "both")
+axs[1].plot(w, phase)
+axs[1].set_ylabel(r"$arg H (\omega) / $ deg")
+axs[1].grid(True, "both")
+axs[1].set_xlim([0, 2000])
+axs[1].set_xlabel("$\omega$")
+plt.show()
+
+
+# %% Cheychev filter F_N plot
+
+w = np.linspace(-1.1,1.1, 1000)
+plt.figure(figsize=(5.5,2.5))
+for N in [3,6,11]:
+    # F_N = np.cos(N * np.arccos(w))
+    F_N = scipy.special.eval_chebyt(N, w)
+    plt.plot(w, F_N, label=f"$N={N}$")
+plt.xlim([-1.2,1.2])
+plt.ylim([-2,2])
+plt.grid()
+plt.xlabel("$w$")
+plt.ylabel("$T_N(w)$")
+plt.legend()
+plt.tight_layout()
+plt.savefig("F_N_chebychev2.pgf")
+plt.show()
+
+# %% Build Chebychev polynomials
+
+N = 11
+
+zeros = (np.arange(N)+0.5) * np.pi
+zeros = np.cos(zeros/N)
+
+x = np.linspace(-1.2,1.2,1000)
+y = np.prod(x[:, None] - zeros[None, :], axis=-1)*2**(N-1)
+
+plt.plot(x, y)
+plt.ylim([-1,1])
+plt.grid()
+plt.show()
diff --git a/buch/papers/ellfilter/python/elliptic.pgf b/buch/papers/ellfilter/python/elliptic.pgf
new file mode 100644
index 0000000..31b77d4
--- /dev/null
+++ b/buch/papers/ellfilter/python/elliptic.pgf
@@ -0,0 +1,709 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%%   \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%%   \usepackage{pgf}
+%%
+%% Also ensure that all the required font packages are loaded; for instance,
+%% the lmodern package is sometimes necessary when using math font.
+%%   \usepackage{lmodern}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%%   \usepackage{import}
+%%
+%% and then include the figures with
+%%   \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
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diff --git a/buch/papers/ellfilter/python/elliptic.py b/buch/papers/ellfilter/python/elliptic.py
new file mode 100644
index 0000000..b3336a1
--- /dev/null
+++ b/buch/papers/ellfilter/python/elliptic.py
@@ -0,0 +1,356 @@
+
+# %%
+
+import scipy.special
+import scipyx as spx
+import numpy as np
+import matplotlib.pyplot as plt
+from matplotlib.patches import Rectangle
+
+import plot_params
+
+def last_color():
+    return plt.gca().lines[-1].get_color()
+
+# define elliptic functions
+
+def ell_int(k):
+    """ Calculate K(k) """
+    m = k**2
+    return scipy.special.ellipk(m)
+
+def sn(z, k):
+    return spx.ellipj(z, k**2)[0]
+
+def cn(z, k):
+    return spx.ellipj(z, k**2)[1]
+
+def dn(z, k):
+    return spx.ellipj(z, k**2)[2]
+
+def cd(z, k):
+    sn, cn, dn, ph = spx.ellipj(z, k**2)
+    return cn / dn
+
+# https://mathworld.wolfram.com/JacobiEllipticFunctions.html eq 3-8
+
+def sn_inv(z, k):
+    m = k**2
+    return scipy.special.ellipkinc(np.arcsin(z), m)
+
+def cn_inv(z, k):
+    m = k**2
+    return scipy.special.ellipkinc(np.arccos(z), m)
+
+def dn_inv(z, k):
+    m = k**2
+    x = np.sqrt((1-z**2) / k**2)
+    return scipy.special.ellipkinc(np.arcsin(x), m)
+
+def cd_inv(z, k):
+    m = k**2
+    x = np.sqrt(((m - 1) * z**2) / (m*z**2 - 1))
+    return scipy.special.ellipkinc(np.arccos(x), m)
+
+
+k = 0.8
+z = 0.5
+
+assert np.allclose(sn_inv(sn(z ,k), k), z)
+assert np.allclose(cn_inv(cn(z ,k), k), z)
+assert np.allclose(dn_inv(dn(z ,k), k), z)
+assert np.allclose(cd_inv(cd(z ,k), k), z)
+
+
+# %% Buttwerworth filter F_N plot
+
+w = np.linspace(0,1.5, 100)
+plt.figure(figsize=(4,2.5))
+
+for N in range(1,5):
+    F_N = w**N
+    plt.plot(w, F_N**2, label=f"$N={N}$")
+plt.gca().add_patch(Rectangle(
+    (0, 0),
+    1, 1,
+    fc ='green',
+    alpha=0.2,
+    lw = 10,
+))
+plt.gca().add_patch(Rectangle(
+    (1, 1),
+    0.5, 1,
+    fc ='orange',
+    alpha=0.2,
+    lw = 10,
+))
+plt.xlim([0,1.5])
+plt.ylim([0,2])
+plt.grid()
+plt.xlabel("$w$")
+plt.ylabel("$F^2_N(w)$")
+plt.legend()
+plt.tight_layout()
+plt.savefig("F_N_butterworth.pgf")
+plt.show()
+
+# %% Cheychev filter F_N plot
+
+w = np.linspace(0,1.5, 100)
+
+plt.figure(figsize=(4,2.5))
+for N in range(1,5):
+    # F_N = np.cos(N * np.arccos(w))
+    F_N = scipy.special.eval_chebyt(N, w)
+    plt.plot(w, F_N**2, label=f"$N={N}$")
+plt.gca().add_patch(Rectangle(
+    (0, 0),
+    1, 1,
+    fc ='green',
+    alpha=0.2,
+    lw = 10,
+))
+plt.gca().add_patch(Rectangle(
+    (1, 1),
+    0.5, 1,
+    fc ='orange',
+    alpha=0.2,
+    lw = 10,
+))
+plt.xlim([0,1.5])
+plt.ylim([0,2])
+plt.grid()
+plt.xlabel("$w$")
+plt.ylabel("$F^2_N(w)$")
+plt.legend()
+plt.tight_layout()
+plt.savefig("F_N_chebychev.pgf")
+plt.show()
+
+
+# %% plot arcsin
+
+def lattice(a1, b1, c1, a2, b2, c2):
+    r1 = np.logspace(a1, b1, c1)
+    x1 = np.concatenate((-np.flip(r1), [0], r1), axis=0)
+    x1 = x1.astype(np.complex128)
+    r2 = np.logspace(a2, b2, c2)
+    x2 = np.concatenate((-np.flip(r2), [0], r2), axis=0)
+    x2 = x2.astype(np.complex128)
+    x = (x1[:, None] + (x2[None, :] * 1j))
+    return x
+
+plt.figure(figsize=(12,12))
+y = np.arcsin(lattice(-1,6,1000, -1,5,10))
+plt.plot(np.real(y), np.imag(y), "-", color="red", lw=0.5)
+y = np.arcsin(lattice(-1,6,10, -1,5,100)).T
+plt.plot(np.real(y), np.imag(y), "-", color="red", lw=0.5)
+y = np.arcsin(lattice(-1,6,10, -1,5,10))
+plt.plot(np.real(y), np.imag(y), ".", color="red", lw=0.5)
+plt.show()
+
+# %% plot cd^-1 TODO complex cd^-1 missing
+
+
+r = np.logspace(-1,8, 50)
+
+
+
+x = np.concatenate((-np.flip(r), [0], r), axis=0)
+y = cd_inv(x, 0.99)
+
+plt.figure(figsize=(12,12))
+plt.plot(np.real(y), np.imag(y), "-")
+plt.show()
+
+# %%plot cd
+plt.figure(figsize=(10,6))
+z = np.linspace(-4,4, 500)
+for k in [0, 0.9, 0.99, 0.999, 0.99999]:
+    w = cd(z*ell_int(k), k)
+    plt.plot(z, w, label=f"$k={k}$")
+plt.grid()
+plt.legend()
+# plt.xlim([-4,4])
+plt.xlabel("$u$")
+plt.ylabel("$cd(uK, k)$")
+plt.show()
+
+# %% Test ????
+
+N = 5
+k = 0.9
+k1 = k**N
+
+assert np.allclose(k**(-N), k1**(-1))
+
+K = ell_int(k)
+Kp = ell_int(np.sqrt(1-k**2))
+
+K1 = ell_int(k1)
+Kp1 = ell_int(np.sqrt(1-k1**2))
+
+print(Kp * (K1 / K) * N, Kp1)
+
+
+# %%
+
+
+k = 0.9
+k_prim = np.sqrt(1 - k**2)
+K = ell_int(k)
+Kp = ell_int(k_prim)
+
+print(K, Kp)
+
+zs = [
+    0 + (K + 0j) * np.linspace(0,1,25),
+    K +  (Kp*1j) * np.linspace(0,1,25),
+    (K + Kp*1j) + (-K) * np.linspace(0,1,25),
+]
+
+
+for z in zs:
+    plt.plot(np.real(z),  np.imag(z))
+plt.show()
+
+
+
+for z in zs:
+    w = cd(z, k)
+    plt.plot(np.real(w),  np.imag(w))
+plt.show()
+
+
+
+
+
+# %%
+
+for i in range(10):
+    x = np.linspace(i*1,i*1+1,10, dtype=np.complex64)
+    w = np.arccos(x)
+
+    x2 = np.cos(w)
+    x4 = np.cos(w+ 2*np.pi)
+    x3 = np.cos(np.conj(w))
+
+    assert np.allclose(x2, x4, rtol=0.001, atol=1e-5)
+
+    assert np.allclose(x2, x3)
+    assert np.allclose(x2, x, rtol=0.001, atol=1e-5)
+
+    plt.plot(np.real(w), np.imag(w), ".-")
+
+for i in range(10):
+    x = -np.linspace(i*1,i*1+1,100, dtype=np.complex64)
+    w = np.arccos(x)
+    plt.plot(np.real(w), np.imag(w), ".-")
+
+plt.grid()
+plt.show()
+
+
+
+
+# %%
+
+plt.plot(omega, np.abs(G))
+plt.show()
+
+
+def cd_inv(u, m):
+    return K(1/2) - F(np.arcsin())
+
+def K(m):
+    return scipy.special.ellipk(m)
+
+def L(n, xi):
+    return 1 #TODO
+
+def R(n, xi, x):
+    cn(n*K(1/L(n, xi))/K(1/xi) * cd_inv(x, 1/xi, 1/L(n, xi)))
+
+epsilon = 0.1
+n = 3
+omega = np.linspace(0, np.pi, 1000)
+omega_0 = 1
+xi = 1.1
+
+G = 1 / np.sqrt(1 + epsilon**2 * R(n, xi, omega/omega_0)**2)
+
+
+plt.plot(omega, np.abs(G))
+plt.show()
+
+
+
+# %% Chebychef
+
+epsilon = 0.5
+omega = np.linspace(0, np.pi, 1000)
+omega_0 = 1
+n = 4
+
+def chebychef_poly(n, x):
+    x = x.astype(np.complex64)
+    y = np.cos(n* np.arccos(x))
+    return np.real(y)
+
+F_omega = chebychef_poly
+
+for n in (1,2,3,4):
+    plt.plot(omega, F_omega(n, omega/omega_0)**2)
+plt.ylim([0,5])
+plt.xlim([0,1.5])
+plt.grid()
+plt.show()
+
+for n in (1,2,3,4):
+    G = 1 / np.sqrt(1 + epsilon**2 * F_omega(n, omega/omega_0)**2)
+    plt.plot(omega, np.abs(G))
+plt.grid()
+plt.show()
+
+
+
+
+# %%
+
+
+k = np.concatenate(([0.00001,0.0001,0.001], np.linspace(0,1,101)[1:-1], [0.999,0.9999, 0.99999]), axis=0)
+K = ell_int(k)
+K_prime = ell_int(np.sqrt(1-k**2))
+
+
+f, axs = plt.subplots(1,2, figsize=(5,2.5))
+axs[0].plot(k, K, linewidth=0.1)
+axs[0].text(k[30], K[30]+0.1, f"$K$")
+axs[0].plot(k, K_prime, linewidth=0.1)
+axs[0].text(k[30], K_prime[30]+0.1, f"$K^\prime$")
+axs[0].set_xlim([0,1])
+axs[0].set_ylim([0,4])
+axs[0].set_xlabel("$k$")
+
+axs[1].axvline(x=np.pi/2, color="gray", linewidth=0.5)
+axs[1].axhline(y=np.pi/2, color="gray", linewidth=0.5)
+axs[1].text(0.1, np.pi/2 + 0.1, "$\pi/2$")
+axs[1].text(np.pi/2+0.1, 0.1, "$\pi/2$")
+axs[1].plot(K, K_prime, linewidth=1)
+
+k = np.array([0.1,0.2,0.4,0.6,0.9,0.99])
+K = ell_int(k)
+K_prime = ell_int(np.sqrt(1-k**2))
+
+axs[1].plot(K, K_prime, '.', color=last_color(), markersize=2)
+for x, y, n in zip(K, K_prime, k):
+    axs[1].text(x+0.1, y+0.1, f"$k={n:.2f}$", rotation_mode="anchor")
+axs[1].set_ylabel("$K^\prime$")
+axs[1].set_xlabel("$K$")
+axs[1].set_xlim([0,6])
+axs[1].set_ylim([0,5])
+plt.tight_layout()
+plt.savefig("k.pgf")
+plt.show()
+
+print(K[0], K[-1])
diff --git a/buch/papers/ellfilter/python/elliptic2.py b/buch/papers/ellfilter/python/elliptic2.py
new file mode 100644
index 0000000..29c6f47
--- /dev/null
+++ b/buch/papers/ellfilter/python/elliptic2.py
@@ -0,0 +1,149 @@
+# %%
+
+import matplotlib.pyplot as plt
+import scipy.signal
+import numpy as np
+import matplotlib
+from matplotlib.patches import Rectangle
+
+import plot_params
+
+def ellip_filter(N):
+
+    order = N
+    passband_ripple_db = 3
+    stopband_attenuation_db = 20
+    omega_c = 1
+
+    a, b = scipy.signal.ellip(
+        order,
+        passband_ripple_db,
+        stopband_attenuation_db,
+        omega_c,
+        btype='low',
+        analog=True,
+        output='ba',
+        fs=None
+    )
+
+    w, mag_db, phase = scipy.signal.bode((a, b), w=np.linspace(0*omega_c,2*omega_c, 4000))
+
+    mag = 10**(mag_db/20)
+
+    passband_ripple = 10**(-passband_ripple_db/20)
+    epsilon2 = (1/passband_ripple)**2 - 1
+
+    FN2 = ((1/mag**2) - 1)
+
+    return w/omega_c, FN2 / epsilon2, mag, a, b
+
+
+plt.figure(figsize=(4,2.5))
+
+for N in [5]:
+    w, FN2, mag, a, b = ellip_filter(N)
+    plt.semilogy(w, FN2, label=f"$N={N}, k=0.1$", linewidth=1)
+
+plt.gca().add_patch(Rectangle(
+    (0, 0),
+    1, 1,
+    fc ='green',
+    alpha=0.2,
+    lw = 10,
+))
+plt.gca().add_patch(Rectangle(
+    (1, 1),
+    0.01, 1e2-1,
+    fc ='orange',
+    alpha=0.2,
+    lw = 10,
+))
+
+plt.gca().add_patch(Rectangle(
+    (1.01, 100),
+    1, 1e6,
+    fc ='red',
+    alpha=0.2,
+    lw = 10,
+))
+
+zeros = [0,0.87,1]
+poles = [1.01,1.155]
+
+import matplotlib.transforms
+plt.plot( # mark errors as vertical bars
+    zeros,
+    np.zeros_like(zeros),
+    "o",
+    mfc='none',
+    color='black',
+    transform=matplotlib.transforms.blended_transform_factory(
+        plt.gca().transData,
+        plt.gca().transAxes,
+    ),
+)
+plt.plot( # mark errors as vertical bars
+    poles,
+    np.ones_like(poles),
+    "x",
+    mfc='none',
+    color='black',
+    transform=matplotlib.transforms.blended_transform_factory(
+        plt.gca().transData,
+        plt.gca().transAxes,
+    ),
+)
+
+plt.xlim([0,2])
+plt.ylim([1e-4,1e6])
+plt.grid()
+plt.xlabel("$w$")
+plt.ylabel("$F^2_N(w)$")
+plt.legend()
+plt.tight_layout()
+plt.savefig("F_N_elliptic.pgf")
+plt.show()
+
+
+
+plt.figure(figsize=(4,2.5))
+plt.plot(w, mag, linewidth=1)
+
+plt.gca().add_patch(Rectangle(
+    (0, np.sqrt(2)/2),
+    1, 1,
+    fc ='green',
+    alpha=0.2,
+    lw = 10,
+))
+plt.gca().add_patch(Rectangle(
+    (1, 0.1),
+    0.01, np.sqrt(2)/2 - 0.1,
+    fc ='orange',
+    alpha=0.2,
+    lw = 10,
+))
+
+plt.gca().add_patch(Rectangle(
+    (1.01, 0),
+    1, 0.1,
+    fc ='red',
+    alpha=0.2,
+    lw = 10,
+))
+
+plt.grid()
+plt.xlim([0,2])
+plt.ylim([0,1])
+plt.xlabel("$w$")
+plt.ylabel("$|H(w)|$")
+plt.tight_layout()
+plt.savefig("elliptic.pgf")
+plt.show()
+
+print("zeros", a)
+print("poles", b)
+
+
+
+
diff --git a/buch/papers/ellfilter/python/k.pgf b/buch/papers/ellfilter/python/k.pgf
new file mode 100644
index 0000000..95d61d4
--- /dev/null
+++ b/buch/papers/ellfilter/python/k.pgf
@@ -0,0 +1,1157 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%%   \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%%   \usepackage{pgf}
+%%
+%% Also ensure that all the required font packages are loaded; for instance,
+%% the lmodern package is sometimes necessary when using math font.
+%%   \usepackage{lmodern}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%%   \usepackage{import}
+%%
+%% and then include the figures with
+%%   \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
+\begingroup%
+\makeatletter%
+\begin{pgfpicture}%
+\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.000000in}{2.500000in}}%
+\pgfusepath{use as bounding box, clip}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetstrokeopacity{0.000000}%
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+\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathclose%
+\pgfusepath{}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
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+\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}%
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+\pgfpathclose%
+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
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+}%
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+\end{pgfscope}%
+\begin{pgfscope}%
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+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
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+}%
+\begin{pgfscope}%
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+\end{pgfscope}%
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+\end{pgfscope}%
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+\end{pgfpicture}%
+\makeatother%
+\endgroup%
diff --git a/buch/papers/ellfilter/python/plot_params.py b/buch/papers/ellfilter/python/plot_params.py
new file mode 100644
index 0000000..4ddd1d8
--- /dev/null
+++ b/buch/papers/ellfilter/python/plot_params.py
@@ -0,0 +1,9 @@
+import matplotlib
+
+matplotlib.rcParams.update({
+    "pgf.texsystem": "pdflatex",
+    'font.family': 'serif',
+    'font.size': 9,
+    'text.usetex': True,
+    'pgf.rcfonts': False,
+})
diff --git a/buch/papers/ellfilter/references.bib b/buch/papers/ellfilter/references.bib
index 81b3577..8f21971 100644
--- a/buch/papers/ellfilter/references.bib
+++ b/buch/papers/ellfilter/references.bib
@@ -4,32 +4,19 @@
 % (c) 2020 Autor, Hochschule Rapperswil
 %
 
-@online{ellfilter:bibtex,
-	title = {BibTeX},
-	url = {https://de.wikipedia.org/wiki/BibTeX},
-	date = {2020-02-06},
-	year = {2020},
-	month = {2},
-	day = {6}
-}
-
-@book{ellfilter:numerical-analysis,
-	title = {Numerical Analysis},
-	author = {David Kincaid and Ward Cheney},
-	publisher = {American Mathematical Society},
-	year = {2002},
-	isbn = {978-8-8218-4788-6},
-	inseries = {Pure and applied undegraduate texts},
-	volume = {2}
-}
-
-@article{ellfilter:mendezmueller,
-        author = { Tabea Méndez and Andreas Müller },
-        title = { Noncommutative harmonic analysis and image registration },
-        journal = { Appl. Comput. Harmon. Anal.},
-        year = 2019,
-        volume = 47,
-        pages = {607--627},
-        url = {https://doi.org/10.1016/j.acha.2017.11.004}
+@online{ellfilter:bib:orfanidis,
+        author = { Sophocles J. Orfanidis},
+        title = { LECTURE NOTES ON ELLIPTIC FILTER DESIGN },
+        year = 2006,
+        url = {https://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf}
 }
 
+% Schwalm
+% https://en.wikipedia.org/wiki/Elliptic_rational_functions
+% https://en.wikipedia.org/wiki/Rational_function
+% https://en.wikipedia.org/wiki/Jacobi_elliptic_functions
+% https://de.wikipedia.org/wiki/Elliptisches_Integral
+% https://de.wikipedia.org/wiki/Tschebyschow-Polynom
+% https://en.wikipedia.org/wiki/Chebyshev_filter
+% https://mathworld.wolfram.com/JacobiEllipticFunctions.html
+% https://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html
diff --git a/buch/papers/ellfilter/teil0.tex b/buch/papers/ellfilter/teil0.tex
deleted file mode 100644
index fd04ba9..0000000
--- a/buch/papers/ellfilter/teil0.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-%
-% einleitung.tex -- Beispiel-File für die Einleitung
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 0\label{ellfilter:section:teil0}}
-\rhead{Teil 0}
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{ellfilter:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum.  Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
-
-
diff --git a/buch/papers/ellfilter/teil1.tex b/buch/papers/ellfilter/teil1.tex
deleted file mode 100644
index 7e62a2f..0000000
--- a/buch/papers/ellfilter/teil1.tex
+++ /dev/null
@@ -1,55 +0,0 @@
-%
-% teil1.tex -- Beispiel-File für das Paper
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 1
-\label{ellfilter:section:teil1}}
-\rhead{Problemstellung}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo.
-Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit
-aut fugit, sed quia consequuntur magni dolores eos qui ratione
-voluptatem sequi nesciunt
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{ellfilter:equation1}
-\end{equation}
-Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet,
-consectetur, adipisci velit, sed quia non numquam eius modi tempora
-incidunt ut labore et dolore magnam aliquam quaerat voluptatem.
-
-Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis
-suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur?
-Quis autem vel eum iure reprehenderit qui in ea voluptate velit
-esse quam nihil molestiae consequatur, vel illum qui dolorem eum
-fugiat quo voluptas nulla pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{ellfilter:subsection:finibus}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}.
-
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{ellfilter:section:loesung}.
-Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil
-impedit quo minus id quod maxime placeat facere possimus, omnis
-voluptas assumenda est, omnis dolor repellendus
-\ref{ellfilter:section:folgerung}.
-Temporibus autem quibusdam et aut officiis debitis aut rerum
-necessitatibus saepe eveniet ut et voluptates repudiandae sint et
-molestiae non recusandae.
-Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis
-voluptatibus maiores alias consequatur aut perferendis doloribus
-asperiores repellat.
-
-
diff --git a/buch/papers/ellfilter/teil2.tex b/buch/papers/ellfilter/teil2.tex
deleted file mode 100644
index 71fdc6d..0000000
--- a/buch/papers/ellfilter/teil2.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil2.tex -- Beispiel-File für teil2 
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2 
-\label{ellfilter:section:teil2}}
-\rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{ellfilter:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/ellfilter/teil3.tex b/buch/papers/ellfilter/teil3.tex
deleted file mode 100644
index 79a5f3d..0000000
--- a/buch/papers/ellfilter/teil3.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil3.tex -- Beispiel-File für Teil 3
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 3
-\label{ellfilter:section:teil3}}
-\rhead{Teil 3}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{ellfilter:subsection:malorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/ellfilter/tikz/arccos.tikz.tex b/buch/papers/ellfilter/tikz/arccos.tikz.tex
new file mode 100644
index 0000000..2772620
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/arccos.tikz.tex
@@ -0,0 +1,66 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \draw[gray, ->] (0,-2) -- (0,2) node[anchor=south]{$\mathrm{Im}~z$};
+    \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+    \begin{scope}[xscale=0.6]
+
+        \clip(-7.5,-2) rectangle (7.5,2);
+
+        \draw[thick, ->, darkgreen] (0, 0) -- (0,1.5);
+        \draw[thick, ->, orange] (1, 0) -- (0,0);
+        \draw[thick, ->, red] (2, 0) -- (1,0);
+        \draw[thick, ->, blue] (2,1.5) -- (2, 0);
+
+        \foreach \i in {-2,...,1} {
+            \begin{scope}[opacity=0.5, xshift=\i*4cm]
+                \draw[->, orange] (-1, 0) -- (0,0);
+                \draw[->, darkgreen] (0, 0) -- (0,1.5);
+                \draw[->, darkgreen] (0, 0) -- (0,-1.5);
+                \draw[->, orange] (1, 0) -- (0,0);
+                \draw[->, red] (2, 0) -- (1,0);
+                \draw[->, blue] (2,1.5) -- (2, 0);
+                \draw[->, blue] (2,-1.5) -- (2, 0);
+                \draw[->, red] (2, 0) -- (3,0);
+
+                \node[zero] at (1,0) {};
+                \node[zero] at (3,0) {};
+            \end{scope}
+        }
+
+        \node[gray, anchor=north] at (-6,0) {$-3\pi$};
+        \node[gray, anchor=north] at (-4,0) {$-2\pi$};
+        \node[gray, anchor=north] at (-2,0) {$-\pi$};
+        % \node[gray, anchor=north] at (0,0) {$0$};
+        \node[gray, anchor=north] at (2,0) {$\pi$};
+        \node[gray, anchor=north] at (4,0) {$2\pi$};
+        \node[gray, anchor=north] at (6,0) {$3\pi$};
+
+        \node[gray, anchor=east] at (0,-1.5) {$-\infty$};
+        % \node[gray, anchor=south east] at (0, 0) {$0$};
+        \node[gray, anchor=east] at (0, 1.5) {$\infty$};
+
+    \end{scope}
+
+    \begin{scope}[yshift=-2.5cm]
+
+        \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$w$};
+
+        \draw[thick, ->, blue]      (-4, 0) -- (-2, 0);
+        \draw[thick, ->, red]       (-2, 0) -- (0, 0);
+        \draw[thick, ->, orange]    (0, 0) -- (2, 0);
+        \draw[thick, ->, darkgreen] (2, 0) -- (4, 0);
+
+        \node[anchor=south] at (-4,0) {$-\infty$};
+        \node[anchor=south] at (-2,0) {$-1$};
+        \node[anchor=south] at (0,0) {$0$};
+        \node[anchor=south] at (2,0) {$1$};
+        \node[anchor=south] at (4,0) {$\infty$};
+
+    \end{scope}
+
+
+\end{tikzpicture}
+\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/arccos2.tikz.tex b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
new file mode 100644
index 0000000..3fc3cc6
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
@@ -0,0 +1,45 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=0.5]
+
+        \draw[gray, ->] (0,-2) -- (0,2) node[anchor=south]{$\mathrm{Im}~z_1$};
+        \draw[gray, ->] (-10,0) -- (10,0) node[anchor=west]{$\mathrm{Re}~z_1$};
+
+        \begin{scope}
+
+            \draw[>->, line width=0.05, thick, blue]   (2, 1.5) -- (2,0.05)  -- node[anchor=south, pos=0.5]{$N=1$} (0.1,0.05) -- (0.1,1.5);
+            \draw[>->, line width=0.05, thick, orange] (4, 1.5) -- (4,0)     -- node[anchor=south, pos=0.25]{$N=2$} (0,0) -- (0,1.5);
+            \draw[>->, line width=0.05, thick, red]    (6, 1.5) node[anchor=north west]{$-\infty$} -- (6,-0.05) node[anchor=west]{$-1$} -- node[anchor=north]{$0$} node[anchor=south, pos=0.1666]{$N=3$} (-0.1,-0.05) node[anchor=east]{$1$}  -- (-0.1,1.5) node[anchor=north east]{$\infty$};
+
+
+            \node[zero] at (-7,0) {};
+            \node[zero] at (-5,0) {};
+            \node[zero] at (-3,0) {};
+            \node[zero] at (-1,0) {};
+            \node[zero] at (1,0) {};
+            \node[zero] at (3,0) {};
+            \node[zero] at (5,0) {};
+            \node[zero] at (7,0) {};
+
+
+        \end{scope}
+
+        \node[gray, anchor=north] at (-8,0) {$-4\pi$};
+        \node[gray, anchor=north] at (-6,0) {$-3\pi$};
+        \node[gray, anchor=north] at (-4,0) {$-2\pi$};
+        \node[gray, anchor=north] at (-2,0) {$-\pi$};
+        \node[gray, anchor=north] at (2,0) {$\pi$};
+        \node[gray, anchor=north] at (4,0) {$2\pi$};
+        \node[gray, anchor=north] at (6,0) {$3\pi$};
+        \node[gray, anchor=north] at (8,0) {$4\pi$};
+
+
+        \node[gray, anchor=east] at (0,-1.5) {$-\infty$};
+        \node[gray, anchor=east] at (0, 1.5) {$\infty$};
+
+    \end{scope}
+
+\end{tikzpicture}
+\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/cd.tikz.tex b/buch/papers/ellfilter/tikz/cd.tikz.tex
new file mode 100644
index 0000000..7155a85
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/cd.tikz.tex
@@ -0,0 +1,87 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=1, yscale=2]
+
+        \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \draw[gray] ( 1,0) +(0,0.1) -- +(0, -0.1) node[inner sep=0, anchor=north] {\small $K$};
+
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+
+        \begin{scope}
+
+            \begin{scope}[xshift=0cm]
+
+                \clip(-4.5,-1.25) rectangle (4.5,1.25);
+
+                \fill[yellow!30] (0,0) rectangle (1, 0.5);
+
+
+                \draw[thick, ->, darkgreen] (0, 0) -- (0,0.5);
+                \draw[thick, ->, orange] (1, 0) -- (0,0);
+                \draw[thick, ->, red] (2, 0) -- (1,0);
+                \draw[thick, ->, blue] (2,0.5) -- (2, 0);
+                \draw[thick, ->, purple] (1, 0.5) -- (2,0.5);
+                \draw[thick, ->, cyan] (0, 0.5) -- (1,0.5);
+
+
+
+                \foreach \i in {-2,...,1} {
+                    \foreach \j in {-2,...,1} {
+                        \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
+                            \draw[opacity=0.5, ->, darkgreen] (0, 0) -- (0,0.5);
+                            \draw[opacity=0.5, ->, orange] (1, 0) -- (0,0);
+                            \draw[opacity=0.5, ->, red] (2, 0) -- (1,0);
+                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 0);
+                            \draw[opacity=0.5, ->, purple] (1, 0.5) -- (2,0.5);
+                            \draw[opacity=0.5, ->, cyan] (0, 0.5) -- (1,0.5);
+                            \draw[opacity=0.5, ->, darkgreen] (0,1) -- (0,0.5);
+                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 1);
+                            \draw[opacity=0.5, ->, purple] (3, 0.5) -- (2,0.5);
+                            \draw[opacity=0.5, ->, cyan] (4, 0.5) -- (3,0.5);
+                            \draw[opacity=0.5, ->, red] (2, 0) -- (3,0);
+                            \draw[opacity=0.5, ->, orange] (3, 0) -- (4,0);
+
+                            \node[zero] at ( 1, 0) {};
+                            \node[zero] at ( 3, 0) {};
+                            \node[pole] at ( 1,0.5) {};
+                            \node[pole] at ( 3,0.5) {};
+
+                        \end{scope}
+                    }
+                }
+
+            \end{scope}
+
+        \end{scope}
+
+    \end{scope}
+
+    \begin{scope}[yshift=-3.5cm, xscale=0.75]
+
+        \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$};
+
+        \draw[thick, ->, purple] (-5, 0) -- (-3, 0);
+        \draw[thick, ->, blue]      (-3, 0) -- (-2, 0);
+        \draw[thick, ->, red]       (-2, 0) -- (0, 0);
+        \draw[thick, ->, orange]    (0, 0) -- (2, 0);
+        \draw[thick, ->, darkgreen] (2, 0) -- (3, 0);
+        \draw[thick, ->, cyan] (3, 0) -- (5, 0);
+
+        \node[anchor=south] at (-5,0) {$-\infty$};
+        \node[anchor=south] at (-3,0) {$-1/k$};
+        \node[anchor=south] at (-2,0) {$-1$};
+        \node[anchor=south] at (0,0) {$0$};
+        \node[anchor=south] at (2,0) {$1$};
+        \node[anchor=south] at (3,0) {$1/k$};
+        \node[anchor=south] at (5,0) {$\infty$};
+
+    \end{scope}
+
+\end{tikzpicture}
+\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/cd2.tikz.tex b/buch/papers/ellfilter/tikz/cd2.tikz.tex
new file mode 100644
index 0000000..0743f7d
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/cd2.tikz.tex
@@ -0,0 +1,84 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+    \tikzstyle{dot} = [fill, circle, inner sep =0, minimum height=0.1cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=1.25, yscale=2.5]
+
+        \draw[gray, ->] (0,-0.75) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z_1$};
+        \draw[gray, ->] (-1.5,0) -- (6,0) node[anchor=west]{$\mathrm{Re}~z_1$};
+
+        \draw[gray] ( 1,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K_1$};
+        \draw[gray] ( 5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $5K_1$};
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime_1$};
+
+        \begin{scope}
+
+            \clip(-1.5,-0.75) rectangle (6.8,1.25);
+
+            % \draw[>->, line width=0.05, thick, blue]   (1, 0.45) -- (2, 0.45) -- (2, 0.05) -- ( 0.1, 0.05) -- ( 0.1,0.45) -- (1, 0.45);
+            % \draw[>->, line width=0.05, thick, orange] (2, 0.5 ) -- (4, 0.5 ) -- (4, 0   ) -- ( 0  , 0   ) -- ( 0  ,0.5 ) -- (2, 0.5 );
+            % \draw[>->, line width=0.05, thick, red]    (3, 0.55) -- (6, 0.55) -- (6,-0.05) -- (-0.1,-0.05) -- (-0.1,0.55) -- (3, 0.55);
+            % \node[blue] at (1, 0.25) {$N=1$};
+            % \node[orange] at (3, 0.25) {$N=2$};
+            % \node[red] at (5, 0.25) {$N=3$};
+
+
+
+            % \draw[line width=0.1cm, fill, red!50] (0,0) rectangle (3, 0.5);
+            % \draw[line width=0.05cm, fill, orange!50] (0,0) rectangle (2, 0.5);
+            % \fill[yellow!50] (0,0) rectangle (1, 0.5);
+            % \node[] at (0.5, 0.25) {\small $N=1$};
+            % \node[] at (1.5, 0.25) {\small $N=2$};
+            % \node[] at (2.5, 0.25) {\small $N=3$};
+
+            \fill[orange!30] (0,0) rectangle (5, 0.5);
+            \fill[yellow!30] (0,0) rectangle (1, 0.5);
+            \node[] at (2.5, 0.25) {\small $N=5$};
+
+
+            \draw[decorate,decoration={brace,amplitude=3pt,mirror}, yshift=0.05cm]
+                (5,0.5) node(t_k_unten){} -- node[above, yshift=0.1cm]{$NK$}
+                (0,0.5) node(t_k_opt_unten){};
+
+            \draw[decorate,decoration={brace,amplitude=3pt,mirror}, xshift=0.1cm]
+                (5,0) node(t_k_unten){} -- node[right, xshift=0.1cm]{$K^\prime \frac{K_1N}{K} = K^\prime_1$}
+                (5,0.5) node(t_k_opt_unten){};
+
+            \foreach \i in {-2,...,1} {
+                \foreach \j in {-2,...,1} {
+                    \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
+
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( 3, 0) {};
+                        \node[pole] at ( 1,0.5) {};
+                        \node[pole] at ( 3,0.5) {};
+
+                    \end{scope}
+                }
+            }
+
+
+
+
+            \draw[thick, ->, darkgreen] (5, 0) -- node[yshift=-0.5cm]{Durchlassbereich} (0,0);
+            \draw[thick, ->, orange] (-0, 0) --  node[align=center]{Übergangs-\\berech} (0,0.5);
+            \draw[thick, ->, red] (0,0.5) -- node[align=center, yshift=0.5cm]{Sperrbereich} (5, 0.5);
+
+            \draw (4,0  )  node[dot]{} node[anchor=south]      {\small $1$};
+            \draw (2,0  )  node[dot]{} node[anchor=south]      {\small $-1$};
+            \draw (0,0  )  node[dot]{} node[anchor=south west] {\small $1$};
+            \draw (0,0.5)  node[dot]{} node[anchor=north west] {\small $1/k$};
+            \draw (2,0.5)  node[dot]{} node[anchor=north]      {\small $-1/k$};
+            \draw (4,0.5)  node[dot]{} node[anchor=north]      {\small $1/k$};
+
+
+
+        \end{scope}
+
+
+    \end{scope}
+
+\end{tikzpicture}
+\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex b/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex
new file mode 100644
index 0000000..921dbfa
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex
@@ -0,0 +1,26 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=2, yscale=2]
+
+        \draw[gray, ->] (0,-0.25) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-0.25,0) -- (1.5,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \draw[gray] ( 1,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K$};
+
+        \draw[gray]  (0, 1) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+        \fill[yellow!50] (0,0) rectangle (1, 1);
+
+        \node[anchor=south east] at ( 1,0) {$c$};
+        \node[anchor=north east] at ( 1,1) {$d$};
+        \node[anchor=north west] at ( 0,1) {$n$};
+        \node[anchor=south west] at ( 0,0) {$s$};
+
+    \end{scope}
+
+
+\end{tikzpicture}
+\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/sn.tikz.tex b/buch/papers/ellfilter/tikz/sn.tikz.tex
new file mode 100644
index 0000000..87c63c0
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/sn.tikz.tex
@@ -0,0 +1,86 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=1, yscale=2]
+
+        \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \begin{scope}
+
+            \clip(-4.5,-1.25) rectangle (4.5,1.25);
+
+            \fill[yellow!30] (0,0) rectangle (1, 0.5);
+
+            \begin{scope}[xshift=-1cm]
+
+                \draw[thick, ->, darkgreen] (0, 0) -- (0,0.5);
+                \draw[thick, ->, orange] (1, 0) -- (0,0);
+                \draw[thick, ->, red] (2, 0) -- (1,0);
+                \draw[thick, ->, blue] (2,0.5) -- (2, 0);
+                \draw[thick, ->, purple] (1, 0.5) -- (2,0.5);
+                \draw[thick, ->, cyan] (0, 0.5) -- (1,0.5);
+
+
+                \foreach \i in {-2,...,2} {
+                    \foreach \j in {-2,...,1} {
+                        \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
+                            \draw[opacity=0.5, ->, darkgreen] (0, 0) -- (0,0.5);
+                            \draw[opacity=0.5, ->, orange] (1, 0) -- (0,0);
+                            \draw[opacity=0.5, ->, red] (2, 0) -- (1,0);
+                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 0);
+                            \draw[opacity=0.5, ->, purple] (1, 0.5) -- (2,0.5);
+                            \draw[opacity=0.5, ->, cyan] (0, 0.5) -- (1,0.5);
+                            \draw[opacity=0.5, ->, darkgreen] (0,1) -- (0,0.5);
+                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 1);
+                            \draw[opacity=0.5, ->, purple] (3, 0.5) -- (2,0.5);
+                            \draw[opacity=0.5, ->, cyan] (4, 0.5) -- (3,0.5);
+                            \draw[opacity=0.5, ->, red] (2, 0) -- (3,0);
+                            \draw[opacity=0.5, ->, orange] (3, 0) -- (4,0);
+
+                            \node[zero] at ( 1, 0) {};
+                            \node[zero] at ( 3, 0) {};
+                            \node[pole] at ( 1,0.5) {};
+                            \node[pole] at ( 3,0.5) {};
+
+                        \end{scope}
+                    }
+                }
+
+            \end{scope}
+
+        \end{scope}
+
+        \draw[gray] ( 1,0) +(0,0.1) -- +(0, -0.1) node[inner sep=0, anchor=north] {\small $K$};
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+
+
+    \end{scope}
+
+    \begin{scope}[yshift=-3.5cm, xscale=0.75]
+
+        \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$};
+
+        \draw[thick, ->, purple] (-5, 0) -- (-3, 0);
+        \draw[thick, ->, blue]      (-3, 0) -- (-2, 0);
+        \draw[thick, ->, red]       (-2, 0) -- (0, 0);
+        \draw[thick, ->, orange]    (0, 0) -- (2, 0);
+        \draw[thick, ->, darkgreen] (2, 0) -- (3, 0);
+        \draw[thick, ->, cyan] (3, 0) -- (5, 0);
+
+        \node[anchor=south] at (-5,0) {$-\infty$};
+        \node[anchor=south] at (-3,0) {$-1/k$};
+        \node[anchor=south] at (-2,0) {$-1$};
+        \node[anchor=south] at (0,0) {$0$};
+        \node[anchor=south] at (2,0) {$1$};
+        \node[anchor=south] at (3,0) {$1/k$};
+        \node[anchor=south] at (5,0) {$\infty$};
+
+    \end{scope}
+
+
+\end{tikzpicture}
+\ No newline at end of file
diff --git a/buch/papers/ellfilter/tschebyscheff.tex b/buch/papers/ellfilter/tschebyscheff.tex
new file mode 100644
index 0000000..7d426b6
--- /dev/null
+++ b/buch/papers/ellfilter/tschebyscheff.tex
@@ -0,0 +1,133 @@
+\section{Tschebyscheff-Filter}
+
+Als Einstieg betrachent Wir das Tschebyscheff-Filter, welches sehr verwand ist mit dem elliptischen Filter.
+Genauer ausgedrückt sind die Tschebyscheff-1 und -2 Filter Spezialfälle davon.
+
+Der Name des Filters deutet schon an, dass die Tschebyscheff-Polynome $T_N$ für das Filter relevant sind:
+\begin{align}
+    T_{0}(x)&=1\\
+    T_{1}(x)&=x\\
+    T_{2}(x)&=2x^{2}-1\\
+    T_{3}(x)&=4x^{3}-3x\\
+    T_{n+1}(x)&=2x~T_{n}(x)-T_{n-1}(x).
+\end{align}
+Bemerkenswert ist, dass die Polynome im Intervall $[-1, 1]$ mit der trigonometrischen Funktion
+\begin{align} \label{ellfilter:eq:chebychef_polynomials}
+    T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\
+           &= \cos \left(N~z \right), \quad w= \cos(z)
+\end{align}
+übereinstimmt.
+Der Zusammenhang lässt sich mit den Doppel- und Mehrfachwinkelfunktionen der trigonometrischen Funktionen erklären.
+Abbildung \ref{ellfilter:fig:chebychef_polynomials} zeigt einige Tschebyscheff-Polynome.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/python/F_N_chebychev2.pgf}
+    \caption{Die Tschebyscheff-Polynome $C_N$.}
+    \label{ellfilter:fig:chebychef_polynomials}
+\end{figure}
+Da der Kosinus begrenzt zwischen $-1$ und $1$ ist, sind auch die Tschebyscheff-Polynome begrenzt.
+Geht man aber über das Intervall $[-1, 1]$ hinaus, divergieren die Funktionen mit zunehmender Ordnung immer steiler gegen $\pm \infty$.
+Diese Eigenschaft ist sehr nützlich für ein Filter.
+Wenn wir die Tschebyscheff-Polynome quadrieren, passen sie perfekt in die Voraussetzungen für Filterfunktionen, wie es Abbildung \ref{ellfiter:fig:chebychef} demonstriert.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/python/F_N_chebychev.pgf}
+    \caption{Die Tschebyscheff-Polynome füllen den erlaubten Bereich besser, und erhalten dadurch eine steilere Flanke im Sperrbereich.}
+    \label{ellfiter:fig:chebychef}
+\end{figure}
+
+
+Die analytische Fortsetzung von \eqref{ellfilter:eq:chebychef_polynomials} über das Intervall $[-1,1]$ hinaus stimmt mit den Polynomen überein, wie es zu erwarten ist.
+Die genauere Betrachtung wird uns dann helfen die elliptischen Filter besser zu verstehen.
+
+Starten wir mit der Funktion, die als erstes auf $w$ angewendet wird, dem Arcuscosinus.
+Die invertierte Funktion des Kosinus kann als definites Integral dargestellt werden:
+\begin{align}
+    \cos^{-1}(x)
+    &=
+    \int_{x}^{1}
+    \frac{
+        dz
+    }{
+        \sqrt{
+            1-z^2
+        }
+    }\\
+    &=
+    \int_{0}^{x}
+    \frac{
+        -1
+    }{
+        \sqrt{
+            1-z^2
+        }
+    }
+    ~dz
+    + \frac{\pi}{2}
+\end{align}
+Der Integrand oder auch die Ableitung
+\begin{equation}
+    \frac{
+        -1
+    }{
+        \sqrt{
+            1-z^2
+        }
+    }
+\end{equation}
+bestimmt dabei die Richtung, in der die Funktion verläuft.
+Der reelle Arcuscosinus is bekanntlich nur für $|z| \leq 1$ definiert.
+Hier bleibt der Wert unter der Wurzel positiv und das Integral liefert reelle Werte.
+Doch wenn $|z|$ über 1 hinausgeht, wird der Term unter der Wurzel negativ.
+Durch die Quadratwurzel entstehen für den Integranden zwei rein komplexe Lösungen.
+Der Wert des Arcuscosinus verlässt also bei $z= \pm 1$ den reellen Zahlenstrahl und knickt in die komplexe Ebene ab.
+Abbildung \ref{ellfilter:fig:arccos} zeigt den $\arccos$ in der komplexen Ebene.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/arccos.tikz.tex}
+    \caption{Die Funktion $z = \cos^{-1}(w)$ dargestellt in der komplexen ebene.}
+    \label{ellfilter:fig:arccos}
+\end{figure}
+Wegen der Periodizität des Kosinus ist auch der Arcuscosinus $2\pi$-periodisch und es entstehen periodische Nullstellen.
+% \begin{equation}
+%     \frac{
+%         1
+%     }{
+%         \sqrt{
+%             1-z^2
+%         }
+%     }
+%     \in \mathbb{R}
+%     \quad
+%     \forall
+%     \quad
+%     -1  \leq z \leq 1
+% \end{equation}
+% \begin{equation}
+%     \frac{
+%         1
+%     }{
+%         \sqrt{
+%             1-z^2
+%         }
+%     }
+%     = i \xi \quad | \quad \xi \in \mathbb{R}
+%     \quad
+%     \forall
+%     \quad
+%     z \leq -1 \cup z \geq 1
+% \end{equation}
+
+Die Tschebyscheff-Polynome skalieren diese Nullstellen mit dem Ordnungsfaktor $N$, wie dargestellt in Abbildung \ref{ellfilter:fig:arccos2}.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/arccos2.tikz.tex}
+    \caption{
+        $z_1=N \cos^{-1}(w)$-Ebene der Tschebyscheff-Funktion.
+        Die eingefärbten Pfade sind Verläufe von $w~\forall~[-\infty, \infty]$ für verschiedene Ordnungen $N$.
+        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert.
+    }
+    \label{ellfilter:fig:arccos2}
+\end{figure}
+Somit passert $\cos( N~\cos^{-1}(w))$ im Intervall $[-1, 1]$ $N$ Nullstellen.
+Durch die spezielle Anordnung der Nullstellen hat die Funktion Equirippel-Verhalten und ist dennoch ein Polynom, was sich perfekt für linear Filter eignet.
diff --git a/buch/papers/kugel/images/Makefile b/buch/papers/kugel/images/Makefile
new file mode 100644
index 0000000..4226dab
--- /dev/null
+++ b/buch/papers/kugel/images/Makefile
@@ -0,0 +1,30 @@
+#
+# Makefile -- build images
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all: curvature.jpg spherecurve.jpg
+
+curvature.inc:	curvgraph.m
+	octave curvgraph.m
+
+curvature.png:	curvature.pov curvature.inc
+	povray +A0.1 +W1920 +H1080 +Ocurvature.png curvature.pov
+
+curvature.jpg:	curvature.png
+	convert curvature.png -density 300 -units PixelsPerInch curvature.jpg
+
+spherecurve2.inc:	spherecurve.m
+	octave spherecurve.m
+
+spherecurve.png:	spherecurve.pov spherecurve.inc
+	povray +A0.1 +W1080 +H1080 +Ospherecurve.png spherecurve.pov
+
+spherecurve.jpg:	spherecurve.png
+	convert spherecurve.png -density 300 -units PixelsPerInch spherecurve.jpg
+
+spherecurve:	spherecurve.cpp
+	g++ -o spherecurve -g -Wall -O spherecurve.cpp
+
+spherecurve.inc:	spherecurve
+	./spherecurve
diff --git a/buch/papers/kugel/images/curvature.maxima b/buch/papers/kugel/images/curvature.maxima
new file mode 100644
index 0000000..6313642
--- /dev/null
+++ b/buch/papers/kugel/images/curvature.maxima
@@ -0,0 +1,6 @@
+
+f: exp(-r^2/sigma^2)/sigma;
+laplacef: ratsimp(diff(r * diff(f,r), r) / r);
+f: exp(-r^2/(2*sigma^2))/(sqrt(2)*sigma);
+laplacef: ratsimp(diff(r * diff(f,r), r) / r);
+
diff --git a/buch/papers/kugel/images/curvature.pov b/buch/papers/kugel/images/curvature.pov
new file mode 100644
index 0000000..3b15d77
--- /dev/null
+++ b/buch/papers/kugel/images/curvature.pov
@@ -0,0 +1,139 @@
+//
+// curvature.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.09;
+
+camera {
+	location <10, 10, -40>
+	look_at <0, 0, 0>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-10, 10, -40> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+//
+// draw an arrow from <from> to <to> with thickness <arrowthickness> with
+// color <c>
+//
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+arrow(<-3.1,0,0>, <3.1,0,0>, 0.01, White)
+arrow(<0,-1,0>, <0,1,0>, 0.01, White)
+arrow(<0,0,-2.1>, <0,0,2.1>, 0.01, White)
+
+#include "curvature.inc"
+
+#declare sigma = 1;
+#declare s = 1.4;
+#declare N0 = 0.4;
+#declare funktion = function(r) {
+	(exp(-r*r/(sigma*sigma)) / sigma
+	-
+	exp(-r*r/(2*sigma*sigma)) / (sqrt(2)*sigma)) / N0
+};
+#declare hypot = function(xx, yy) { sqrt(xx*xx+yy*yy) };
+
+#declare Funktion = function(x,y) { funktion(hypot(x+s,y)) - funktion(hypot(x-s,y)) };
+#macro punkt(xx,yy)
+	<xx, Funktion(xx, yy), yy>
+#end
+
+#declare griddiameter = 0.006;
+union {
+	#declare xmin = -3;
+	#declare xmax = 3;
+	#declare ymin = -2;
+	#declare ymax = 2;
+
+
+	#declare xstep = 0.2;
+	#declare ystep = 0.02;
+	#declare xx = xmin;
+	#while (xx < xmax + xstep/2)
+		#declare yy = ymin;
+		#declare P = punkt(xx, yy);
+		#while (yy < ymax - ystep/2)
+			#declare yy = yy + ystep;
+			#declare Q = punkt(xx, yy);
+			sphere { P, griddiameter }
+			cylinder { P, Q, griddiameter }
+			#declare P = Q;
+		#end
+		sphere { P, griddiameter }
+		#declare xx = xx + xstep;
+	#end
+
+	#declare xstep = 0.02;
+	#declare ystep = 0.2;
+	#declare yy = ymin;
+	#while (yy < ymax + ystep/2)
+		#declare xx = xmin;
+		#declare P = punkt(xx, yy);
+		#while (xx < xmax - xstep/2)
+			#declare xx = xx + xstep;
+			#declare Q = punkt(xx, yy);
+			sphere { P, griddiameter }
+			cylinder { P, Q, griddiameter }
+			#declare P = Q;
+		#end
+		sphere { P, griddiameter }
+		#declare yy = yy + ystep;
+	#end
+
+	pigment {
+		color rgb<0.8,0.8,0.8>
+	}
+	finish {
+		metallic
+		specular 0.8
+	}
+}
+
diff --git a/buch/papers/kugel/images/curvgraph.m b/buch/papers/kugel/images/curvgraph.m
new file mode 100644
index 0000000..75effd6
--- /dev/null
+++ b/buch/papers/kugel/images/curvgraph.m
@@ -0,0 +1,140 @@
+#
+# curvature.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+global N;
+N = 10;
+
+global sigma2;
+sigma2 = 1;
+
+global s;
+s = 1.4;
+
+global cmax;
+cmax = 0.9;
+global cmin;
+cmin = -0.9;
+
+global Cmax;
+global Cmin;
+Cmax = 0;
+Cmin = 0;
+
+xmin = -3;
+xmax = 3;
+xsteps = 200;
+hx = (xmax - xmin) / xsteps;
+
+ymin = -2;
+ymax = 2;
+ysteps = 200;
+hy = (ymax - ymin) / ysteps;
+
+function retval = f0(r)
+	global sigma2;
+	retval = exp(-r^2/sigma2)/sqrt(sigma2) - exp(-r^2/(2*sigma2))/(sqrt(2*sigma2));
+end
+
+global N0;
+N0 = f0(0)
+N0 = 0.4;
+
+function retval = f1(x,y)
+	global N0;
+	retval = f0(hypot(x, y)) / N0;
+endfunction
+
+function retval = f(x, y)
+	global s;
+	retval = f1(x+s, y) - f1(x-s, y);
+endfunction
+
+function retval = curvature0(r)
+	global sigma2;
+	retval = (
+		-4*(sigma2-r^2)*exp(-r^2/sigma2)
+		+
+		(2*sigma2-r^2)*exp(-r^2/(2*sigma2))
+	) / (sigma2^(5/2));
+endfunction
+
+function retval = curvature1(x, y)
+	retval = curvature0(hypot(x, y));
+endfunction
+
+function retval = curvature(x, y)
+	global s;
+	retval = curvature1(x+s, y) - curvature1(x-s, y);
+endfunction
+
+function retval = farbe(x, y)
+	global Cmax;
+	global Cmin;
+	global cmax;
+	global cmin;
+	c = curvature(x, y);
+	if (c < Cmin) 
+		Cmin = c
+	endif
+	if (c > Cmax) 
+		Cmax = c
+	endif
+	u = (c - cmin) / (cmax - cmin);
+	if (u > 1)
+		u = 1;
+	endif
+	if (u < 0)
+		u = 0;
+	endif
+	color = [ u, 0.5, 1-u ];
+	color = color/max(color);
+	color(1,4) = c/2;
+	retval = color;
+endfunction
+
+function dreieck(fn, A, B, C)
+	fprintf(fn, "\ttriangle {\n");
+	fprintf(fn, "\t    <%.4f,%.4f,%.4f>,\n", A(1,1), A(1,3), A(1,2));
+	fprintf(fn, "\t    <%.4f,%.4f,%.4f>,\n", B(1,1), B(1,3), B(1,2));
+	fprintf(fn, "\t    <%.4f,%.4f,%.4f>\n",  C(1,1), C(1,3), C(1,2));
+	fprintf(fn, "\t}\n");
+endfunction
+
+function viereck(fn, punkte)
+	color = farbe(mean(punkte(:,1)), mean(punkte(:,2)));
+	fprintf(fn, "    mesh {\n");
+	dreieck(fn, punkte(1,:), punkte(2,:), punkte(3,:));
+	dreieck(fn, punkte(2,:), punkte(3,:), punkte(4,:));
+	fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> } // %.4f\n",
+		color(1,1), color(1,2), color(1,3), color(1,4));
+	fprintf(fn, "    }\n");
+endfunction
+
+fn = fopen("curvature.inc", "w");
+punkte = zeros(4,3);
+for ix = (0:xsteps-1)
+	x = xmin + ix * hx;
+	punkte(1,1) = x;      
+	punkte(2,1) = x;
+	punkte(3,1) = x + hx;
+	punkte(4,1) = x + hx;
+	for iy = (0:ysteps-1)
+		y = ymin + iy * hy;
+		punkte(1,2) = y;
+		punkte(2,2) = y + hy;
+		punkte(3,2) = y;
+		punkte(4,2) = y + hy;
+		for i = (1:4)
+			punkte(i,3) = f(punkte(i,1), punkte(i,2));
+		endfor
+		viereck(fn, punkte);
+	end
+end
+#fprintf(fn, "    finish { metallic specular 0.5 }\n");
+fclose(fn);
+
+printf("Cmax = %.4f\n", Cmax);
+printf("Cmin = %.4f\n", Cmin);
diff --git a/buch/papers/kugel/images/spherecurve.cpp b/buch/papers/kugel/images/spherecurve.cpp
new file mode 100644
index 0000000..8ddf5e5
--- /dev/null
+++ b/buch/papers/kugel/images/spherecurve.cpp
@@ -0,0 +1,292 @@
+/*
+ * spherecurve.cpp
+ *
+ * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <cstdio>
+#include <cstdlib>
+#include <cmath>
+#include <string>
+#include <iostream>
+
+inline double	sqr(double x) { return x * x; }
+
+/**
+ * \brief Class for 3d vectors (also used as colors)
+ */
+class vector {
+	double	X[3];
+public:
+	vector() { X[0] = X[1] = X[2] = 0; }
+	vector(double a) { X[0] = X[1] = X[2] = a; }
+	vector(double x, double y, double z) {
+		X[0] = x; X[1] = y; X[2] = z;
+	}
+	vector(double theta, double phi) {
+		double	s = sin(theta);
+		X[0] = cos(phi) * s;
+		X[1] = sin(phi) * s;
+		X[2] = cos(theta);
+	}
+	vector(const vector& other) {
+		for (int i = 0; i < 3; i++) {
+			X[i] = other.X[i];
+		}
+	}
+	vector	operator+(const vector& other) const {
+		return vector(X[0] + other.X[0],
+				X[1] + other.X[1],
+				X[2] + other.X[2]);
+	}
+	vector	operator*(double l) const {
+		return vector(X[0] * l, X[1] * l, X[2] * l);
+	}
+	double	operator*(const vector& other) const {
+		double	s = 0;
+		for (int i = 0; i < 3; i++) {
+			s += X[i] * other.X[i];
+		}
+		return s;
+	}
+	double	norm() const {
+		double	s = 0;
+		for (int i = 0; i < 3; i++) {
+			s += sqr(X[i]);
+		}
+		return sqrt(s);
+	}
+	vector	normalize() const {
+		double	l = norm();
+		return vector(X[0]/l, X[1]/l, X[2]/l);
+	}
+	double	max() const {
+		return std::max(X[0], std::max(X[1], X[2]));
+	}
+	double	l0norm() const {
+		double	l = 0;
+		for (int i = 0; i < 3; i++) {
+			if (fabs(X[i]) > l) {
+				l = fabs(X[i]);
+			}
+		}
+		return l;
+	}
+	vector	l0normalize() const {
+		double	l = l0norm();
+		vector	result(X[0]/l, X[1]/l, X[2]/l);
+		return result;
+	}
+	const double&	operator[](int i) const { return X[i]; }
+	double&	operator[](int i) { return X[i]; }
+};
+
+/**
+ * \brief Derived 3d vector class implementing color
+ *
+ * The constructor in this class converts a single value into a
+ * color on a suitable gradient.
+ */
+class color : public vector {
+public:
+	static double	utop;
+	static double	ubottom;
+	static double	green;
+public:
+	color(double u) {
+		u = (u - ubottom) / (utop - ubottom);
+		if (u > 1) {
+			u = 1;
+		}
+		if (u < 0) {
+			u = 0;
+		}
+		u = pow(u,2);
+		(*this)[0] = u;
+		(*this)[1] = green * u * (1 - u);
+		(*this)[2] = 1-u;
+		double	l = l0norm();
+		for (int i = 0; i < 3; i++) {
+			(*this)[i] /= l;
+		}
+	}
+};
+
+double	color::utop = 12;
+double	color::ubottom = -31;
+double	color::green = 0.5;
+
+/**
+ * \brief Surface model
+ *
+ * This class contains the definitions of the functions to plot
+ * and the parameters to 
+ */
+class surfacefunction {
+	static vector	axes[6];
+
+	double	_a;
+	double	_A;
+
+	double	_umin;
+	double	_umax;
+public:
+	double	a() const { return _a; }
+	double	A() const { return _A; }
+
+	double	umin() const { return _umin; }
+	double	umax() const { return _umax; }
+
+	surfacefunction(double a, double A) : _a(a), _A(A), _umin(0), _umax(0) {
+	}
+
+	double	f(double z) {
+		return A() * exp(a() * (sqr(z) - 1));
+	}
+
+	double	g(double z) {
+		return -f(z) * 2*a() * ((2*a()*sqr(z) + (3-2*a()))*sqr(z) - 1);
+	}
+
+	double	F(const vector& v) {
+		double	s = 0;
+		for (int i = 0; i < 6; i++) {
+			s += f(axes[i] * v);
+		}
+		return s / 6;
+	}
+
+	double	G(const vector& v) {
+		double	s = 0;
+		for (int i = 0; i < 6; i++) {
+			s += g(axes[i] * v);
+		}
+		return s / 6;
+	}
+protected:
+	color	farbe(const vector& v) {
+		double	u = G(v);
+		if (u < _umin) {
+			_umin = u;
+		}
+		if (u > _umax) {
+			_umax = u;
+		}
+		return color(u);
+	}
+};
+
+static double	phi = (1 + sqrt(5)) / 2;
+static double	sl = sqrt(sqr(phi) + 1);
+vector	surfacefunction::axes[6] = {
+	vector(   0.   , -1./sl, phi/sl ),
+	vector(   0.   ,  1./sl, phi/sl ),
+	vector(   1./sl, phi/sl,  0.    ),
+	vector(  -1./sl, phi/sl,  0.    ),
+	vector(  phi/sl,  0.   ,  1./sl ),
+	vector( -phi/sl,  0.   ,  1./sl )
+};
+
+/**
+ * \brief Class to construct the plot
+ */
+class surface : public surfacefunction {
+	FILE	*outfile;
+
+	int	_phisteps;
+	int	_thetasteps;
+	double	_hphi;
+	double	_htheta;
+public:
+	int	phisteps() const { return _phisteps; }
+	int	thetasteps() const { return _thetasteps; }
+	double	hphi() const { return _hphi; }
+	double	htheta() const { return _htheta; }
+	void	phisteps(int s) { _phisteps = s; _hphi = 2 * M_PI / s; }
+	void	thetasteps(int s) { _thetasteps = s; _htheta = M_PI / s; }
+
+	surface(const std::string& filename, double a, double A)
+		: surfacefunction(a, A) {
+		outfile = fopen(filename.c_str(), "w");
+		phisteps(400);
+		thetasteps(200);
+	}
+
+	~surface() {
+		fclose(outfile);
+	}
+
+private:
+	void	triangle(const vector& v0, const vector& v1, const vector& v2) {
+		fprintf(outfile, "    mesh {\n");
+		vector	c = (v0 + v1 + v2) * (1./3.);
+		vector	color = farbe(c.normalize());
+		vector	V0 = v0 * (1 + F(v0));
+		vector	V1 = v1 * (1 + F(v1));
+		vector	V2 = v2 * (1 + F(v2));
+		fprintf(outfile, "\ttriangle {\n");
+		fprintf(outfile, "\t    <%.6f,%.6f,%.6f>,\n",
+			V0[0], V0[2], V0[1]);
+		fprintf(outfile, "\t    <%.6f,%.6f,%.6f>,\n",
+			V1[0], V1[2], V1[1]);
+		fprintf(outfile, "\t    <%.6f,%.6f,%.6f>\n",
+			V2[0], V2[2], V2[1]);
+		fprintf(outfile, "\t}\n");
+		fprintf(outfile, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n",
+			color[0], color[1], color[2]);
+		fprintf(outfile, "\tfinish { metallic specular 0.5 }\n");
+		fprintf(outfile, "    }\n");
+	}
+
+	void	northcap() {
+		vector	v0(0, 0, 1);
+		for (int i = 1; i <= phisteps(); i++) {
+			fprintf(outfile, "    // northcap i = %d\n", i);
+			vector	v1(htheta(), (i - 1) * hphi());
+			vector	v2(htheta(),  i      * hphi());
+			triangle(v0, v1, v2);
+		}
+	}
+
+	void	southcap() {
+		vector	v0(0, 0, -1);
+		for (int i = 1; i <= phisteps(); i++) {
+			fprintf(outfile, "    // southcap i = %d\n", i);
+			vector	v1(M_PI - htheta(), (i - 1) * hphi());
+			vector	v2(M_PI - htheta(),  i      * hphi());
+			triangle(v0, v1, v2);
+		}
+	}
+
+	void	zone() {
+		for (int j = 1; j < thetasteps() - 1; j++) {
+			for (int i = 1; i <= phisteps(); i++) {
+				fprintf(outfile, "    // zone j = %d, i = %d\n",
+					j, i);
+				vector v0( j    * htheta(), (i-1) * hphi());
+				vector v1((j+1) * htheta(), (i-1) * hphi());
+				vector v2( j    * htheta(),  i    * hphi());
+				vector v3((j+1) * htheta(),  i    * hphi());
+				triangle(v0, v1, v2);
+				triangle(v1, v2, v3);
+			}
+		}
+	}
+public:
+	void	draw() {
+		northcap();
+		southcap();
+		zone();
+	}
+};
+
+/**
+ * \brief main function
+ */
+int	main(int argc, char *argv[]) {
+	surface	S("spherecurve.inc", 5, 10);
+	color::green = 1.0;
+	S.draw();
+	std::cout << "umin: " << S.umin() << std::endl;
+	std::cout << "umax: " << S.umax() << std::endl;
+	return EXIT_SUCCESS;
+}
diff --git a/buch/papers/kugel/images/spherecurve.m b/buch/papers/kugel/images/spherecurve.m
new file mode 100644
index 0000000..99d5c9a
--- /dev/null
+++ b/buch/papers/kugel/images/spherecurve.m
@@ -0,0 +1,160 @@
+#
+# spherecurve.m
+#
+# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global a;
+a = 5;
+global A;
+A = 10;
+
+phisteps = 400;
+hphi = 2 * pi / phisteps;
+thetasteps = 200;
+htheta = pi / thetasteps;
+
+function retval = f(z)
+	global a;
+	global A;
+	retval = A * exp(a * (z^2 - 1));
+endfunction
+
+function retval = g(z)
+	global a;
+	retval = -f(z) * 2 * a * (2 * a * z^4 + (3 - 2*a) * z^2 - 1);
+	#                                                              2
+	#                   - a     2  4             2   2          a z
+	#(%o6)          - %e    (4 a  z  + (6 a - 4 a ) z  - 2 a) %e
+endfunction
+
+phi = (1 + sqrt(5)) / 2;
+
+global axes;
+axes = [
+     0,   0,   1,  -1, phi, -phi;
+     1,  -1, phi, phi,   0,    0;
+   phi, phi,   0,   0,   1,    1;
+];
+axes = axes / (sqrt(phi^2+1));
+
+function retval = kugel(theta, phi)
+	retval = [
+	    cos(phi) * sin(theta);
+	    sin(phi) * sin(theta);
+	               cos(theta)
+	];
+endfunction
+
+function retval = F(v)
+	global axes;
+	s = 0;
+	for i = (1:6)
+		z = axes(:,i)' * v;
+		s = s + f(z);
+	endfor
+	retval = s / 6;
+endfunction
+
+function retval = F2(theta, phi)
+	v = kugel(theta, phi);
+	retval = F(v);
+endfunction
+
+function retval = G(v)
+	global axes;
+	s = 0;
+	for i = (1:6)
+		s = s + g(axes(:,i)' * v);
+	endfor
+	retval = s / 6;
+endfunction
+
+function retval = G2(theta, phi)
+	v = kugel(theta, phi);
+	retval = G(v);
+endfunction
+
+function retval = cnormalize(u)
+	utop = 11;
+	ubottom = -30;
+	retval = (u - ubottom)  / (utop - ubottom);
+	if (retval > 1) 
+		retval = 1;
+	endif
+	if (retval < 0)
+		retval = 0;
+	endif
+endfunction
+
+global umin;
+umin = 0;
+global umax;
+umax = 0;
+
+function color = farbe(v)
+	global umin;
+	global umax;
+	u = G(v);
+	if (u < umin) 
+		umin = u;
+	endif
+	if (u > umax)
+		umax = u;
+	endif
+	u = cnormalize(u);
+	color = [ u, 0.5, 1-u ];
+	color = color/max(color);
+endfunction
+
+function dreieck(fn, v0, v1, v2)
+	fprintf(fn, "    mesh {\n");
+	c = (v0 + v1 + v2) / 3;
+	c = c / norm(c);
+	color = farbe(c);
+	v0 = v0 * (1 + F(v0));
+	v1 = v1 * (1 + F(v1));
+	v2 = v2 * (1 + F(v2));
+	fprintf(fn, "\ttriangle {\n");
+	fprintf(fn, "\t    <%.6f,%.6f,%.6f>,\n", v0(1,1), v0(3,1), v0(2,1));
+	fprintf(fn, "\t    <%.6f,%.6f,%.6f>,\n", v1(1,1), v1(3,1), v1(2,1));
+	fprintf(fn, "\t    <%.6f,%.6f,%.6f>\n",  v2(1,1), v2(3,1), v2(2,1));
+	fprintf(fn, "\t}\n");
+	fprintf(fn, "\tpigment { color rgb<%.4f,%.4f,%.4f> }\n",
+		color(1,1), color(1,2), color(1,3));
+	fprintf(fn, "\tfinish { metallic specular 0.5 }\n");
+	fprintf(fn, "    }\n");
+endfunction
+
+fn = fopen("spherecurve2.inc", "w");
+
+	for i = (1:phisteps)
+		# Polkappe nord
+		v0 = [ 0; 0; 1 ];
+		v1 = kugel(htheta, (i-1) * hphi);
+		v2 = kugel(htheta,  i    * hphi);
+		fprintf(fn, "    // i = %d\n", i);
+		dreieck(fn, v0, v1, v2);
+
+		# Polkappe sued
+		v0 = [ 0; 0; -1 ];
+		v1 = kugel(pi-htheta, (i-1) * hphi);
+		v2 = kugel(pi-htheta,  i    * hphi);
+		dreieck(fn, v0, v1, v2);
+	endfor
+
+	for j = (1:thetasteps-2)
+		for i = (1:phisteps)
+			v0 = kugel( j    * htheta, (i-1) * hphi);
+			v1 = kugel((j+1) * htheta, (i-1) * hphi);
+			v2 = kugel( j    * htheta,  i    * hphi);
+			v3 = kugel((j+1) * htheta,  i    * hphi);
+			fprintf(fn, "    // i = %d, j = %d\n", i, j);
+			dreieck(fn, v0, v1, v2);
+			dreieck(fn, v1, v2, v3);
+		endfor
+	endfor
+
+fclose(fn);
+
+umin
+umax
diff --git a/buch/papers/kugel/images/spherecurve.maxima b/buch/papers/kugel/images/spherecurve.maxima
new file mode 100644
index 0000000..1e9077c
--- /dev/null
+++ b/buch/papers/kugel/images/spherecurve.maxima
@@ -0,0 +1,13 @@
+/*
+ * spherecurv.maxima
+ *
+ * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+f: exp(-a * sin(theta)^2);
+
+g: ratsimp(diff(sin(theta) * diff(f, theta), theta)/sin(theta));
+g: subst(z, cos(theta), g);
+g: subst(sqrt(1-z^2), sin(theta), g);
+ratsimp(g);
+
+f: ratsimp(subst(sqrt(1-z^2), sin(theta), f));
diff --git a/buch/papers/kugel/images/spherecurve.pov b/buch/papers/kugel/images/spherecurve.pov
new file mode 100644
index 0000000..b1bf4b8
--- /dev/null
+++ b/buch/papers/kugel/images/spherecurve.pov
@@ -0,0 +1,73 @@
+//
+// curvature.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.13;
+
+camera {
+	location <10, 10, -40>
+	look_at <0, 0, 0>
+	right x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-10, 10, -40> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+//
+// draw an arrow from <from> to <to> with thickness <arrowthickness> with
+// color <c>
+//
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+arrow(<-2.7,0,0>, <2.7,0,0>, 0.03, White)
+arrow(<0,-2.7,0>, <0,2.7,0>, 0.03, White)
+arrow(<0,0,-2.7>, <0,0,2.7>, 0.03, White)
+
+#include "spherecurve.inc"
+
diff --git a/buch/papers/nav/beispiel.txt b/buch/papers/nav/beispiel.txt
new file mode 100644
index 0000000..12d9e9e
--- /dev/null
+++ b/buch/papers/nav/beispiel.txt
@@ -0,0 +1,40 @@
+Datum: 28. 5. 2022
+Zeit: 15:29:49 UTC
+Sternzeit: 7h 54m 26.593s  7.90738694h
+
+Deneb
+
+RA    20h 42m 12.14s      20.703372h
+DEC   45g 21' 40.3"       45.361194
+
+H     50g 15' 21.7"       50.256027
+Azi   59g 36' 02.0"       59.600555
+
+Altair
+
+RA    19h 51' 53.39"      19.864831h
+DEC    8g 55' 42.3         8.928416
+
+H     45g 27' 48.1"       45.463361
+Azi  117g 16' 14.1"      117.270583
+
+Arktur
+
+RA    14h 16' 42.14"      14.278372
+DEC   19g 03' 47.6        19.063222
+
+H     47g 25' 38.8"       47.427444
+Azi  259g 09' 38.4"      259.160666
+
+Spica
+
+RA     13h 26m 23.44s     13.439844h
+DEC   -11g 16' 46.8"     -11.279666
+
+H      18g 27' 30.0"      18.458333
+Azi   240g 23' 52.5"     240.397916
+
+Position:
+
+l = 140 14' 00.01" E     140.233336 E
+b =  35 43' 00.02" N      35.716672 N
diff --git a/buch/papers/nav/bilder/ephe.png b/buch/papers/nav/bilder/ephe.png
index 0aeef6f..3f99a36 100644
--- a/buch/papers/nav/bilder/ephe.png
+++ b/buch/papers/nav/bilder/ephe.png
diff --git a/buch/papers/nav/bilder/recht.jpg b/buch/papers/nav/bilder/recht.jpg
new file mode 100644
index 0000000..3f60370
--- /dev/null
+++ b/buch/papers/nav/bilder/recht.jpg
diff --git a/buch/papers/nav/bilder/sextant.jpg b/buch/papers/nav/bilder/sextant.jpg
new file mode 100644
index 0000000..472e61f
--- /dev/null
+++ b/buch/papers/nav/bilder/sextant.jpg
diff --git a/buch/papers/nav/einleitung.tex b/buch/papers/nav/einleitung.tex
index 8d8c5c1..8eb4481 100644
--- a/buch/papers/nav/einleitung.tex
+++ b/buch/papers/nav/einleitung.tex
@@ -3,7 +3,7 @@
 \section{Einleitung}
 Heutzutage ist die Navigation ein Teil des Lebens. 
 Man sendet dem Kollegen seinen eigenen Standort, um sich das ewige Erklären zu sparen oder gibt die Adresse des Ziels ein, damit man seinen Aufenthaltsort zum Beispiel auf einer riesigen Wiese am See findet. 
-Dies wird durch Technologien wie Funknavigation, welches ein auf Langzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt.
-Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf kleineren Schiffen benötigt wird im Falle eines Stromausfalls. 
+Dies wird durch Technologien wie Funknavigation, welches ein auf Laufzeitmessung beruhendes Hyperbelverfahren mit Langwellen ist, oder die verbreitete Satellitennavigation, welche vier Satelliten für eine Messung zur Standortbestimmung nutzt.
+Vor all diesen technologischen Fortschritten gab es lediglich die Astronavigation, welche heute noch auf Schiffen verwendet wird im Falle eines Stromausfalls. 
 Aber wie funktioniert die Navigation mit den Sternen? Welche Hilfsmittel benötigt man, welche Rolle spielt die Mathematik und weshalb kann die Erde nicht flach sein? 
 In diesem Kapitel werden genau diese Fragen mithilfe des nautischen Dreiecks, der sphärischen Trigonometrie und einigen Hilfsmitteln und Messgeräten beantwortet.
 \ No newline at end of file
diff --git a/buch/papers/nav/flatearth.tex b/buch/papers/nav/flatearth.tex
index bec242e..3b08e8d 100644
--- a/buch/papers/nav/flatearth.tex
+++ b/buch/papers/nav/flatearth.tex
@@ -9,19 +9,19 @@
 	\end{center}	
 \end{figure}
 
-Es gibt heut zu Tage viele Beweise dafür, dass die Erde eine Kugel ist. 
+Es gibt heutzutage viele Beweise dafür, dass die Erde eine Kugel ist. 
 Die Fotos von unserem	Planeten oder die Berichte der Astronauten. 
-Aber schon vor ca. 2300 Jahren hat Aristotoles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist oder der Erdschatten bei einer Mondfinsternis immer rund ist. 
+Aber schon vor ca. 2300 Jahren hat Aristoteles bemerkt, dass Schiffe im Horizont verschwinden und die einzige Erklärung dafür die Kugelgestalt der Erde ist.
+Auch der Erdschatten bei einer Mondfinsternis ist immer rund.
 Eratosthenes konnte etwa 100 Jahre später den Erdumfang berechnen. 
 Er beobachtete, dass die Sonne in Syene mittags im Zenit steht und gleichzeitig in Alexandria unter einem Winkel einfällt. 
 Mithilfe der Trigonometrie konnte er mit dem Abstand der Städte und dem Einfallswinkel den Umfang berechnen.
 
-Auch in der Navigation würden grobe Fehler passieren, wenn man davon ausgeht, dass die Erde eine Scheibe ist. 
-Man sieht es zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. 
+Der Kartograph Gerhard Mercator projizierte die Erdkugel wie in Abbildung 21.1 dargestellt auf ein Papier und erstellte so eine winkeltreue Karte. 
+Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. 
+Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen.
+Dies sieht man zum Beispiel sehr gut, wenn man die Anwendung Google Earth und eine Weltkarte vergleicht. 
 Grönland ist auf der Weltkarte so gross wie Afrika. 
 In der Anwendung Google Earth jedoch ist Grönland etwa so gross wie Algerien. 
 Das liegt daran, das man die 3D – Weltkarte nicht einfach auslegen kann. 
 
-Der Kartograph Gerhard Mercator projizierte die Erdkugel auf ein Papier und erstellte so eine winkeltreue Karte. 
-Jedoch wurden die Länder, die einen grösseren Abstand zum Äquator haben vergrössert, damit die Winkel stimmen können. 
-Wurde man also nun davon ausgehen, dass die Erde flach ist so würden wir nie dort ankommen wo wir es wollen.
-\ No newline at end of file
diff --git a/buch/papers/nav/images/2k_earth_daymap.png b/buch/papers/nav/images/2k_earth_daymap.png
new file mode 100644
index 0000000..4d55da8
--- /dev/null
+++ b/buch/papers/nav/images/2k_earth_daymap.png
diff --git a/buch/papers/nav/images/Makefile b/buch/papers/nav/images/Makefile
index da4defa..39bfbcf 100644
--- a/buch/papers/nav/images/Makefile
+++ b/buch/papers/nav/images/Makefile
@@ -51,73 +51,80 @@ DREIECKE3D =								\
 	dreieck3d5.pdf							\
 	dreieck3d6.pdf							\
 	dreieck3d7.pdf							\
-	dreieck3d8.pdf
+	dreieck3d8.pdf	
 
 dreiecke3d: $(DREIECKE3D)
 
 POVRAYOPTIONS = -W1080 -H1080
 #POVRAYOPTIONS = -W480 -H480
 
-dreieck3d1.png:	dreieck3d1.pov common.inc
+dreieck3d1.png:	dreieck3d1.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d1.png dreieck3d1.pov
 dreieck3d1.jpg:	dreieck3d1.png
 	convert dreieck3d1.png -density 300 -units PixelsPerInch dreieck3d1.jpg
 dreieck3d1.pdf:	dreieck3d1.tex dreieck3d1.jpg
 	pdflatex dreieck3d1.tex
 
-dreieck3d2.png:	dreieck3d2.pov common.inc
+dreieck3d2.png:	dreieck3d2.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d2.png dreieck3d2.pov
 dreieck3d2.jpg:	dreieck3d2.png
 	convert dreieck3d2.png -density 300 -units PixelsPerInch dreieck3d2.jpg
 dreieck3d2.pdf:	dreieck3d2.tex dreieck3d2.jpg
 	pdflatex dreieck3d2.tex
 
-dreieck3d3.png:	dreieck3d3.pov common.inc
+dreieck3d3.png:	dreieck3d3.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d3.png dreieck3d3.pov
 dreieck3d3.jpg:	dreieck3d3.png
 	convert dreieck3d3.png -density 300 -units PixelsPerInch dreieck3d3.jpg
 dreieck3d3.pdf:	dreieck3d3.tex dreieck3d3.jpg
 	pdflatex dreieck3d3.tex
 
-dreieck3d4.png:	dreieck3d4.pov common.inc
+dreieck3d4.png:	dreieck3d4.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d4.png dreieck3d4.pov
 dreieck3d4.jpg:	dreieck3d4.png
 	convert dreieck3d4.png -density 300 -units PixelsPerInch dreieck3d4.jpg
 dreieck3d4.pdf:	dreieck3d4.tex dreieck3d4.jpg
 	pdflatex dreieck3d4.tex
 
-dreieck3d5.png:	dreieck3d5.pov common.inc
+dreieck3d5.png:	dreieck3d5.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d5.png dreieck3d5.pov
 dreieck3d5.jpg:	dreieck3d5.png
 	convert dreieck3d5.png -density 300 -units PixelsPerInch dreieck3d5.jpg
 dreieck3d5.pdf:	dreieck3d5.tex dreieck3d5.jpg
 	pdflatex dreieck3d5.tex
 
-dreieck3d6.png:	dreieck3d6.pov common.inc
+dreieck3d6.png:	dreieck3d6.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d6.png dreieck3d6.pov
 dreieck3d6.jpg:	dreieck3d6.png
 	convert dreieck3d6.png -density 300 -units PixelsPerInch dreieck3d6.jpg
 dreieck3d6.pdf:	dreieck3d6.tex dreieck3d6.jpg
 	pdflatex dreieck3d6.tex
 
-dreieck3d7.png:	dreieck3d7.pov common.inc
+dreieck3d7.png:	dreieck3d7.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d7.png dreieck3d7.pov
 dreieck3d7.jpg:	dreieck3d7.png
 	convert dreieck3d7.png -density 300 -units PixelsPerInch dreieck3d7.jpg
 dreieck3d7.pdf:	dreieck3d7.tex dreieck3d7.jpg
 	pdflatex dreieck3d7.tex
 
-dreieck3d8.png:	dreieck3d8.pov common.inc
+dreieck3d8.png:	dreieck3d8.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d8.png dreieck3d8.pov
 dreieck3d8.jpg:	dreieck3d8.png
 	convert dreieck3d8.png -density 300 -units PixelsPerInch dreieck3d8.jpg
 dreieck3d8.pdf:	dreieck3d8.tex dreieck3d8.jpg
 	pdflatex dreieck3d8.tex
 
-dreieck3d9.png:	dreieck3d9.pov common.inc
+dreieck3d9.png:	dreieck3d9.pov common.inc macros.inc
 	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d9.png dreieck3d9.pov
 dreieck3d9.jpg:	dreieck3d9.png
 	convert dreieck3d9.png -density 300 -units PixelsPerInch dreieck3d9.jpg
 dreieck3d9.pdf:	dreieck3d9.tex dreieck3d9.jpg
 	pdflatex dreieck3d9.tex
 
+dreieck3d10.png:	dreieck3d10.pov common.inc macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Odreieck3d10.png dreieck3d10.pov
+dreieck3d10.jpg:	dreieck3d10.png
+	convert dreieck3d10.png -density 300 -units PixelsPerInch dreieck3d10.jpg
+dreieck3d10.pdf:	dreieck3d10.tex dreieck3d10.jpg macros.inc
+	pdflatex dreieck3d10.tex
+
diff --git a/buch/papers/nav/images/beispiele/2k_earth_daymap.png b/buch/papers/nav/images/beispiele/2k_earth_daymap.png
new file mode 100644
index 0000000..4d55da8
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/2k_earth_daymap.png
diff --git a/buch/papers/nav/images/beispiele/Makefile b/buch/papers/nav/images/beispiele/Makefile
new file mode 100644
index 0000000..9546c8e
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/Makefile
@@ -0,0 +1,38 @@
+#
+# Makefile to build images 
+#
+# (c) 2022
+#
+all:	beispiele
+
+POSITION =								\
+	beispiele1.pdf							\
+	beispiele2.pdf							\
+	beispiele3.pdf	
+
+beispiele: $(POSITION)
+
+POVRAYOPTIONS = -W1080 -H1080
+#POVRAYOPTIONS = -W480 -H480
+
+beispiele1.png:	beispiele1.pov common.inc geometrie.inc ../macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Obeispiele1.png beispiele1.pov
+beispiele1.jpg:	beispiele1.png
+	convert beispiele1.png -density 300 -units PixelsPerInch beispiele1.jpg
+beispiele1.pdf:	beispiele1.tex common.tex beispiele1.jpg
+	pdflatex beispiele1.tex
+
+beispiele2.png:	beispiele2.pov common.inc geometrie.inc ../macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Obeispiele2.png beispiele2.pov
+beispiele2.jpg:	beispiele2.png
+	convert beispiele2.png -density 300 -units PixelsPerInch beispiele2.jpg
+beispiele2.pdf:	beispiele2.tex common.tex beispiele2.jpg
+	pdflatex beispiele2.tex
+
+beispiele3.png:	beispiele3.pov common.inc geometrie.inc ../macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Obeispiele3.png beispiele3.pov
+beispiele3.jpg:	beispiele3.png
+	convert beispiele3.png -density 300 -units PixelsPerInch beispiele3.jpg
+beispiele3.pdf:	beispiele3.tex common.tex beispiele3.jpg
+	pdflatex beispiele3.tex
+
diff --git a/buch/papers/nav/images/beispiele/beispiele1.pdf b/buch/papers/nav/images/beispiele/beispiele1.pdf
new file mode 100644
index 0000000..d0fe3dc
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele1.pdf
diff --git a/buch/papers/nav/images/beispiele/beispiele1.pov b/buch/papers/nav/images/beispiele/beispiele1.pov
new file mode 100644
index 0000000..7fb3de2
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele1.pov
@@ -0,0 +1,12 @@
+//
+// beispiele1.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+#declare Stern1 = Deneb;
+#declare Stern2 = Arktur;
+
+#include "geometrie.inc"
+
diff --git a/buch/papers/nav/images/beispiele/beispiele1.tex b/buch/papers/nav/images/beispiele/beispiele1.tex
new file mode 100644
index 0000000..5666ba6
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele1.tex
@@ -0,0 +1,49 @@
+%
+% beispiele1.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math,calc}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{beispiele1.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelP
+\labelDeneb
+\labelArktur
+\labelhDeneb
+\labelhArktur
+\labellone
+\labeldDeneb
+\labeldArktur
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/beispiele/beispiele2.pdf b/buch/papers/nav/images/beispiele/beispiele2.pdf
new file mode 100644
index 0000000..8579ee5
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele2.pdf
diff --git a/buch/papers/nav/images/beispiele/beispiele2.pov b/buch/papers/nav/images/beispiele/beispiele2.pov
new file mode 100644
index 0000000..b69f0f9
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele2.pov
@@ -0,0 +1,12 @@
+//
+// beispiele1.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+#declare Stern1 = Altair;
+#declare Stern2 = Spica;
+
+#include "geometrie.inc"
+
diff --git a/buch/papers/nav/images/beispiele/beispiele2.tex b/buch/papers/nav/images/beispiele/beispiele2.tex
new file mode 100644
index 0000000..c9b70bd
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele2.tex
@@ -0,0 +1,50 @@
+%
+% beispiele2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math,calc}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{beispiele2.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelP
+\labelAltair
+\labelSpica
+\labelhAltair
+\labelhSpica
+\labelltwo
+\labeldAltair
+\labeldSpica
+
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/beispiele/beispiele3.pdf b/buch/papers/nav/images/beispiele/beispiele3.pdf
new file mode 100644
index 0000000..a7189dd
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele3.pdf
diff --git a/buch/papers/nav/images/beispiele/beispiele3.pov b/buch/papers/nav/images/beispiele/beispiele3.pov
new file mode 100644
index 0000000..af9a468
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele3.pov
@@ -0,0 +1,12 @@
+//
+// beispiele1.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+#declare Stern1 = Deneb;
+#declare Stern2 = Altair;
+
+#include "geometrie.inc"
+
diff --git a/buch/papers/nav/images/beispiele/beispiele3.tex b/buch/papers/nav/images/beispiele/beispiele3.tex
new file mode 100644
index 0000000..2573199
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/beispiele3.tex
@@ -0,0 +1,49 @@
+%
+% beispiele3.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math,calc}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{beispiele3.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelP
+\labelDeneb
+\labelAltair
+\labelhDeneb
+\labelhAltair
+\labellone
+%\labeldDeneb
+%\labeldAltair
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/beispiele/common.inc b/buch/papers/nav/images/beispiele/common.inc
new file mode 100644
index 0000000..51fbd1f
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/common.inc
@@ -0,0 +1,50 @@
+//
+// common.inc -- 3d Darstellung
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+#include "../macros.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.034;
+
+camera {
+	location <40, 20, -20>
+	look_at <0, 0.24, -0.20>
+	right x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<30, 10, -40> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+erde(0)
+achse(fein, White)
+koordinatennetz(gitterfarbe, 9, 0.001)
+
+union {
+        punkt(Sakura, fett)
+        pigment {
+                color rot
+        }
+        finish {
+                metallic
+                specular 0.9
+        }
+}
+
diff --git a/buch/papers/nav/images/beispiele/common.tex b/buch/papers/nav/images/beispiele/common.tex
new file mode 100644
index 0000000..b7b3dac
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/common.tex
@@ -0,0 +1,79 @@
+%
+% common.tex
+%
+% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+
+\def\labelA{\node at (0.7,3.8) {$A$};}
+
+\def\labelSpica{
+	\node at (-3.6,-2.8) {Spica};
+}
+\def\labelAltair{
+	\node at (3.0,-2.3) {Altair};
+}
+\def\labelArktur{
+	\node at (-3.3,-0.7) {Arktur};
+}
+\def\labelDeneb{
+	\node at (3.4,0.9) {Deneb};
+}
+
+\def\labelP{\node at (0,-0.2) {$P$};}
+
+\def\labellone{\node at (0.1,1.9) {$l$};}
+\def\labelltwo{\node at (0.1,2.0) {$l$};}
+
+\def\labelhSpica{
+	\coordinate (Spica) at (-1.8,-0.3);
+	\node at (Spica) {$h_{\text{Spica}}\mathstrut$};
+}
+\def\labelhAltair{
+	\coordinate (Altair) at (1.1,-1.0);
+	\node at (Altair) {$h_{\text{Altair}}\mathstrut$};
+}
+\def\labelhArktur{
+	\coordinate (Arktur) at (-1.3,-0.3);
+	\node at (Arktur) {$h_{\text{Arktur}}\mathstrut$};
+}
+\def\labelhDeneb{
+	\coordinate (Deneb) at (1.6,0.45);
+	\node at (Deneb) {$h_{\text{Deneb}}\mathstrut$};
+}
+
+\def\labeldSpica{
+	\coordinate (dSpica) at (-1.5,2.6);
+	\fill[color=white,opacity=0.5]
+		($(dSpica)+(-1.8,0.08)$)
+		rectangle
+		($(dSpica)+(-0.06,0.55)$);
+	\node at (dSpica) [above left]
+		{$90^\circ-\delta_{\text{Spica}}\mathstrut$};
+}
+\def\labeldAltair{
+	\coordinate (dAltair) at (2.0,2.1);
+	\fill[color=white,opacity=0.5]
+		($(dAltair)+(0.10,0.05)$)
+		rectangle
+		($(dAltair)+(1.8,0.5)$);
+	\node at (dAltair) [above right]
+		{$90^\circ-\delta_{\text{Altair}}\mathstrut$};
+}
+\def\labeldArktur{
+	\coordinate (dArktur) at (-1.2,2.5);
+	\fill[color=white,opacity=0.5]
+		($(dArktur)+(-1.8,0.05)$)
+		rectangle
+		($(dArktur)+(-0.06,0.5)$);
+	\node at (dArktur) [above left]
+		{$90^\circ-\delta_{\text{Arktur}}\mathstrut$};
+}
+\def\labeldDeneb{
+	\coordinate (dDeneb) at (2.0,2.8);
+	\fill[color=white,opacity=0.5]
+		($(dDeneb)+(0.05,0.5)$)
+		rectangle
+		($(dDeneb)+(1.87,0.05)$);
+	\node at (dDeneb) [above right]
+		{$90^\circ-\delta_{\text{Deneb}}\mathstrut$};
+}
diff --git a/buch/papers/nav/images/beispiele/geometrie.inc b/buch/papers/nav/images/beispiele/geometrie.inc
new file mode 100644
index 0000000..2f6084e
--- /dev/null
+++ b/buch/papers/nav/images/beispiele/geometrie.inc
@@ -0,0 +1,41 @@
+union {
+	punkt(A, fett)
+	punkt(Stern1, fein)
+	punkt(Stern2, fein)
+	seite(Stern1, Stern2, fein)
+	pigment {
+		color kugelfarbe
+	}
+	finish {
+		metallic
+		specular 0.9
+	}
+}
+
+union {
+	seite(A, Stern1, fein)
+	seite(A, Stern2, fein)
+	seite(Stern1, Sakura, fein)
+	seite(Stern2, Sakura, fein)
+	winkel(A, Stern1, Stern2, 0.5*fein, gross)
+	pigment {
+		color bekannt
+	}
+	finish {
+		metallic
+		specular 0.9
+	}
+}
+
+union {
+	seite(A, Sakura, fein)
+	winkel(A, Sakura, Stern1, 0.5*fett, klein)
+	pigment {
+		color unbekannt
+	}
+	finish {
+		metallic
+		specular 0.9
+	}
+}
+
diff --git a/buch/papers/nav/images/common.inc b/buch/papers/nav/images/common.inc
index 2c0ae6e..7b861de 100644
--- a/buch/papers/nav/images/common.inc
+++ b/buch/papers/nav/images/common.inc
@@ -5,6 +5,7 @@
 //
 #version 3.7;
 #include "colors.inc"
+#include "macros.inc"
 
 global_settings {
 	assumed_gamma 1
@@ -12,12 +13,6 @@ global_settings {
 
 #declare imagescale = 0.034;
 
-#declare O = <0, 0, 0>;
-#declare A = vnormalize(< 0, 1, 0>);
-#declare B = vnormalize(< 1, 2, -8>);
-#declare C = vnormalize(< 5, 1, 0>);
-#declare P = vnormalize(< 5, -1, -7>);
-
 camera {
 	location <40, 20, -20>
 	look_at <0, 0.24, -0.20>
@@ -26,7 +21,7 @@ camera {
 }
 
 light_source {
-	<10, 10, -40> color White
+	<30, 10, -40> color White
 	area_light <1,0,0> <0,0,1>, 10, 10
 	adaptive 1
 	jitter
@@ -38,150 +33,3 @@ sky_sphere {
 	}
 }
 
-//
-// draw an arrow from <from> to <to> with thickness <arrowthickness> with
-// color <c>
-//
-#macro arrow(from, to, arrowthickness, c)
-#declare arrowdirection = vnormalize(to - from);
-#declare arrowlength = vlength(to - from);
-union {
-        sphere {
-                from, 1.1 * arrowthickness
-        }
-        cylinder {
-                from,
-                from + (arrowlength - 5 * arrowthickness) * arrowdirection,
-                arrowthickness
-        }
-        cone {
-                from + (arrowlength - 5 * arrowthickness) * arrowdirection,
-                2 * arrowthickness,
-                to,
-                0
-        }
-        pigment {
-                color c
-        }
-        finish {
-                specular 0.9
-                metallic
-        }
-}
-#end
-
-#macro grosskreis(normale, staerke)
-union {
-	#declare v1 = vcross(normale, <normale.x, normale.z, normale.y>);
-	#declare v1 = vnormalize(v1);
-	#declare v2 = vnormalize(vcross(v1, normale));
-	#declare phisteps = 100;
-	#declare phistep = pi / phisteps;
-	#declare phi = 0;
-	#declare p1 = v1;
-	#while (phi < 2 * pi - phistep/2)
-		sphere { p1, staerke }
-		#declare phi = phi + phistep;
-		#declare p2 = v1 * cos(phi) + v2 * sin(phi);
-		cylinder { p1, p2, staerke }
-		#declare p1 = p2;
-	#end
-}
-#end
-
-#macro seite(p, q, staerke)
-	#declare n = vcross(p, q);
-	intersection {
-		grosskreis(n, staerke)
-		plane { -vcross(n, q) * vdot(vcross(n, q), p), 0 }
-		plane { -vcross(n, p) * vdot(vcross(n, p), q), 0 }
-	}
-#end
-
-#macro winkel(w, p, q, staerke, r)
-	#declare n = vnormalize(w);
-	#declare pp = vnormalize(p - vdot(n, p) * n);
-	#declare qq = vnormalize(q - vdot(n, q) * n);
-	intersection {
-		sphere { O, 1 + staerke }
-		cone { O, 0, 1.2 * vnormalize(w), r }
-		plane { -vcross(n, qq) * vdot(vcross(n, qq), pp), 0 }
-		plane { -vcross(n, pp) * vdot(vcross(n, pp), qq), 0 }
-	}
-#end
-
-#macro punkt(p, staerke)
-	sphere { p, 1.5 * staerke }
-#end
-
-#macro dreieck(p, q, r, farbe)
-	#declare n1 = vnormalize(vcross(p, q));
-	#declare n2 = vnormalize(vcross(q, r));
-	#declare n3 = vnormalize(vcross(r, p));
-	intersection {
-		plane { n1, 0 }
-		plane { n2, 0 }
-		plane { n3, 0 }
-		sphere { <0, 0, 0>, 1 + 0.001 }
-		pigment {
-			color farbe
-		}
-		finish {
-			metallic
-			specular 0.4
-		}
-	}
-#end
-
-#macro ebenerwinkel(a, p, q, s, r, farbe)
-	#declare n = vnormalize(-vcross(p, q));
-	#declare np = vnormalize(-vcross(p, n));
-	#declare nq = -vnormalize(-vcross(q, n));
-//	arrow(a, a + n, 0.02, White)
-//	arrow(a, a + np, 0.01, Red)
-//	arrow(a, a + nq, 0.01, Blue)
-	intersection {
-		cylinder { a - (s/2) * n, a + (s/2) * n, r }
-		plane { np, vdot(np, a) }
-		plane { nq, vdot(nq, a) }
-		pigment {
-			farbe
-		}
-		finish {
-			metallic
-			specular 0.5
-		}
-	}
-#end
-
-#macro komplement(a, p, q, s, r, farbe)
-	#declare n = vnormalize(-vcross(p, q));
-//	arrow(a, a + n, 0.015, Orange)
-	#declare m = vnormalize(-vcross(q, n));
-//	arrow(a, a + m, 0.015, Pink)
-	ebenerwinkel(a, p, m, s, r, farbe)
-#end
-
-#declare fett = 0.015;
-#declare fein = 0.010;
-
-#declare klein = 0.3;
-#declare gross = 0.4;
-
-#declare dreieckfarbe = rgb<0.6,0.6,0.6>;
-#declare rot = rgb<0.8,0.2,0.2>;
-#declare gruen = rgb<0,0.6,0>;
-#declare blau = rgb<0.2,0.2,0.8>;
-
-#declare kugelfarbe = rgb<0.8,0.8,0.8>;
-#declare kugeltransparent = rgbt<0.8,0.8,0.8,0.5>;
-
-#macro kugel(farbe)
-sphere {
-	<0, 0, 0>, 1
-	pigment {
-		color farbe
-	}
-}
-#end
-
diff --git a/buch/papers/nav/images/dreieck3d1.pdf b/buch/papers/nav/images/dreieck3d1.pdf
index 015bce7..fecaece 100644
--- a/buch/papers/nav/images/dreieck3d1.pdf
+++ b/buch/papers/nav/images/dreieck3d1.pdf
diff --git a/buch/papers/nav/images/dreieck3d1.pov b/buch/papers/nav/images/dreieck3d1.pov
index e491075..336161c 100644
--- a/buch/papers/nav/images/dreieck3d1.pov
+++ b/buch/papers/nav/images/dreieck3d1.pov
@@ -3,8 +3,11 @@
 //
 // (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 //
+#version 3.7;
 #include "common.inc"
 
+kugel(kugeldunkel)
+
 union {
 	seite(A, B, fett)
 	seite(B, C, fett)
diff --git a/buch/papers/nav/images/dreieck3d10.pov b/buch/papers/nav/images/dreieck3d10.pov
new file mode 100644
index 0000000..2dd7c79
--- /dev/null
+++ b/buch/papers/nav/images/dreieck3d10.pov
@@ -0,0 +1,46 @@
+//
+// dreiecke3d10.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+erde()
+
+#declare Stern1 = Deneb;
+#declare Stern2 = Spica;
+
+koordinatennetz(gitterfarbe, 9, 0.001)
+
+union {
+	seite(A, Stern1, 0.5*fein)
+	seite(A, Stern2, 0.5*fein)
+	seite(A, Sakura, 0.5*fein)
+	seite(Stern1, Sakura, 0.5*fein)
+	seite(Stern2, Sakura, 0.5*fein)
+	seite(Stern1, Stern2, 0.5*fein)
+
+	punkt(A, fein)
+	punkt(Sakura, fett)
+	punkt(Deneb, fein)
+	punkt(Spica, fein)
+	punkt(Altair, fein)
+	punkt(Arktur, fein)
+	pigment {
+		color Red
+	}
+}
+
+//arrow(<-1.3,0,0>, <1.3,0,0>, fein, White)
+arrow(<0,-1.3,0>, <0,1.3,0>, fein, White)
+//arrow(<0,0,-1.3>, <0,0,1.3>, fein, White)
+
+#declare imagescale = 0.044;
+
+camera {
+	location <40, 20, -20>
+	look_at <0, 0.24, -0.20>
+	right x * imagescale
+	up y * imagescale
+}
+
diff --git a/buch/papers/nav/images/dreieck3d2.pdf b/buch/papers/nav/images/dreieck3d2.pdf
index 6b3f09d..28af5fe 100644
--- a/buch/papers/nav/images/dreieck3d2.pdf
+++ b/buch/papers/nav/images/dreieck3d2.pdf
diff --git a/buch/papers/nav/images/dreieck3d2.pov b/buch/papers/nav/images/dreieck3d2.pov
index c0625ce..9e57d22 100644
--- a/buch/papers/nav/images/dreieck3d2.pov
+++ b/buch/papers/nav/images/dreieck3d2.pov
@@ -5,6 +5,8 @@
 //
 #include "common.inc"
 
+kugel(kugeldunkel)
+
 union {
 	seite(A, B, fett)
 	seite(B, C, fett)
diff --git a/buch/papers/nav/images/dreieck3d3.pdf b/buch/papers/nav/images/dreieck3d3.pdf
index 7d79455..4fc4fc1 100644
--- a/buch/papers/nav/images/dreieck3d3.pdf
+++ b/buch/papers/nav/images/dreieck3d3.pdf
diff --git a/buch/papers/nav/images/dreieck3d3.pov b/buch/papers/nav/images/dreieck3d3.pov
index b6f64d5..bde780b 100644
--- a/buch/papers/nav/images/dreieck3d3.pov
+++ b/buch/papers/nav/images/dreieck3d3.pov
@@ -5,6 +5,8 @@
 //
 #include "common.inc"
 
+kugel(kugeldunkel)
+
 union {
 	seite(A, B, fett)
 	seite(B, C, fett)
diff --git a/buch/papers/nav/images/dreieck3d4.pdf b/buch/papers/nav/images/dreieck3d4.pdf
index e1ea757..0d57fc2 100644
--- a/buch/papers/nav/images/dreieck3d4.pdf
+++ b/buch/papers/nav/images/dreieck3d4.pdf
diff --git a/buch/papers/nav/images/dreieck3d4.pov b/buch/papers/nav/images/dreieck3d4.pov
index b6f17e3..08f266b 100644
--- a/buch/papers/nav/images/dreieck3d4.pov
+++ b/buch/papers/nav/images/dreieck3d4.pov
@@ -5,6 +5,8 @@
 //
 #include "common.inc"
 
+kugel(kugelfarbe)
+
 union {
 	seite(A, B, fein)
 	seite(A, C, fein)
diff --git a/buch/papers/nav/images/dreieck3d5.pdf b/buch/papers/nav/images/dreieck3d5.pdf
index 0c86d36..a5dd0ae 100644
--- a/buch/papers/nav/images/dreieck3d5.pdf
+++ b/buch/papers/nav/images/dreieck3d5.pdf
diff --git a/buch/papers/nav/images/dreieck3d5.pov b/buch/papers/nav/images/dreieck3d5.pov
index 188f181..1aac0dc 100644
--- a/buch/papers/nav/images/dreieck3d5.pov
+++ b/buch/papers/nav/images/dreieck3d5.pov
@@ -5,6 +5,8 @@
 //
 #include "common.inc"
 
+kugel(kugeldunkel)
+
 union {
 	seite(A, B, fein)
 	seite(A, C, fein)
diff --git a/buch/papers/nav/images/dreieck3d6.pov b/buch/papers/nav/images/dreieck3d6.pov
index 191a1e7..6bbd1a9 100644
--- a/buch/papers/nav/images/dreieck3d6.pov
+++ b/buch/papers/nav/images/dreieck3d6.pov
@@ -5,6 +5,8 @@
 //
 #include "common.inc"
 
+kugel(kugeldunkel)
+
 union {
 	seite(A, B, fett)
 	seite(A, C, fett)
diff --git a/buch/papers/nav/images/dreieck3d7.pov b/buch/papers/nav/images/dreieck3d7.pov
index aae5c6c..45dc5d6 100644
--- a/buch/papers/nav/images/dreieck3d7.pov
+++ b/buch/papers/nav/images/dreieck3d7.pov
@@ -5,6 +5,8 @@
 //
 #include "common.inc"
 
+kugel(kugeldunkel)
+
 union {
 	seite(A, C, fett)
 	seite(A, P, fett)
diff --git a/buch/papers/nav/images/dreieck3d8.jpg b/buch/papers/nav/images/dreieck3d8.jpg
index 52bd25e..f24ea33 100644
--- a/buch/papers/nav/images/dreieck3d8.jpg
+++ b/buch/papers/nav/images/dreieck3d8.jpg
diff --git a/buch/papers/nav/images/dreieck3d8.pdf b/buch/papers/nav/images/dreieck3d8.pdf
index 9d630aa..da3b110 100644
--- a/buch/papers/nav/images/dreieck3d8.pdf
+++ b/buch/papers/nav/images/dreieck3d8.pdf
diff --git a/buch/papers/nav/images/dreieck3d8.pov b/buch/papers/nav/images/dreieck3d8.pov
index 9e9921a..dae7f67 100644
--- a/buch/papers/nav/images/dreieck3d8.pov
+++ b/buch/papers/nav/images/dreieck3d8.pov
@@ -93,4 +93,5 @@ object {
 
 dreieck(A, B, C, White)
 
+kugel(kugeldunkel)
 
diff --git a/buch/papers/nav/images/macros.inc b/buch/papers/nav/images/macros.inc
new file mode 100644
index 0000000..20cb2ff
--- /dev/null
+++ b/buch/papers/nav/images/macros.inc
@@ -0,0 +1,345 @@
+//
+// macros.inc -- 3d Darstellung
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+//
+// Dimensions
+//
+#declare fett = 0.015;
+#declare fein = 0.010;
+
+#declare klein = 0.3;
+#declare gross = 0.4;
+
+//
+// colors
+//
+#declare dreieckfarbe = rgb<0.6,0.6,0.6>;
+#declare rot = rgb<0.8,0.2,0.2>;
+#declare gruen = rgb<0,0.6,0>;
+#declare blau = rgb<0.2,0.2,0.8>;
+
+#declare bekannt = rgb<0.2,0.6,1>;
+#declare unbekannt = rgb<1.0,0.6,0.8>;
+
+#declare kugelfarbe = rgb<0.8,0.8,0.8>;
+#declare kugeldunkel = rgb<0.4,0.4,0.4>;
+#declare kugeltransparent = rgbt<0.8,0.8,0.8,0.5>;
+
+#declare gitterfarbe = rgb<0.2,0.6,1>;
+#declare gitterfarbe = rgb<1.0,0.8,0>;
+
+//
+// Points Points
+//
+#declare O = <0, 0, 0>;
+#declare Nordpol = vnormalize(< 0, 1, 0>);
+#declare A = vnormalize(< 0, 1, 0>);
+#declare B = vnormalize(< 1, 2, -8>);
+#declare C = vnormalize(< 5, 1, 0>);
+#declare P = vnormalize(< 5, -1, -7>);
+
+//
+// \brief convert spherical coordinates to recctangular coordinates
+//
+// \param phi 
+// \param theta
+//
+#macro kugelpunkt(phi, theta)
+	< sin(theta) * cos(phi - pi), cos(theta), sin(theta) * sin(phi - pi) >
+#end
+
+#declare Sakura = kugelpunkt(radians(140.2325498), radians(90 -   35.71548014));
+#declare Deneb  = kugelpunkt(radians(191.9397759), radians(90 -   45.361194));
+#declare Spica  = kugelpunkt(radians(82.9868559),  radians(90 - (-11.279666)));
+#declare Altair = kugelpunkt(radians(179.3616609), radians(90 -    8.928416));
+#declare Arktur = kugelpunkt(radians(95.5647759),  radians(90 -   19.063222));
+
+//
+// draw an arrow from <from> to <to> with thickness <arrowthickness> with
+// color <c>
+//
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+        sphere {
+                from, 1.1 * arrowthickness
+        }
+        cylinder {
+                from,
+                from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+                arrowthickness
+        }
+        cone {
+                from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+                2 * arrowthickness,
+                to,
+                0
+        }
+        pigment {
+                color c
+        }
+        finish {
+                specular 0.9
+                metallic
+        }
+}
+#end
+
+#declare ntsteps = 100;
+
+//
+// \brief Draw a circle
+//
+// \param b1	basis vector for a coordinate system of the plane containing
+//		the circle
+// \param b2	the other basis vector
+// \param o	center of the circle
+// \param thick	diameter of the circular tube
+//
+#macro kreis(b1, b2, o, thick, maxwinkel)
+	#declare tpstep = pi / ntsteps;
+	#declare tp = tpstep;
+	#declare p1 = b1 + o;
+	sphere { p1, thick }
+	#declare tpstep = pi/ntsteps;
+	#while (tp < (maxwinkel - tpstep/2))
+		#declare p2 = cos(tp) * b1 + sin(tp) * b2 + o;
+		cylinder { p1, p2, thick }
+		sphere { p2, thick }
+		#declare p1 = p2;
+		#declare tp = tp + tpstep;
+	#end
+	#if ((tp - tpstep) < maxwinkel)
+		#declare p2 = cos(maxwinkel) * b1 + sin(maxwinkel) * b2 + o;
+		cylinder { p1, p2, thick }
+		sphere { p2, thick }
+	#end
+#end
+
+//
+// \brief Draw a great circle
+//
+// \param normale	the normal of the plane containing the great circle
+// \param thick		diameter
+//
+#macro grosskreis(normale, thick)
+	#declare other = < normale.y, -normale.x, normale.z >;
+	#declare b1 = vnormalize(vcross(other, normale));
+	#declare b2 = vnormalize(vcross(normale, b1));
+	kreis(b1, b2, <0,0,0>, thick, 2*pi)
+#end
+
+//
+// \brief Draw a circle of latitude
+//
+// \param theta		latitude
+// \param thick		diameter
+//
+#macro breitenkreis(theta, thick)
+	#declare b1 = sin(theta) * kugelpunkt(0, pi/2);
+	#declare b2 = sin(theta) * kugelpunkt(pi/2, pi/2);
+	#declare o = < 0, cos(theta), 0 >;
+	kreis(b1, b2, o, thick, 2*pi)
+#end
+
+//
+// \brief Draw the great circle connecting the two points
+//
+// \param p		first point
+// \param q		second point
+// \param staerke	diameter
+//
+
+#macro seite(p, q, staerke)
+	#declare s1 = vnormalize(p);
+	#declare s2 = vnormalize(q);
+	#declare w = acos(vdot(s1, s2));
+	#declare n = vnormalize(vcross(p, q));
+	#declare s2 = vnormalize(vcross(n, s1));
+	kreis(s1, s2, O, staerke, w)
+#end
+
+//
+// \brief Draw an angle
+//
+// \param w		the edge where the angle is located
+// \param p		point on the first leg
+// \param q		point on the second leg
+// \param r		diameter of the angle
+//
+#macro winkel(w, p, q, staerke, r)
+	#declare n = vnormalize(w);
+	#declare pp = vnormalize(p - vdot(n, p) * n);
+	#declare qq = vnormalize(q - vdot(n, q) * n);
+	intersection {
+		sphere { O, 1 + staerke }
+		cone { O, 0, 1.2 * vnormalize(w), r }
+		plane { -vcross(n, qq) * vdot(vcross(n, qq), pp), 0 }
+		plane { -vcross(n, pp) * vdot(vcross(n, pp), qq), 0 }
+	}
+#end
+
+//
+// \brief Draw a point on the sphere as a circle
+//
+// \param p		the point
+// \param staerke	the diameter of the point
+//
+#macro punkt(p, staerke)
+	sphere { p, 1.5 * staerke }
+#end
+
+//
+// \brief Draw a circle as a part of the differently colored cutout from 
+//        the sphere
+//
+// \param p		first point of the triangle
+// \param q		second point of the triangle
+// \param r		third point of the triangle
+// \param farbe		color
+//
+#macro dreieck(p, q, r, farbe)
+	#declare n1 = vnormalize(vcross(p, q));
+	#declare n2 = vnormalize(vcross(q, r));
+	#declare n3 = vnormalize(vcross(r, p));
+	intersection {
+		plane { n1, 0 }
+		plane { n2, 0 }
+		plane { n3, 0 }
+		sphere { <0, 0, 0>, 1 + 0.001 }
+		pigment {
+			color farbe
+		}
+		finish {
+			metallic
+			specular 0.4
+		}
+	}
+#end
+
+//
+// \brief 
+//
+// \param a		axis of the angle
+// \param p		first leg
+// \param q		second leg
+// \param s		thickness of the angle disk
+// \param r		radius of the angle disk
+// \param farbe		color
+//
+#macro ebenerwinkel(a, p, q, s, r, farbe)
+	#declare n = vnormalize(-vcross(p, q));
+	#declare np = vnormalize(-vcross(p, n));
+	#declare nq = -vnormalize(-vcross(q, n));
+//	arrow(a, a + n, 0.02, White)
+//	arrow(a, a + np, 0.01, Red)
+//	arrow(a, a + nq, 0.01, Blue)
+	intersection {
+		cylinder { a - (s/2) * n, a + (s/2) * n, r }
+		plane { np, vdot(np, a) }
+		plane { nq, vdot(nq, a) }
+		pigment {
+			farbe
+		}
+		finish {
+			metallic
+			specular 0.5
+		}
+	}
+#end
+
+//
+// \brief Show the complement angle
+//
+//
+#macro komplement(a, p, q, s, r, farbe)
+	#declare n = vnormalize(-vcross(p, q));
+//	arrow(a, a + n, 0.015, Orange)
+	#declare m = vnormalize(-vcross(q, n));
+//	arrow(a, a + m, 0.015, Pink)
+	ebenerwinkel(a, p, m, s, r, farbe)
+#end
+
+//
+// \brief Show a coordinate grid on the sphere
+//
+// \param farbe		the color of the grid
+// \param thick		the line thickness
+//
+#macro koordinatennetz(farbe, netzschritte, thick)
+union {
+	// circles of latitude
+	#declare theta = pi/(2*netzschritte);
+	#declare thetastep = pi/(2*netzschritte);
+	#while (theta < pi - thetastep/2)
+		breitenkreis(theta, thick)
+		#declare theta = theta + thetastep;
+	#end
+	// cirles of longitude
+	#declare phi = 0;
+	#declare phistep = pi/(2*netzschritte);
+	#while (phi < pi-phistep/2)
+		grosskreis(kugelpunkt(phi, pi/2), thick)
+		#declare phi = phi + phistep;
+	#end
+	pigment {
+		color farbe
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+//
+// \brief Display a color of given color
+//
+// \param farbe		the color
+//
+#macro kugel(farbe)
+sphere {
+	<0, 0, 0>, 1
+	pigment {
+		color farbe
+	}
+}
+#end
+
+//
+// \brief Display the earth 
+//
+#macro erde(winkel)
+sphere {
+	<0, 0, 0>, 1
+	pigment {
+		image_map {
+			png "2k_earth_daymap.png" gamma 1.0
+			map_type 1
+		}
+	}
+	rotate <0,winkel,0>
+}
+#end
+
+//
+// achse
+//
+#macro achse(durchmesser, farbe)
+	cylinder {
+		< 0, -1.2, 0 >, <0, 1.2, 0 >, durchmesser
+		pigment {
+			color farbe
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
diff --git a/buch/papers/nav/images/position/2k_earth_daymap.png b/buch/papers/nav/images/position/2k_earth_daymap.png
new file mode 100644
index 0000000..4d55da8
--- /dev/null
+++ b/buch/papers/nav/images/position/2k_earth_daymap.png
diff --git a/buch/papers/nav/images/position/Makefile b/buch/papers/nav/images/position/Makefile
new file mode 100644
index 0000000..eed2e56
--- /dev/null
+++ b/buch/papers/nav/images/position/Makefile
@@ -0,0 +1,69 @@
+#
+# Makefile to build images 
+#
+# (c) 2022
+#
+all:	position
+
+POSITION =								\
+	position1.pdf position1-small.pdf				\
+	position2.pdf position2-small.pdf				\
+	position3.pdf position3-small.pdf				\
+	position4.pdf position4-small.pdf				\
+	position5.pdf position5-small.pdf	
+
+position: $(POSITION)
+
+POVRAYOPTIONS = -W1080 -H1080
+#POVRAYOPTIONS = -W480 -H480
+
+position1.png:	position1.pov common.inc ../macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Oposition1.png position1.pov
+position1.jpg:	position1.png
+	convert position1.png -density 300 -units PixelsPerInch position1.jpg
+position1.pdf:	position1.tex common.tex position1.jpg
+	pdflatex position1.tex
+
+position2.png:	position2.pov common.inc ../macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Oposition2.png position2.pov
+position2.jpg:	position2.png
+	convert position2.png -density 300 -units PixelsPerInch position2.jpg
+position2.pdf:	position2.tex common.tex position2.jpg
+	pdflatex position2.tex
+
+position3.png:	position3.pov common.inc ../macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Oposition3.png position3.pov
+position3.jpg:	position3.png
+	convert position3.png -density 300 -units PixelsPerInch position3.jpg
+position3.pdf:	position3.tex common.tex position3.jpg
+	pdflatex position3.tex
+
+position4.png:	position4.pov common.inc ../macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Oposition4.png position4.pov
+position4.jpg:	position4.png
+	convert position4.png -density 300 -units PixelsPerInch position4.jpg
+position4.pdf:	position4.tex common.tex position4.jpg
+	pdflatex position4.tex
+
+position5.png:	position5.pov common.inc ../macros.inc
+	povray +A0.1 $(POVRAYOPTIONS) -Oposition5.png position5.pov
+position5.jpg:	position5.png
+	convert position5.png -density 300 -units PixelsPerInch position5.jpg
+position5.pdf:	position5.tex common.tex position5.jpg
+	pdflatex position5.tex
+
+position1-small.pdf:	position1-small.tex common.tex position1.jpg
+	pdflatex position1-small.tex
+position2-small.pdf:	position2-small.tex common.tex position2.jpg
+	pdflatex position2-small.tex
+position3-small.pdf:	position3-small.tex common.tex position3.jpg
+	pdflatex position3-small.tex
+position4-small.pdf:	position4-small.tex common.tex position4.jpg
+	pdflatex position4-small.tex
+position5-small.pdf:	position5-small.tex common.tex position5.jpg
+	pdflatex position5-small.tex
+
+test:	test.pdf
+
+test.pdf:	test.tex $(POSITION)
+	pdflatex test.tex
diff --git a/buch/papers/nav/images/position/common-small.tex b/buch/papers/nav/images/position/common-small.tex
new file mode 100644
index 0000000..9430608
--- /dev/null
+++ b/buch/papers/nav/images/position/common-small.tex
@@ -0,0 +1,32 @@
+%
+% common.tex
+%
+% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+
+\def\labelA{\node at (0.7,3.8) {$A$};}
+\def\labelB{\node at (-3.4,-0.8) {$B$};}
+\def\labelC{\node at (3.3,-2.1) {$C$};}
+\def\labelP{\node at (-1.4,-3.5) {$P$};}
+
+\def\labelc{\node at (-1.9,2.1) {$c$};}
+\def\labela{\node at (-0.2,-1.2) {$a$};}
+\def\labelb{\node at (2.6,1.5) {$b$};}
+
+\def\labelhb{\node at (-2.6,-2.2) {$h_B$};}
+\def\labelhc{\node at (1,-2.9) {$h_C$};}
+\def\labell{\node at (-0.7,0.3) {$l$};}
+
+\def\labelalpha{\node at (0.6,2.85) {$\alpha$};}
+\def\labelbeta{\node at (-2.5,-0.5) {$\beta$};}
+\def\labelgamma{\node at (2.3,-1.2) {$\gamma$};}
+\def\labelomega{\node at (0.85,3.3) {$\omega$};}
+
+\def\labelgammaone{\node at (2.1,-2.0) {$\gamma_1$};}
+\def\labelgammatwo{\node at (2.3,-1.3) {$\gamma_2$};}
+\def\labelbetaone{\node at (-2.4,-1.4) {$\beta_1$};}
+\def\labelbetatwo{\node at (-2.5,-0.8) {$\beta_2$};}
+
+\def\labelomegalinks{\node at (0.25,3.25) {$\omega$};}
+\def\labelomegarechts{\node at (0.85,3.1) {$\omega$};}
+
diff --git a/buch/papers/nav/images/position/common.inc b/buch/papers/nav/images/position/common.inc
new file mode 100644
index 0000000..56e2836
--- /dev/null
+++ b/buch/papers/nav/images/position/common.inc
@@ -0,0 +1,39 @@
+//
+// common.inc -- 3d Darstellung
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+#include "../macros.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.034;
+
+camera {
+	location <40, 20, -20>
+	look_at <0, 0.24, -0.20>
+	right x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<30, 10, -40> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+//kugel(kugeldunkel)
+erde(-100)
+koordinatennetz(gitterfarbe, 9, 0.001)
+achse(fein, White)
diff --git a/buch/papers/nav/images/position/common.tex b/buch/papers/nav/images/position/common.tex
new file mode 100644
index 0000000..9430608
--- /dev/null
+++ b/buch/papers/nav/images/position/common.tex
@@ -0,0 +1,32 @@
+%
+% common.tex
+%
+% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+
+\def\labelA{\node at (0.7,3.8) {$A$};}
+\def\labelB{\node at (-3.4,-0.8) {$B$};}
+\def\labelC{\node at (3.3,-2.1) {$C$};}
+\def\labelP{\node at (-1.4,-3.5) {$P$};}
+
+\def\labelc{\node at (-1.9,2.1) {$c$};}
+\def\labela{\node at (-0.2,-1.2) {$a$};}
+\def\labelb{\node at (2.6,1.5) {$b$};}
+
+\def\labelhb{\node at (-2.6,-2.2) {$h_B$};}
+\def\labelhc{\node at (1,-2.9) {$h_C$};}
+\def\labell{\node at (-0.7,0.3) {$l$};}
+
+\def\labelalpha{\node at (0.6,2.85) {$\alpha$};}
+\def\labelbeta{\node at (-2.5,-0.5) {$\beta$};}
+\def\labelgamma{\node at (2.3,-1.2) {$\gamma$};}
+\def\labelomega{\node at (0.85,3.3) {$\omega$};}
+
+\def\labelgammaone{\node at (2.1,-2.0) {$\gamma_1$};}
+\def\labelgammatwo{\node at (2.3,-1.3) {$\gamma_2$};}
+\def\labelbetaone{\node at (-2.4,-1.4) {$\beta_1$};}
+\def\labelbetatwo{\node at (-2.5,-0.8) {$\beta_2$};}
+
+\def\labelomegalinks{\node at (0.25,3.25) {$\omega$};}
+\def\labelomegarechts{\node at (0.85,3.1) {$\omega$};}
+
diff --git a/buch/papers/nav/images/position/position1-small.pdf b/buch/papers/nav/images/position/position1-small.pdf
new file mode 100644
index 0000000..ba7755f
--- /dev/null
+++ b/buch/papers/nav/images/position/position1-small.pdf
diff --git a/buch/papers/nav/images/position/position1-small.tex b/buch/papers/nav/images/position/position1-small.tex
new file mode 100644
index 0000000..05fad44
--- /dev/null
+++ b/buch/papers/nav/images/position/position1-small.tex
@@ -0,0 +1,55 @@
+%
+% position1-small.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common-small.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick,scale=0.625]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=5cm]{position1.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelB
+\labelC
+\labelP
+
+\labelc
+\labela
+\labelb
+\labell
+
+\labelhb
+\labelhc
+
+\labelalpha
+\labelomega
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position1.pdf b/buch/papers/nav/images/position/position1.pdf
new file mode 100644
index 0000000..fc4f760
--- /dev/null
+++ b/buch/papers/nav/images/position/position1.pdf
diff --git a/buch/papers/nav/images/position/position1.pov b/buch/papers/nav/images/position/position1.pov
new file mode 100644
index 0000000..a79a9f1
--- /dev/null
+++ b/buch/papers/nav/images/position/position1.pov
@@ -0,0 +1,71 @@
+//
+// position1.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "common.inc"
+
+union {
+	seite(B, C, fett)
+	punkt(A, fett)
+	punkt(B, fett)
+	punkt(C, fett)
+	punkt(P, fett)
+	pigment {
+		color dreieckfarbe
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	seite(A, P, fett)
+	pigment {
+		color rot
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+
+union {
+	seite(A, B, fett)
+	seite(A, C, fett)
+	seite(B, P, fett)
+	seite(C, P, fett)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+object {
+	winkel(A, B, C, fein, gross)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+object {
+	winkel(A, P, C, fett, klein)
+	pigment {
+		color rot
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
diff --git a/buch/papers/nav/images/position/position1.tex b/buch/papers/nav/images/position/position1.tex
new file mode 100644
index 0000000..d6c21c3
--- /dev/null
+++ b/buch/papers/nav/images/position/position1.tex
@@ -0,0 +1,55 @@
+%
+% dreieck3d1.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{position1.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelB
+\labelC
+\labelP
+
+\labelc
+\labela
+\labelb
+\labell
+
+\labelhb
+\labelhc
+
+\labelalpha
+\labelomega
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position2-small.pdf b/buch/papers/nav/images/position/position2-small.pdf
new file mode 100644
index 0000000..3333dd4
--- /dev/null
+++ b/buch/papers/nav/images/position/position2-small.pdf
diff --git a/buch/papers/nav/images/position/position2-small.tex b/buch/papers/nav/images/position/position2-small.tex
new file mode 100644
index 0000000..e5c33cf
--- /dev/null
+++ b/buch/papers/nav/images/position/position2-small.tex
@@ -0,0 +1,53 @@
+%
+% position2-small.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common-small.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick,scale=0.625]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=5cm]{position2.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelB
+\labelC
+
+\labelc
+\labela
+\labelb
+
+\begin{scope}[yshift=0.3cm,xshift=0.1cm]
+\labelalpha
+\end{scope}
+\labelbeta
+\labelgamma
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position2.pdf b/buch/papers/nav/images/position/position2.pdf
new file mode 100644
index 0000000..dbd2ea9
--- /dev/null
+++ b/buch/papers/nav/images/position/position2.pdf
diff --git a/buch/papers/nav/images/position/position2.pov b/buch/papers/nav/images/position/position2.pov
new file mode 100644
index 0000000..2abcd94
--- /dev/null
+++ b/buch/papers/nav/images/position/position2.pov
@@ -0,0 +1,70 @@
+//
+// position3.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "common.inc"
+
+dreieck(A, B, C, kugelfarbe)
+
+union {
+	punkt(A, fett)
+	punkt(B, fett)
+	punkt(C, fett)
+	pigment {
+		color dreieckfarbe
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	seite(A, B, fett)
+	seite(A, C, fett)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	seite(B, C, fett)
+	pigment {
+		color unbekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+object {
+	winkel(A, B, C, fein, gross)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	winkel(B, C, A, fein, gross)
+	winkel(C, A, B, fein, gross)
+	pigment {
+		color unbekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+
diff --git a/buch/papers/nav/images/position/position2.tex b/buch/papers/nav/images/position/position2.tex
new file mode 100644
index 0000000..339592c
--- /dev/null
+++ b/buch/papers/nav/images/position/position2.tex
@@ -0,0 +1,53 @@
+%
+% position2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{position2.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelB
+\labelC
+
+\labelc
+\labela
+\labelb
+
+\begin{scope}[yshift=0.3cm,xshift=0.1cm]
+\labelalpha
+\end{scope}
+\labelbeta
+\labelgamma
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position3-small.pdf b/buch/papers/nav/images/position/position3-small.pdf
new file mode 100644
index 0000000..fae0b85
--- /dev/null
+++ b/buch/papers/nav/images/position/position3-small.pdf
diff --git a/buch/papers/nav/images/position/position3-small.tex b/buch/papers/nav/images/position/position3-small.tex
new file mode 100644
index 0000000..4f7b0e9
--- /dev/null
+++ b/buch/papers/nav/images/position/position3-small.tex
@@ -0,0 +1,51 @@
+%
+% position3-small.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common-small.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick,scale=0.625]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=5cm]{position3.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelB
+\labelC
+\labelP
+
+\labela
+
+\labelhb
+\labelhc
+
+\labelbetaone
+\labelgammaone
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position3.pdf b/buch/papers/nav/images/position/position3.pdf
new file mode 100644
index 0000000..2c940d2
--- /dev/null
+++ b/buch/papers/nav/images/position/position3.pdf
diff --git a/buch/papers/nav/images/position/position3.pov b/buch/papers/nav/images/position/position3.pov
new file mode 100644
index 0000000..f6823eb
--- /dev/null
+++ b/buch/papers/nav/images/position/position3.pov
@@ -0,0 +1,48 @@
+//
+// position3.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "common.inc"
+
+dreieck(B, P, C, kugelfarbe)
+
+union {
+	punkt(B, fett)
+	punkt(C, fett)
+	punkt(P, fett)
+	pigment {
+		color dreieckfarbe
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	seite(B, C, fett)
+	seite(B, P, fett)
+	seite(C, P, fett)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	winkel(B, P, C, fein, gross)
+	winkel(C, B, P, fein, gross)
+	pigment {
+		color unbekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
diff --git a/buch/papers/nav/images/position/position3.tex b/buch/papers/nav/images/position/position3.tex
new file mode 100644
index 0000000..d5480da
--- /dev/null
+++ b/buch/papers/nav/images/position/position3.tex
@@ -0,0 +1,51 @@
+%
+% dreieck3d1.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{position3.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelB
+\labelC
+\labelP
+
+\labela
+
+\labelhb
+\labelhc
+
+\labelbetaone
+\labelgammaone
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position4-small.pdf b/buch/papers/nav/images/position/position4-small.pdf
new file mode 100644
index 0000000..ac80c46
--- /dev/null
+++ b/buch/papers/nav/images/position/position4-small.pdf
diff --git a/buch/papers/nav/images/position/position4-small.tex b/buch/papers/nav/images/position/position4-small.tex
new file mode 100644
index 0000000..e06523b
--- /dev/null
+++ b/buch/papers/nav/images/position/position4-small.tex
@@ -0,0 +1,50 @@
+%
+% position4-small.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common-small.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick,scale=0.625]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=5cm]{position4.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelB
+\labelP
+
+\labelc
+\labell
+\labelhb
+
+\labelomegalinks
+\labelbetatwo
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position4.pdf b/buch/papers/nav/images/position/position4.pdf
new file mode 100644
index 0000000..8eeeaac
--- /dev/null
+++ b/buch/papers/nav/images/position/position4.pdf
diff --git a/buch/papers/nav/images/position/position4.pov b/buch/papers/nav/images/position/position4.pov
new file mode 100644
index 0000000..80628f9
--- /dev/null
+++ b/buch/papers/nav/images/position/position4.pov
@@ -0,0 +1,69 @@
+//
+// position4.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "common.inc"
+
+dreieck(A, B, P, kugelfarbe)
+
+union {
+	punkt(A, fett)
+	punkt(B, fett)
+	punkt(P, fett)
+	pigment {
+		color dreieckfarbe
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	seite(A, P, fett)
+	pigment {
+		color unbekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+
+union {
+	seite(A, B, fett)
+	seite(B, P, fett)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+object {
+	winkel(B, P, A, fein, gross)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+object {
+	winkel(A, B, P, fein, gross)
+	pigment {
+		color unbekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
diff --git a/buch/papers/nav/images/position/position4.tex b/buch/papers/nav/images/position/position4.tex
new file mode 100644
index 0000000..27c1757
--- /dev/null
+++ b/buch/papers/nav/images/position/position4.tex
@@ -0,0 +1,50 @@
+%
+% position4.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{position4.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelB
+\labelP
+
+\labelc
+\labell
+\labelhb
+
+\labelomegalinks
+\labelbetatwo
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position5-small.pdf b/buch/papers/nav/images/position/position5-small.pdf
new file mode 100644
index 0000000..afe120e
--- /dev/null
+++ b/buch/papers/nav/images/position/position5-small.pdf
diff --git a/buch/papers/nav/images/position/position5-small.tex b/buch/papers/nav/images/position/position5-small.tex
new file mode 100644
index 0000000..0a0e229
--- /dev/null
+++ b/buch/papers/nav/images/position/position5-small.tex
@@ -0,0 +1,50 @@
+%
+% position5-small.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common-small.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick,scale=0.625]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=5cm]{position5.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelC
+\labelP
+
+\labelb
+\labell
+\labelhc
+
+\labelomegarechts
+\labelgammatwo
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/position5.pdf b/buch/papers/nav/images/position/position5.pdf
new file mode 100644
index 0000000..05a64cb
--- /dev/null
+++ b/buch/papers/nav/images/position/position5.pdf
diff --git a/buch/papers/nav/images/position/position5.pov b/buch/papers/nav/images/position/position5.pov
new file mode 100644
index 0000000..7ed33c5
--- /dev/null
+++ b/buch/papers/nav/images/position/position5.pov
@@ -0,0 +1,69 @@
+//
+// position5.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "common.inc"
+
+dreieck(A, P, C, kugelfarbe)
+
+union {
+	punkt(A, fett)
+	punkt(C, fett)
+	punkt(P, fett)
+	pigment {
+		color dreieckfarbe
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	seite(A, P, fett)
+	pigment {
+		color unbekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+
+union {
+	seite(A, C, fett)
+	seite(C, P, fett)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+object {
+	winkel(C, P, A, fein, gross)
+	pigment {
+		color bekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+object {
+	winkel(A, C, P, fein, gross)
+	pigment {
+		color unbekannt
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
diff --git a/buch/papers/nav/images/position/position5.tex b/buch/papers/nav/images/position/position5.tex
new file mode 100644
index 0000000..b234429
--- /dev/null
+++ b/buch/papers/nav/images/position/position5.tex
@@ -0,0 +1,50 @@
+%
+% position5.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{txfonts}
+\usepackage[utf8]{inputenc}
+\usepackage{graphics}
+\usetikzlibrary{arrows,intersections,math}
+\usepackage{ifthen}
+\begin{document}
+
+\input{common.tex}
+
+\newboolean{showgrid}
+\setboolean{showgrid}{false}
+\def\breite{4}
+\def\hoehe{4}
+
+\begin{tikzpicture}[>=latex,thick]
+
+% Povray Bild
+\node at (0,0) {\includegraphics[width=8cm]{position5.jpg}};
+
+% Gitter
+\ifthenelse{\boolean{showgrid}}{
+\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe);
+\draw                            (-\breite,-\hoehe) grid (\breite, \hoehe);
+\fill (0,0) circle[radius=0.05];
+}{}
+
+\labelA
+\labelC
+\labelP
+
+\labelb
+\labell
+\labelhc
+
+\labelomegarechts
+\labelgammatwo
+
+\end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/nav/images/position/test.tex b/buch/papers/nav/images/position/test.tex
new file mode 100644
index 0000000..8f4b341
--- /dev/null
+++ b/buch/papers/nav/images/position/test.tex
@@ -0,0 +1,135 @@
+%
+% test.tex
+%
+% (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[12pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{etex}
+\usepackage[ngerman]{babel}
+\usepackage{times}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{graphicx}
+\usepackage{wrapfig}
+\begin{document}
+
+\begin{wrapfigure}{R}{5.2cm}
+\includegraphics{position1-small.pdf}
+\end{wrapfigure}
+Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
+Aenean
+commodo ligula eget dolor.
+Aenean massa.
+Cum sociis natoque penatibus
+et magnis dis parturient montes, nascetur ridiculus mus.
+Donec quam
+felis, ultricies nec, pellentesque eu, pretium quis, sem.
+Nulla
+consequat massa quis enim.
+Donec pede justo, fringilla vel, aliquet
+nec, vulputate eget, arcu.
+In enim justo, rhoncus ut, imperdiet a,
+venenatis vitae, justo.
+Nullam dictum felis eu pede mollis pretium.
+Integer tincidunt.
+Cras dapibus.
+Vivamus elementum semper nisi.
+Aenean vulputate eleifend tellus.
+Aenean leo ligula, porttitor eu,
+consequat vitae, eleifend ac, enim.
+Aliquam lorem ante, dapibus in,
+viverra quis, feugiat a, tellus.
+
+\begin{wrapfigure}{R}{5.2cm}
+\includegraphics{position2-small.pdf}
+\end{wrapfigure}
+Maecenas tempus, tellus eget condimentum rhoncus, sem quam semper
+libero, sit amet adipiscing sem neque sed ipsum. Nam quam nunc,
+blandit vel, luctus pulvinar, hendrerit id, lorem. Maecenas nec
+odio et ante tincidunt tempus. Donec vitae sapien ut libero venenatis
+faucibus. Nullam quis ante. Etiam sit amet orci eget eros faucibus
+tincidunt. Duis leo. Sed fringilla mauris sit amet nibh. Donec
+sodales sagittis magna. Sed consequat, leo eget bibendum sodales,
+augue velit cursus nunc, quis gravida magna mi a libero. Fusce
+vulputate eleifend sapien. Vestibulum purus quam, scelerisque ut,
+mollis sed, nonummy id, metus. Nullam accumsan lorem in dui. Cras
+ultricies mi eu turpis hendrerit fringilla. Vestibulum ante ipsum
+primis in faucibus orci luctus et ultrices posuere cubilia Curae;
+
+\pagebreak
+
+\begin{wrapfigure}{R}{5.2cm}
+\includegraphics{position3-small.pdf}
+\end{wrapfigure}
+Integer ante arcu, accumsan a, consectetuer eget, posuere ut, mauris.
+Praesent adipiscing. Phasellus ullamcorper ipsum rutrum nunc. Nunc
+nonummy metus. Vestibulum volutpat pretium libero. Cras id dui.
+Aenean ut eros et nisl sagittis vestibulum. Nullam nulla eros,
+ultricies sit amet, nonummy id, imperdiet feugiat, pede. Sed lectus.
+Donec mollis hendrerit risus. Phasellus nec sem in justo pellentesque
+facilisis. Etiam imperdiet imperdiet orci. Nunc nec neque. Phasellus
+leo dolor, tempus non, auctor et, hendrerit quis, nisi. Curabitur
+ligula sapien, tincidunt non, euismod vitae, posuere imperdiet,
+leo. Maecenas malesuada. Praesent congue erat at massa. Sed cursus
+turpis vitae tortor. Donec posuere vulputate arcu. Phasellus accumsan
+cursus velit. Vestibulum ante ipsum primis in faucibus orci luctus
+et ultrices posuere cubilia Curae; Sed aliquam, nisi quis porttitor
+congue, elit erat euismod orci, ac placerat dolor lectus quis orci.
+Phasellus consectetuer vestibulum elit.
+
+\begin{wrapfigure}{R}{5.2cm}
+\includegraphics{position4-small.pdf}
+\end{wrapfigure}
+Aenean tellus metus, bibendum sed, posuere ac, mattis non, nunc.
+Vestibulum fringilla pede sit amet augue. In turpis. Pellentesque
+posuere. Praesent turpis. Aenean posuere, tortor sed cursus feugiat,
+nunc augue blandit nunc, eu sollicitudin urna dolor sagittis lacus.
+Donec elit libero, sodales nec, volutpat a, suscipit non, turpis.
+Nullam sagittis. Suspendisse pulvinar, augue ac venenatis condimentum,
+sem libero volutpat nibh, nec pellentesque velit pede quis nunc.
+Vestibulum ante ipsum primis in faucibus orci luctus et ultrices
+posuere cubilia Curae; Fusce id purus. Ut varius tincidunt libero.
+Phasellus dolor. Maecenas vestibulum mollis diam. Pellentesque ut
+neque. Pellentesque habitant morbi tristique senectus et netus et
+malesuada fames ac turpis egestas. In dui magna, posuere eget,
+vestibulum et, tempor auctor, justo. In ac felis quis tortor malesuada
+pretium. Pellentesque auctor neque nec urna.
+
+\pagebreak
+
+\begin{wrapfigure}{R}{5.2cm}
+\includegraphics{position5-small.pdf}
+\end{wrapfigure}
+Proin sapien ipsum, porta a, auctor quis, euismod ut, mi. Aenean
+viverra rhoncus pede. Pellentesque habitant morbi tristique senectus
+et netus et malesuada fames ac turpis egestas. Ut non enim eleifend
+felis pretium feugiat. Vivamus quis mi. Phasellus a est. Phasellus
+magna. In hac habitasse platea dictumst. Curabitur at lacus ac velit
+ornare lobortis. Curabitur a felis in nunc fringilla tristique.
+Morbi mattis ullamcorper velit. Phasellus gravida semper nisi.
+Nullam vel sem. Pellentesque libero tortor, tincidunt et, tincidunt
+eget, semper nec, quam. Sed hendrerit. Morbi ac felis. Nunc egestas,
+augue at pellentesque laoreet, felis eros vehicula leo, at malesuada
+velit leo quis pede. Donec interdum, metus et hendrerit aliquet,
+dolor diam sagittis ligula, eget egestas libero turpis vel mi. Nunc
+nulla. Fusce risus nisl, viverra et, tempor et, pretium in, sapien.
+Donec venenatis vulputate lorem. Morbi nec metus. Phasellus blandit
+leo ut odio. Maecenas ullamcorper, dui et placerat feugiat, eros
+pede varius nisi, condimentum viverra felis nunc et lorem. Sed magna
+purus, fermentum eu, tincidunt eu, varius ut, felis. In auctor
+lobortis lacus. Quisque libero metus, condimentum nec, tempor a,
+commodo mollis, magna. Vestibulum ullamcorper mauris at ligula.
+Fusce fermentum. Nullam cursus lacinia erat. Praesent blandit laoreet
+nibh. Fusce convallis metus id felis luctus adipiscing. Pellentesque
+egestas, neque sit amet convallis pulvinar, justo nulla eleifend
+augue, ac auctor orci leo non est. Quisque id mi. Ut tincidunt
+tincidunt erat. Etiam feugiat lorem non metus. Vestibulum dapibus
+nunc ac augue. Curabitur vestibulum aliquam leo. Praesent egestas
+neque eu enim. In hac habitasse platea dictumst. Fusce a quam. Etiam
+ut purus mattis mauris
+
+\end{document}
diff --git a/buch/papers/nav/main.tex b/buch/papers/nav/main.tex
index 47764e8..4c52547 100644
--- a/buch/papers/nav/main.tex
+++ b/buch/papers/nav/main.tex
@@ -3,7 +3,7 @@
 %
 % (c) 2020 Hochschule Rapperswil
 %
-\chapter{Spährische Navigation\label{chapter:nav}}
+\chapter{Sphärische Navigation\label{chapter:nav}}
 \lhead{Sphärische Navigation}
 \begin{refsection}
 \chapterauthor{Enez Erdem und Marc Kühne}
@@ -19,3 +19,4 @@
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
+
diff --git a/buch/papers/nav/nautischesdreieck.tex b/buch/papers/nav/nautischesdreieck.tex
index c1ad38a..d8a14af 100644
--- a/buch/papers/nav/nautischesdreieck.tex
+++ b/buch/papers/nav/nautischesdreieck.tex
@@ -1,53 +1,29 @@
 \section{Das Nautische Dreieck}
 \subsection{Definition des Nautischen Dreiecks}
-Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel. 
 Die Himmelskugel ist eine gedachte Kugel, welche die Erde und dessen Beobachter umgibt und als Rechenfläche für Koordinaten in der Astronomie und Geodäsie dient.
-Das nautische Dreieck definiert sich durch folgende Ecken: Zenit, Gestirn und Himmelspol.
-
 Der Zenit ist jener Punkt, der vom Erdmittelpunkt durch denn eigenen Standort an die Himmelskugel verlängert wird.
 Ein Gestirn ist ein Planet oder ein Fixstern, zu welchen es diverse Jahrbücher mit allen astronomischen Eigenschaften gibt. 
 Der Himmelspol ist der Nordpol an die Himmelskugel projiziert.
+Das nautische Dreieck hat die Ecken Zenit, Gestirn und Himmelspol, wie man in der Abbildung 21.5 sehen kann.
 
-Zur Anwendung der Formeln der sphärischen Trigonometrie gelten folgende einfache Zusammenhänge:
-\begin{itemize}
-	\item Seitenlänge Zenit zu Himmelspol $= \frac{\pi}{2} - \phi $
-	\item Seitenlänge Himmelspol zu Gestirn $= \frac{\pi}{2} - \delta$
-	\item Seitenlänge Zenit zu Gestirn $= \frac{\pi}{2} - h$
-	\item Winkel von Zenit zu Himmelsnordpol zu Gestirn$=\pi - \alpha$
-	\item Winkel von Himmelsnordpol zu Zenit und Gestirn$= \tau$
-\end{itemize}
-Um mit diesen Zusammenhängen zu rechnen benötigt man folgende Legende:
-\begin{center}
-	\begin{tabular}{ c c c }
-		Winkel && Name / Beziehung \\ 
-		\hline
-		$\alpha$ && Rektaszension \\  
-		$\delta$ && Deklination \\
-		$\theta$ && Sternzeit von Greenwich\\
-		$\phi$  &&  Geographische Breite\\
-		$\tau=\theta-\alpha$ && Stundenwinkel und Längengrad des Gestirns. \\
-		$a$ &&      Azimut\\
-		$h$ &&      Höhe
-	\end{tabular}
-\end{center}
+Ursprünglich ist das nautische Dreieck ein Hilfsmittel der sphärischen Astronomie um die momentane Position eines Fixsterns oder Planeten an der Himmelskugel zu bestimmen. 
 
-\subsection{Zusammenhang des nautischen Dreiecks und des Kugeldreiecks auf der Erdkugel}
+\subsection{Das Bilddreieck}
 \begin{figure}
 	\begin{center}
-		\includegraphics[height=5cm,width=8cm]{papers/nav/bilder/kugel3.png}
+		\includegraphics[width=8cm]{papers/nav/bilder/kugel3.png}
 		\caption[Nautisches Dreieck]{Nautisches Dreieck}
 	\end{center}
 \end{figure}
-
-Wie man im oberen Bild sieht, liegt das nautische Dreieck auf der Himmelskugel mit den Ecken Zenit, Gestirn und Himmelsnordpol. 
-Das selbe Dreieck kann man aber auch auf die Erdkugel projizieren und es hat dann die Ecken Standort, Bildpunkt und Nordpol. 
-Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet.
-
+ Man kann das nautische Dreieck auf die Erdkugel projizieren.
+Dieses Dreieck nennt man dann Bilddreieck. 
+Als Bildpunkt wird in der astronomischen Navigation der Punkt bezeichnet, an dem eine gedachte Linie vom Mittelpunkt eines beobachteten Gestirns zum Mittelpunkt der Erde die Erdoberfläche schneidet. 
+Die Projektion auf der Erdkugel hat die Ecken Nordpol, Standort und Bildpunkt.
 
 \section{Standortbestimmung ohne elektronische Hilfsmittel}
-Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion Nautische Dreieck auf der Erdkugel zur Hilfe genommen. 
-Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen.
-
+Um den eigenen Standort herauszufinden, wird in diesem Kapitel die Projektion des nautische Dreiecks auf die Erdkugel zur Hilfe genommen. 
+Mithilfe eines Sextanten, einem Jahrbuch und der sphärischen Trigonometrie kann man dann die Längen- und Breitengrade des eigenen Standortes bestimmen.
+Was ein Sextant und ein Jahrbuch ist, wird im Abschnitt 21.6.3 erklärt.
 \begin{figure}
 	\begin{center}
 		\includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf}
@@ -60,57 +36,63 @@ Mithilfe einiger Hilfsmittel und der Sphärischen Trigonometrie kann man dann di
 
 \subsection{Ecke $P$ und $A$}
 Unser eigener Standort ist der gesuchte Ecke $P$ und die Ecke $A$ ist in unserem Fall der Nordpol.
-Der Vorteil ander Idee des Nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. 
-Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so simpel.
+Der Vorteil an der Idee des nautischen Dreiecks ist, dass eine Ecke immer der Nordpol ist. 
+Somit ist diese Ecke immer bekannt und nur deswegen sind die Zusammenhänge von Rektaszension, Sternzeit und Deklination so einfach.
 
-\subsection{Ecke $B$ und $C$ - Bildpunkt X und Y}
+\subsection{Ecke $B$ und $C$ - Bildpunkt von $X$ und $Y$}
 Für die Standortermittlung benötigt man als weiteren Punkt ein Gestirn bzw. seinen Bildpunkt auf der Erdkugel. 
 Damit das trigonometrische Rechnen einfacher wird, werden hier zwei Gestirne zur Hilfe genommen.
 Es gibt diverse Gestirne, die man nutzen kann wie zum Beispiel die Sonne, der Mond oder die vier Navigationsplaneten Venus, Mars, Jupiter und Saturn.
 
+Die Bildpunkte von den beiden Gestirnen $X$ und $Y$ bilden die beiden Ecken $B$ und $C$ im Dreieck der Abbildung 21.5.
 \subsection{Ephemeriden}
-Zu all diesen Gestirnen gibt es Ephemeriden, die man auch Jahrbücher nennt. 
-In diesen findet man Begriffe wie Rektaszension, Deklination und Sternzeit.
-Da diese Angaben in Stundenabständen gegeben sind, muss man für die minutengenaue Bestimmung zwischen den Stunden interpolieren.
-Was diese Begriffe bedeuten, wird in den kommenden beiden Abschnitten erklärt.
+Zu all diesen Gestirnen gibt es Ephemeriden. 
+Diese enthalten die Rektaszensionen und Deklinationen in Abhängigkeit von der Zeit.
 
 \begin{figure}
 	\begin{center}
-		\includegraphics[width=18cm]{papers/nav/bilder/ephe.png}
-		\caption[Astrodienst - Ephemeriden Januar 2022]{Astrodienst - Ephemeriden Januar 2022}
+		\includegraphics[width=\textwidth]{papers/nav/bilder/ephe.png}
+		\caption[Nautical Almanac Mai 2002]{Nautical Almanac Mai 2002}
 	\end{center}
 \end{figure}
 
 \subsubsection{Deklination}
-Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und ergibt schlussendlich den Breitengrad.
+Die Deklination $\delta$ beschreibt den Winkel zwischen dem Himmelsäquator und Gestirn und entspricht dem Breitengrad des Gestirns.
+
+\subsubsection{Rektaszension und Sternzeit}
+Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt, welcher der Nullpunkt auf dem Himmelsäquator ist, steht und geht vom Koordinatensystem der Himmelskugel aus.
 
-\subsubsection{Sternzeit und Rektaszension}
-Die Rektaszension $\alpha$ gibt an, in welchem Winkel das Gestirn zum Frühlingspunkt steht und geht vom Koordinatensystem der Himmelskugel aus.
-Der Frühlungspunkt ist der Nullpunkt auf dem Himmelsäquator. 
 Die Tatsache, dass sich die Himmelskugel  ca. vier Minuten schneller um die eigene Achse dreht als die Erdkugel, stellt hier ein kleines Problem dar.
 Die Lösung ist die Sternzeit. 
-Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die 
-Am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit
-$\theta = 0$. 
+Mit dieser können wir die schnellere Drehung der Himmelskugel ausgleichen und können die am Frühlingspunkt (21. März) 12:00 Uhr ist die Sternzeit $\theta = 0$. 
 
 Die Sternzeit geht vom Frühlungspunkt aus, an welchem die Sonne den Himmelsäquator schneidet.
 Für die Standortermittlung auf der Erdkugel ist es am einfachsten, wenn man die Sternzeit von Greenwich berechnet. 
-Für die Sternzeit von Greenwich $\theta $braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht recherchieren lässt.
+Für die Sternzeit von Greenwich $\theta$ braucht man als erstes das Julianische Datum $T$ vom aktuellen Tag, welches sich leicht nachschlagen lässt.
 Im Anschluss berechnet man die Sternzeit von Greenwich
 
-$\theta = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3$.
+\[\theta = 6^h 41^m 50^s,54841 + 8640184^s,812866 \cdot T + 0^s,093104 \cdot T^2 - 0^s,0000062 \cdot T^3.\]
 
-Wenn mann die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ mithilfe der Rektaszension und Sternzeit von Greenwich bestimmen. 
+Wenn man die Sternzeit von Greenwich ausgerechnet hat, kann man den Längengrad des Gestirns $\lambda = \theta - \alpha$ bestimmen, wobei $\alpha$ die Rektaszension und $\theta$ die Sternzeit von Greenwich ist.
 Dies gilt analog auch für das zweite Gestirn.
+\subsubsection{Sextant}
+Ein Sextant ist ein nautisches Messinstrument, mit dem man den Winkel zwischen der Blickrichtung zu weit entfernten Objekten bestimmen kann. Es wird vor allem der Winkelabstand zu Gestirnen gemessen. 
+Man benutzt ihn vor allem für die astronomische Navigation auf See.
 
-\subsection{Bestimmung des eigenen Standortes P}
+\begin{figure}
+	\begin{center}
+		\includegraphics[width=10cm]{papers/nav/bilder/sextant.jpg}
+		\caption[Sextant]{Sextant}
+	\end{center}
+\end{figure}
+\subsection{Bestimmung des eigenen Standortes $P$}
 Nun hat man die Koordinaten der beiden Gestirne und man weiss die Koordinaten des Nordpols.
 Damit wir unseren Standort bestimmen können, bilden wir zuerst das Dreieck $ABC$, dann das Dreieck $BPC$ und zum Schluss noch das Dreieck $ABP$.
 Mithilfe dieser Dreiecken können wir die einfachen Sätze der sphärischen Trigonometrie anwenden und benötigen lediglich ein Ephemeride zu den Gestirnen und einen Sextant.
 
 \begin{figure}
 	\begin{center}
-		\includegraphics[width=10cm]{papers/nav/bilder/dreieck.pdf}
+		\includegraphics[width=8cm]{papers/nav/bilder/dreieck.pdf}
 		\caption[Dreieck für die Standortbestimmung]{Dreieck für die Standortbestimmung}
 	\end{center}
 \end{figure}
@@ -128,15 +110,15 @@ Mithilfe dieser Dreiecken können wir die einfachen Sätze der sphärischen Trig
 	\end{tabular}
 \end{center}
 
-Mithilfe des sphärischen Trigonometrie und den darausfolgenden Zusammenhängen des Nautischen Dreiecks können wir nun alle Seiten des Dreiecks $ABC$ berechnen.
+Mit unserem erlangten Wissen können wir nun alle Seiten des Dreiecks $ABC$ berechnen.
 
-Die Seitenlänge der Seite "Nordpol zum Bildpunkt X" sei $c$. 
+Die Seite vom Nordpol zum Bildpunkt $X$ sei $c$.
 Dann ist $c = \frac{\pi}{2} - \delta_1$. 
 
-Die Seitenlänge der Seite "Nordpol zum Bildpunkt Y" sei $b$.
+Die Seite vom Nordpol zum Bildpunkt $Y$ sei $b$.
 Dann ist $b = \frac{\pi}{2} - \delta_2$. 
 
-Der Innenwinkel beim der Ecke "Nordpol" sei $\alpha$.
+Der Innenwinkel bei der Ecke, wo der Nordpol ist sei $\alpha$.
 Dann ist $ \alpha = |\lambda_1 - \lambda_2|$. 
 
 mit 
@@ -144,55 +126,49 @@ mit
 	\begin{tabular}{ c c c }
 		Ecke && Name  \\ 
 		\hline
-		$\delta_1$ && Deklination Bildpunkt $X$ \\  
-		$\delta_2$ && Deklination Bildpunk $Y$ \\
-		$\lambda_1 $&& Längengrad Bildpunkt $X$\\
-		$\lambda_2$ && Längengrad Bildpunkt $Y$
+		$\delta_1$ && Deklination vom Bildpunkt $X$ \\  
+		$\delta_2$ && Deklination vom Bildpunk $Y$ \\
+		$\lambda_1 $&& Längengrad vom Bildpunkt $X$\\
+		$\lambda_2$ && Längengrad vom Bildpunkt $Y$
 	\end{tabular}
 \end{center}
 
-Wichtig ist: Die Differenz der Längengrade ist gleich der Innenwinkel Alpha, deswegen der Betrag!
-
-Nun haben wir die beiden Seiten $c\ und\ b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. 
+Nun haben wir die beiden Seiten $c$ und $b$ und den Winkel $\alpha$, der sich zwischen diesen Seiten befindet. 
 Mithilfe des Seiten-Kosinussatzes 
 $\cos(a) = \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)$ 
 können wir nun die dritte Seitenlänge bestimmen. 
 Es ist darauf zu achten, dass hier natürlich die Seitenlängen in Bogenmass sind und dementsprechend der Kosinus und Sinus verwendet wird. 
 
-Jetzt fehlen noch die beiden anderen Innenwinkel $\beta \ und\ \gamma$.
-Diese bestimmen wir mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$.
+Jetzt fehlen noch die beiden anderen Innenwinkel $\beta$ und\ $\gamma$.
+Diese bestimmen wir mithilfe des Sinussatzes \[\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}.\]
 Hier muss man aufpassen, dass man Seite von Winkel unterscheiden kann. 
 Im Zähler sind die Seiten, im Nenner die Winkel. 
-Somit ist $\beta =\sin^{-1} [\sin(b) \cdot \frac{\sin(\alpha)}{\sin(a)}] $.
+Somit ist \[\beta =\sin^{-1} [\sin(b) \cdot \frac{\sin(\alpha)}{\sin(a)}].\]
 
-Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha, \beta \ und \ \gamma$ bestimmt und somit das ganze erste Kugeldreieck berechnet.
+Schlussendlich haben wir die Seiten $a,b\ und \ c$, die Ecken A,B und C und die Winkel $\alpha$, $\beta$  und  $\gamma$ bestimmt und somit das ganze Kugeldreieck $ABC$ berechnet.
 
 \subsubsection{Dreieck $BPC$}
-Wir bilden nun ein zweites Dreieck, welches die Ecken B und C des ersten Dreiecks besitzt. 
-Die dritte Ecke ist der eigene Standort P.
+Wir bilden nun ein zweites Dreieck, welches die Ecken $B$ und $C$ des ersten Dreiecks besitzt. 
+Die dritte Ecke ist der eigene Standort $P$.
 Unser Standort definiere sich aus einer geographischen Breite $\delta$ und einer geographischen Länge $\lambda$. 
 
-Die Seite von P zu B sei $pb$ und die Seite von P zu C sei $pc$.
+Die Seite von $P$ zu $B$ sei $pb$ und die Seite von $P$ zu $C$ sei $pc$.
 Die beiden Seitenlängen kann man mit dem Sextant messen und durch eine einfache Formel bestimmen, nämlich $pb=\frac{\pi}{2} - h_{B}$ und $pc=\frac{\pi}{2} - h_{C}$ 
 
-mit $h_B=$ Höhe von Gestirn in B und $h_C=$ Höhe von Gestirn in C mit Sextant gemessen.
+mit $h_B=$ Höhe von Gestirn in $B$ und $h_C=$ Höhe von Gestirn in $C$ mit Sextant gemessen.
 
-Zum Schluss müssen wir noch den Winkel $\beta1$ mithilfe des Seiten-Kosinussatzes  mit den bekannten Seiten $pc$, $pb$ und $a$ bestimmen.
+Zum Schluss müssen wir noch den Winkel $\beta_1$ mithilfe des Seiten-Kosinussatzes \[\cos(pb)=\cos(pc)\cdot\cos(a)+\sin(pc)\cdot\sin(a)\cdot\cos(\beta_1)\]  mit den bekannten Seiten $pc$, $pb$ und $a$ bestimmen. 
 \subsubsection{Dreieck $ABP$}
-Nun muss man eine Verbindungslinie ziehen zwischen P und A. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$, den bekannten Seiten $c\ und \ pb$ und des Seiten-Kosinussatzes berechnen.
-
-Für den Seiten-Kosinussatz benötigt es noch $\kappa=\beta + \beta1$.
-
-Somit ist $\cos(l) = \cos(c)\cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)$
-
+Nun muss man eine Verbindungslinie ziehen zwischen $P$ und $A$. Die Länge $l$ dieser Linie entspricht der gesuchten geographischen Breite $\delta$. Diese lässt sich mithilfe des Dreiecks $ABP$, den bekannten Seiten $c$ und $pb$ und des Seiten-Kosinussatzes berechnen.
+Für den Seiten-Kosinussatz benötigt es noch $\kappa=\beta + \beta_1$.
+Somit ist \[\cos(l) = \cos(c)\cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)\]
 und
-
 \[
 \delta  =\cos^{-1} [\cos(c) \cdot \cos(pb) + \sin(c) \cdot \sin(pb) \cdot \cos(\kappa)].
 \]
 
-Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ACP$ in der Ecke bei $A$ befindet mithilfe des Sinussatzes $\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}$ bestimmen. 
-
+Für die Geographische Länge $\lambda$ des eigenen Standortes muss man den Winkel $\omega$, welcher sich im Dreieck $ACP$ in der Ecke bei $A$ befindet. 
+Mithilfe des Sinussatzes \[\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)}\] können wir das bestimmen. 
 Somit ist \[ \omega=\sin^{-1}[\sin(pc) \cdot \frac{\sin(\gamma)}{\sin(l)}] \]und unsere gesuchte geographische Länge schlussendlich 
 \[\lambda=\lambda_1 - \omega\]
-mit $\lambda_1$=Längengrad Bildpunkt $X$
+wobei $\lambda_1$ die Länge des Bildpunktes $X$ von $C$ ist.
diff --git a/buch/papers/nav/packages.tex b/buch/papers/nav/packages.tex
index 5b87303..f2e6132 100644
--- a/buch/papers/nav/packages.tex
+++ b/buch/papers/nav/packages.tex
@@ -8,4 +8,5 @@
 % following example
 %\usepackage{packagename}
 
-\usepackage{amsmath}
-\ No newline at end of file
+\usepackage{amsmath}
+\usepackage{cancel}
+\ No newline at end of file
diff --git a/buch/papers/nav/sincos.tex b/buch/papers/nav/sincos.tex
index bb7f1e4..a1653e8 100644
--- a/buch/papers/nav/sincos.tex
+++ b/buch/papers/nav/sincos.tex
@@ -6,14 +6,18 @@ Es gibt Hinweise, dass sich schon die Babylonier und Ägypter vor 4000 Jahren si
 Jedoch konnten sie dieses Problem nicht lösen. 
 
 Die Geschichte der sphärischen Trigonometrie ist daher eng mit der Astronomie verknüpft. Ca. 350 vor Christus dachten die Griechen über Kugelgeometrie nach und sie wurde zu einer Hilfswissenschaft der Astronomen. 
-In Folge werden auch die ersten Sätze aufgestellt und wenige Jahrhunderte später wurden Berechnungen mithilfe des Sternkataloges von Hipparchos angestellt und darauffolgend Kartenmaterial erstellt.
-In dieser Zeit wurden auch die ersten Sternenkarten angefertigt, jedoch kannte man damals die Sinusfunktion noch nicht. 
+Zwischen 190 v. Chr. und 120 v. Chr. lebte ein griechischer Astronom names Hipparchos. 
+Dieser entwickelte unter anderem die Chordentafeln, welche die Chord - Funktionen, auch Chord genannt, beinhalten und im Abschnitt 3.1.1 beschrieben sind. 
+Chord ist der Vorgänger der Sinusfunktion und galt damals als wichtigste Grundlage der Trigonometrie.
+In dieser Zeit wurden auch die ersten Sternenkarten angefertigt. Damals kannte man die Sinusfunktionen noch nicht. 
+
 Aus Indien stammten die ersten Ansätze zu den Kosinussätzen.
 Aufbauend auf den indischen und griechischen Forschungen entwickeln die Araber um das 9. Jahrhundert den Sinussatz. 
+Die Definition der trigonometrischen Funktionen ermöglicht nur, rechtwinklige Dreiecke zu berechnen. 
+Die Beziehung zwischen Seiten und Winkeln sind komplizierter und als Sinus- und Kosinussätze bekannt.
 Doch ein paar weitere Jahrhunderte vergingen bis zu diesem Thema wieder verstärkt Forschung betrieben wurde, da im 15. Jahrhundert grosse Entdeckungsreisen, hauptsächlich per Schiff, erfolgten und die Orientierung mit Sternen vermehrt an Wichtigkeit gewann.
 Man nutzte für die Kartographie nun die Kugelgeometrie, um die Genauigkeit zu erhöhen.
 Der Sinussatz, die Tangensfunktion und der neu entwickelte Seitenkosinussatz wurden in dieser Zeit bereits verwendet und im darauffolgenden Jahrhundert folgte der Winkelkosinussatz. 
 
-
 Durch weitere mathematische Entwicklungen wie den Logarithmus wurden im Laufe des nächsten Jahrhunderts viele neue Methoden und kartographische Anwendungen der Kugelgeometrie entdeckt. 
 Im 19. und 20. Jahrhundert wurden weitere nicht-euklidische Geometrien entwickelt und die sphärische Trigonometrie fand auch ihre Anwendung in der Relativitätstheorie.
 \ No newline at end of file
diff --git a/buch/papers/nav/trigo.tex b/buch/papers/nav/trigo.tex
index cf2f242..fa53189 100644
--- a/buch/papers/nav/trigo.tex
+++ b/buch/papers/nav/trigo.tex
@@ -1,34 +1,21 @@
 
 \section{Sphärische Trigonometrie}
-In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat.
-Dabei gibt es folgenden Zusammenhang zwischen der ebenen- und sphärischen Trigonometrie:
-\begin{center}
-
-
-\begin{tabular}{ccc}
-	Eben & $\leftrightarrow$ & sphärisch \\
-	\hline
-	$a$ & $\leftrightarrow$  & $\sin \ a$  \\
-	
-	$a^2$ & $\leftrightarrow$  & $-\cos \ a$ \\
-\end{tabular}
-\end{center}
-
 \subsection{Das Kugeldreieck}
+Damit man die Definition des Kugeldreiecks versteht, müssen wir zuerst Begriffe wie Grosskreisebene und Grosskreisbögen verstehen.
+Ein Grosskreis ist ein grösstmöglicher Kreis auf einer Kugeloberfläche.
+Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften.
+Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise.
+Grosskreisbögen sind die kürzesten Verbindungslinien zwischen zwei Punkten auf der Kugel.
 
-Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden, so entsteht ein Kugeldreieck $ABC$.
+Werden drei voneinander verschiedene Punkte, die sich nicht auf derselben Grosskreisebene befinden, mit Grosskreisbögen verbunden werden, so entsteht ein Kugeldreieck $ABC$.
 Für ein Kugeldreieck gilt, dass die Summe der drei Seiten kleiner als $2\pi$ aber grösser als 0 ist.
-$A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten. 
-Ein Grosskreis ist ein größtmöglicher Kreis auf einer Kugeloberfläche. 
-Sein Mittelpunkt fällt immer mit dem Mittelpunkt der Kugel zusammen und ein Schnitt auf dem Großkreis teilt die Kugel in jedem Fall in zwei gleich grosse Hälften. 
+$A$, $B$ und $C$ sind die Ecken des Dreiecks und dessen Seiten sind die Grosskreisbögen zwischen den Eckpunkten (siehe Abbildung 21.2). 
 
-Da es unendlich viele Möglichkeiten gibt, eine Kugel so zu zerschneiden, dass die Schnittebene den Kugelmittelpunkt trifft, gibt es auch unendlich viele Grosskreise.
 Da die Länge der Grosskreisbögen wegen der Abhängigkeit vom Kugelradius ungeeignet ist, wird die Grösse einer Seite mit dem zugehörigen Mittelpunktwinkel des Grosskreisbogens angegeben. 
-Laut dieser Definition ist die Seite $c$ der Winkel $AMB$. 
+Laut dieser Definition ist die Seite $c$ der Winkel $AMB$, wobei der Punkt $M$ die Erdmitte ist.
 
 Man kann bei Kugeldreiecken nicht so einfach unterscheiden, was Innen oder Aussen ist. 
 Wenn man drei Eckpunkte miteinander verbindet, ergeben sich immer 16 Kugeldreiecke. 
-Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha, \beta, \gamma < \pi$ nennt man Eulersche Dreiecke.
 
 \begin{figure}
 	\begin{center}
@@ -38,11 +25,21 @@ Jenes Kugeldreieck mit den Seitenlängen $a, b, c < \pi$ und den Winkeln $\alpha
 	
 \end{figure}
 
-\subsection{Rechtwinkliges Dreieck und Rechtseitiges Dreieck}
-Wie auch im uns bekannten Dreieck gibt es beim Kugeldreieck auch ein Rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. 
-Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss.
+\subsection{Rechtwinkliges Dreieck und rechtseitiges Dreieck}
+In der sphärischen Trigonometrie gibt es eine Symetrie zwischen Seiten und Winkel, also zu jedem Satz über Seiten und Winkel gibt es einen entsprechenden Satz, mit dem man Winkel durch Seiten und Seiten durch Winkel ersetzt hat.
+
+Wie auch im ebenen Dreieck gibt es beim Kugeldreieck auch ein rechtwinkliges Kugeldreieck, bei dem ein Winkel $\frac{\pi}{2}$ ist. 
+Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine Seitenlänge $\frac{\pi}{2}$ lang sein muss, wie man in der Abbildung 21.3 sehen kann.
+
+\begin{figure}
+	
+	\begin{center}
+		\includegraphics[width=10cm]{papers/nav/bilder/recht.jpg}
+		\caption[Rechtseitiges Kugeldreieck]{Rechtseitiges Kugeldreieck}
+	\end{center}	
+\end{figure}
 
-\subsection{Winkelsumme}
+\subsection{Winkelsumme und Flächeninhalt}
 \begin{figure}
 	
 	\begin{center}
@@ -55,8 +52,9 @@ Ein Rechtseitiges Dreieck gibt es jedoch nur beim Kugeldreieck, weil dort eine S
 Die Winkel eines Kugeldreiecks sind die, welche die Halbtangenten in den Eckpunkten einschliessen. 
 Für die Summe der Innenwinkel gilt
 \begin{align}
-	\alpha+\beta+\gamma &= \frac{A}{r^2} + \pi \ \text{und} \ \alpha+\beta+\gamma > \pi. \nonumber
+	\alpha+\beta+\gamma &= \frac{F}{r^2} + \pi \quad \text{und} \quad \alpha+\beta+\gamma > \pi, \nonumber
 \end{align}
+wobei $F$ der Flächeninhalt des Kugeldreiecks ist.
 \subsubsection{Sphärischer Exzess}
 Der sphärische Exzess
 \begin{align}
@@ -65,34 +63,71 @@ Der sphärische Exzess
 beschreibt die Abweichung der Innenwinkelsumme von $\pi$ und ist proportional zum Flächeninhalt des Kugeldreiecks.
 
 \subsubsection{Flächeninnhalt}
-Der Flächeninhalt $A$ lässt sich aus den Winkeln $\alpha,\ \beta, \ \gamma$ und dem Kugelradius $r$ berechnen.
-\subsection{Sphärischer Sinussatz}
-In jedem Dreieck ist das Verhältnis des Sinus einer Seite zum Sinus des Gegenwinkels konstant. 
+Mithilfe des Radius $r$ und dem sphärischen Exzess $\epsilon$ gilt für den Flächeninhalt 
+\[ F=\frac{\pi \cdot r^2}{\frac{\pi}{2}} \cdot \epsilon\].
+
+\subsection{Seiten und Winkelberechnung}
+Es gibt in der sphärischen Trigonometrie eigentlich gar keinen Satz des Pythagoras, wie man ihn aus der zweidimensionalen Geometrie kennt.
+Es gibt aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks, nicht aber für das rechtseitige Kugeldreieck, in eine Beziehung bringt und zum jetzigen Punkt noch unklar ist, weshalb dieser Satz so aussieht.
+Die Approximation folgt noch. 
+Es gilt nämlich:
+\begin{align}
+	\cos c = \cos a \cdot \cos b \quad \text{wenn}  \nonumber &
+	\quad \alpha = \frac{\pi}{2} \nonumber
+\end{align}
+
+\subsubsection{Approximation von kleinen Dreiecken}
+Die Sätze in der ebenen Trigonometrie sind eigentlich Approximationen der sphärischen Trigonometrie.
+So ist der Sinussatz in der Ebene nur eine Annäherung des sphärischen Sinussatzes. Das Gleiche gilt für den Kosinussatz und dem Satz des Pythagoras.
+So kann mit dem Taylorpolynom 2. Grades den Sinus und den Kosinus vom Sphärischen in die Ebene approximieren: 
+\begin{align}
+	\sin(a) &\approx a \nonumber \intertext{und}
+	\cos(a)&\approx 1-\frac{a^2}{2}.\nonumber
+\end{align}
+Es gibt ebenfalls folgende Approximierung der Seiten von der Sphäre in die Ebene:
+\begin{align}
+	a &\approx \sin(a) \nonumber \intertext{und}
+	\frac{a^2}{2} &\approx 1-\cos(a). \nonumber
+\end{align}
+Die Korrespondenzen zwischen der ebenen- und sphärischen Trigonometrie werden in den kommenden Abschnitten erläutert.
 
-Das bedeutet, dass
+\subsubsection{Sphärischer Satz des Pythagoras}
+Die Korrespondenz \[ a^2 \approx 1- \cos(a)\] liefert unter Anderem einen entsprechenden Satz des Pythagoras, nämlich
 
 \begin{align}
-	\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber \ \text{auch beim Kugeldreieck gilt.}
+	\cos(a)\cdot \cos(b) &= \cos(c)  \\
+	\bigg[1-\frac{a^2}{2}\bigg] \cdot \bigg[1-\frac{b^2}{2}\bigg] &= 1-\frac{c^2}{2} \intertext{Höhere Potenzen vernachlässigen}
+	\xcancel{1}- \frac{a^2}{2} - \frac{b^2}{2} + \xcancel{\frac{a^2b^2}{4}}&= \xcancel{1}- \frac{c^2}{2}  \\
+	-a^2-b^2 &=-c^2\\
+	a^2+b^2&=c^2
 \end{align}
+Dies ist der wohlbekannte ebener Satz des Pythagoras.
 
-\subsection{Sphärischer Kosinussätze}
-Auch in der sphärischen Trigonometrie gibt es den Seitenkosinussatz
+\subsubsection{Sphärischer Sinussatz}
+Den sphärischen Sinussatz 
 \begin{align}
-	cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber
+	\frac{\sin (a)}{\sin (\alpha)} =\frac{\sin (b)}{\sin (\beta)} = \frac{\sin (c)}{\sin (\gamma)} \nonumber
+\end{align}
+kann man ebenfalls mit der Korrespondenz \[a \approx \sin(a) \] zum entsprechenden ebenen Sinussatz \[\frac{a}{\sin (\alpha)} =\frac{b}{\sin (\beta)} = \frac{c}{\sin (\gamma)}\] approximieren.
+
+
+\subsubsection{Sphärische Kosinussätze}
+In der sphärischen Trigonometrie gibt es den Seitenkosinussatz
+\begin{align}
+	\cos \ a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos \alpha \nonumber
 \end{align} %Seitenkosinussatz
 und den Winkelkosinussatz
+\begin{align}
+	\cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c, \nonumber
+\end{align} der nur in der sphärischen Trigonometrie vorhanden ist.
 
+Analog gibt es auch beim Seitenkosinussatz eine Korrespondenz zu \[ a^2 \leftrightarrow 1-\cos(a),\] die den ebenen Kosinussatz herleiten lässt, nämlich
 \begin{align}
-	\cos \gamma = -\cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \cdot \cos c \nonumber
+	\cos(a)&= \cos(b)\cdot \cos(c) + \sin(b) \cdot \sin(c)\cdot \cos(\alpha)  \\
+	1-\frac{a^2}{2} &= \bigg[1-\frac{b^2}{2}\bigg]\bigg[1-\frac{c^2}{2}\bigg]+bc\cdot\cos(\alpha) \intertext{Höhere Potenzen vernachlässigen}
+	\xcancel{1}-\frac{a^2}{2} &= \xcancel{1}-\frac{b^2}{2}-\frac{c^2}{2} \xcancel{+\frac{b^2c^2}{4}}+bc \cdot \cos(\alpha)\\
+	a^2&=b^2+c^2-2bc \cdot \cos(\alpha)
 \end{align}
 
-\subsection{Sphärischer Satz des Pythagoras für das rechtwinklige Kugeldreieck}
-Es gibt in der sphärischen Trigonometrie eigentlich garkeinen "Satz des Pythagoras", wie man ihn aus der zweidimensionalen Geometrie kennt.
-In der sphärischen Trigonometrie gibt es aber auch einen Satz, der alle drei Seiten eines rechtwinkligen Kugeldreiecks in eine Beziehung bringt. 
 
-Es gilt nämlich:
-\begin{align}
-	\cos c = \cos a \cdot \cos b \ \text{wenn} \nonumber &
-	\alpha = \frac{\pi}{2} \nonumber
-\end{align}
  
 \ No newline at end of file
diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex
index bb95b92..0ccc116 100644
--- a/buch/papers/zeta/analytic_continuation.tex
+++ b/buch/papers/zeta/analytic_continuation.tex
@@ -1,7 +1,26 @@
 \section{Analytische Fortsetzung} \label{zeta:section:analytische_fortsetzung}
 \rhead{Analytische Fortsetzung}
 
-%TODO missing Text
+Die analytische Fortsetzung der Riemannschen Zetafunktion ist äusserst interessant.
+Sie ermöglicht die Berechnung von $\zeta(-1)$ und weiterer spannender Werte.
+So liegen zum Beispiel unendlich viele Nullstellen der Zetafunktion bei $\Re(s) = 0.5$.
+Diese sind relevant für die Primzahlverteilung und sind Gegenstand der Riemannschen Vermutung.
+
+Es werden zwei verschiedene Fortsetzungen benötigt.
+Die erste erweitert die Zetafunktion auf $\Re(s) > 0$.
+Die zweite verwendet eine Spiegelung an der $\Re(s) = 0.5$ Linie und erschliesst damit die ganze komplexe Ebene.
+Eine grafische Darstellung dieses Plans ist in Abbildung \ref{zeta:fig:continuation_overview} zu sehen.
+\begin{figure}
+    \centering
+    \input{papers/zeta/continuation_overview.tikz.tex}
+    \caption{
+        Die verschiedenen Abschnitte der Riemannschen Zetafunktion.
+        Die originale Definition von \eqref{zeta:equation1} ist im grünen Bereich gültig.
+        Für den blauen Bereich gilt \eqref{zeta:equation:fortsetzung1}.
+        Um den roten Bereich zu bekommen verwendet die Funktionalgleichung \eqref{zeta:equation:functional} eine Spiegelung an $\Re(s) = 0.5$.
+    }
+    \label{zeta:fig:continuation_overview}
+\end{figure}
 
 \subsection{Fortsetzung auf $\Re(s) > 0$} \label{zeta:subsection:auf_bereich_ge_0}
 Zuerst definieren die Dirichletsche Etafunktion als
@@ -14,8 +33,8 @@ Zuerst definieren die Dirichletsche Etafunktion als
 wobei die Reihe bis auf die alternierenden Vorzeichen die selbe wie in der Zetafunktion ist.
 Diese Etafunktion konvergiert gemäss dem Leibnitz-Kriterium im Bereich $\Re(s) > 0$, da dann die einzelnen Glieder monoton fallend sind.
 
-Wenn wir es nun schaffen, die sehr ähnliche Zetafunktion mit der Etafunktion auszudrücken, dann haben die gesuchte Fortsetzung.
-Die folgenden Schritte zeigen, wie man dazu kommt:
+Wenn wir es nun schaffen, die sehr ähnliche Zetafunktion durch die Etafunktion auszudrücken, dann haben die gesuchte Fortsetzung.
+Zuerst wiederholen wir zweimal die Definition der Zetafunktion \eqref{zeta:equation1}, wobei wir sie einmal durch $2^{s-1}$ teilen
 \begin{align}
     \zeta(s)
     &=
@@ -26,8 +45,10 @@ Die folgenden Schritte zeigen, wie man dazu kommt:
     \zeta(s)
     &=
     \sum_{n=1}^{\infty}
-    \frac{2}{(2n)^s} \label{zeta:align2}
-    \\
+    \frac{2}{(2n)^s}. \label{zeta:align2}
+\end{align}
+Durch Subtraktion der beiden Gleichungen \eqref{zeta:align1} minus \eqref{zeta:align2}, ergibt sich
+\begin{align}
     \left(1 - \frac{1}{2^{s-1}} \right)
     \zeta(s)
     &=
@@ -36,14 +57,15 @@ Die folgenden Schritte zeigen, wie man dazu kommt:
     + \frac{1}{3^s}
     \underbrace{-\frac{2}{4^s} + \frac{1}{4^s}}_{-\frac{1}{4^s}}
     \ldots
-    && \text{\eqref{zeta:align1}} - \text{\eqref{zeta:align2}}
-    \\
-    &= \eta(s)
     \\
+    &= \eta(s).
+\end{align}
+Dies ist die Fortsetzung auf den noch unbekannten Bereich $0 < \Re(s) < 1$
+\begin{equation} \label{zeta:equation:fortsetzung1}
     \zeta(s)
-    &=
+    :=
     \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s).
-\end{align}
+\end{equation}
 
 \subsection{Fortsetzung auf ganz $\mathbb{C}$} \label{zeta:subsection:auf_ganz}
 Für die Fortsetzung auf den Rest von $\mathbb{C}$, verwenden wir den Zusammenhang von Gamma- und Zetafunktion aus \ref{zeta:section:zusammenhang_mit_gammafunktion}.
@@ -54,125 +76,198 @@ Wir beginnen damit, die Gammafunktion für den halben Funktionswert zu berechnen
     \int_0^{\infty} t^{\frac{s}{2}-1} e^{-t} dt.
 \end{equation}
 Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten
-\begin{align}
+\begin{equation}
     \Gamma \left( \frac{s}{2} \right)
-    &=
+    =
     \int_0^{\infty}
     (\pi n^2)^{\frac{s}{2}}
     x^{\frac{s}{2}-1}
     e^{-\pi n^2 x}
-    dx
-    && \text{Division durch } (\pi n^2)^{\frac{s}{2}}
-    \\
+    \,dx.
+\end{equation}
+Analog zum Abschnitt \ref{zeta:section:zusammenhang_mit_gammafunktion} teilen wir durch $(\pi n^2)^{\frac{s}{2}}$
+\begin{equation}
     \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}} n^s}
-    &=
+     =
     \int_0^{\infty}
     x^{\frac{s}{2}-1}
     e^{-\pi n^2 x}
-    dx
-    && \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty}
-    \\
+    \,dx,
+\end{equation}
+und finden Zeta durch die Summenbildung $\sum_{n=1}^{\infty}$
+\begin{equation}
     \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}}
     \zeta(s)
-    &=
+    =
     \int_0^{\infty}
     x^{\frac{s}{2}-1}
     \sum_{n=1}^{\infty}
     e^{-\pi n^2 x}
-    dx. \label{zeta:equation:integral1}
-\end{align}
+    \,dx. \label{zeta:equation:integral1}
+\end{equation}
 Die Summe kürzen wir ab als $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$.
-%TODO Wieso folgendes -> aus Fourier Signal
-Es gilt
+Im Abschnitt \ref{zeta:subsec:poisson_summation} wird die poissonsche Summenformel $\sum f(n) = \sum F(n)$ bewiesen.
+In unserem Problem ist $f(n) =  e^{-\pi n^2 x}$ und die zugehörige Fouriertransformation $F(n)$ ist
+\begin{equation}
+    F(n)
+    =
+    \mathcal{F}
+    (
+    e^{-\pi n^2 x}
+    )
+    =
+    \frac{1}{\sqrt{x}}
+    e^{\frac{-n^2 \pi}{x}}.
+\end{equation}
+Dadurch ergibt sich
 \begin{equation}\label{zeta:equation:psi}
-    \psi(x)
+    \sum_{n=-\infty}^{\infty}
+    e^{-\pi n^2 x}
     =
+    \frac{1}{\sqrt{x}}
+    \sum_{n=-\infty}^{\infty}
+    e^{\frac{-n^2 \pi}{x}},
+\end{equation}
+wobei wir die Summen so verändern müssen, dass sie bei $n=1$ beginnen und wir $\psi(x)$ erhalten als
+\begin{align}
+    2
+    \sum_{n=1}^{\infty}
+    e^{-\pi n^2 x}
+    +
+    1
+    &=
+    \frac{1}{\sqrt{x}}
+    \left(
+    2
+    \sum_{n=1}^{\infty}
+    e^{\frac{-n^2 \pi}{x}}
+    +
+    1
+    \right)
+    \\
+    2
+    \psi(x)
+    +
+    1
+    &=
+    \frac{1}{\sqrt{x}}
+    \left(
+    2
+    \psi\left(\frac{1}{x}\right)
+    +
+    1
+    \right)
+    \\
+    \psi(x)
+    &=
     - \frac{1}{2}
     + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}}
-    + \frac{1}{2 \sqrt{x}}.
-\end{equation}
+    + \frac{1}{2 \sqrt{x}}.\label{zeta:equation:psi}
+\end{align}
+Diese Gleichung wird später wichtig werden.
 
 Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf als
 \begin{equation}\label{zeta:equation:integral2}
     \int_0^{\infty}
     x^{\frac{s}{2}-1}
     \psi(x)
-    dx
+    \,dx
     =
+    \underbrace{
     \int_0^{1}
     x^{\frac{s}{2}-1}
     \psi(x)
-    dx
+    \,dx
+    }_{I_1}
     +
+    \underbrace{
     \int_1^{\infty}
     x^{\frac{s}{2}-1}
     \psi(x)
-    dx,
+    \,dx
+    }_{I_2}
+    =
+    I_1 + I_2,
 \end{equation}
-wobei wir uns nun auf den ersten Teil konzentrieren werden.
-Dabei setzen wir das Wissen aus \eqref{zeta:equation:psi} ein und erhalten
+wobei wir uns nun auf den ersten Teil $I_1$ konzentrieren werden.
+Dabei setzen wir die Definition von $\psi(x)$ aus \eqref{zeta:equation:psi} ein und erhalten
 \begin{align}
+    I_1
+    =
     \int_0^{1}
     x^{\frac{s}{2}-1}
     \psi(x)
-    dx
+    \,dx
     &=
     \int_0^{1}
     x^{\frac{s}{2}-1}
     \left(
     - \frac{1}{2}
     + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}}
-    + \frac{1}{2 \sqrt{x}}.
+    + \frac{1}{2 \sqrt{x}}
     \right)
-    dx
+    \,dx
     \\
     &=
     \int_0^{1}
     x^{\frac{s}{2}-\frac{3}{2}}
     \psi \left( \frac{1}{x} \right)
     + \frac{1}{2}
-    \left(
+    \biggl(
     x^{\frac{s}{2}-\frac{3}{2}}
     -
     x^{\frac{s}{2}-1}
-    \right)
-    dx
+    \biggl)
+    \,dx
     \\
     &=
+    \underbrace{
     \int_0^{1}
     x^{\frac{s}{2}-\frac{3}{2}}
     \psi \left( \frac{1}{x} \right)
-    dx
-    + \frac{1}{2}
+    \,dx
+    }_{I_3}
+    +
+    \underbrace{
+    \frac{1}{2}
     \int_0^1
     x^{\frac{s}{2}-\frac{3}{2}}
     -
     x^{\frac{s}{2}-1}
-    dx. \label{zeta:equation:integral3}
+    \,dx
+    }_{I_4}. \label{zeta:equation:integral3}
 \end{align}
-Dabei kann das zweite Integral gelöst werden als
+Dabei kann das zweite Integral $I_4$ gelöst werden als
 \begin{equation}
+    I_4
+    =
     \frac{1}{2}
     \int_0^1
     x^{\frac{s}{2}-\frac{3}{2}}
     -
     x^{\frac{s}{2}-1}
-    dx
+    \,dx
     =
     \frac{1}{s(s-1)}.
 \end{equation}
-Das erste Integral aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist nicht lösbar in dieser Form.
+Das erste Integral $I_3$ aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist nicht lösbar in dieser Form.
 Deshalb substituieren wir $x = \frac{1}{u}$ und $dx = -\frac{1}{u^2}du$.
 Die untere Integralgrenze wechselt ebenfalls zu $x_0 = 0 \rightarrow u_0 = \infty$.
 Dies ergibt
 \begin{align}
+    I_3
+    =
     \int_{\infty}^{1}
-    {\frac{1}{u}}^{\frac{s}{2}-\frac{3}{2}}
+    \left(
+    \frac{1}{u}
+    \right)^{\frac{s}{2}-\frac{3}{2}}
     \psi(u)
     \frac{-du}{u^2}
     &=
     \int_{1}^{\infty}
-    {\frac{1}{u}}^{\frac{s}{2}-\frac{3}{2}}
+    \left(
+    \frac{1}{u}
+    \right)^{\frac{s}{2}-\frac{3}{2}}
     \psi(u)
     \frac{du}{u^2}
     \\
@@ -180,21 +275,23 @@ Dies ergibt
     \int_{1}^{\infty}
     x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)}
     \psi(x)
-    dx,
+    \,dx,
 \end{align}
 wobei wir durch Multiplikation mit $(-1)$ die Integralgrenzen tauschen dürfen.
 Es ist zu beachten das diese Grenzen nun identisch mit den Grenzen des zweiten Integrals von \eqref{zeta:equation:integral2} sind.
 Wir setzen beide Lösungen ein in Gleichung \eqref{zeta:equation:integral3} und erhalten
 \begin{equation}
+    I_1
+    =
     \int_0^{1}
     x^{\frac{s}{2}-1}
     \psi(x)
-    dx
+    \,dx
     =
     \int_{1}^{\infty}
     x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)}
     \psi(x)
-    dx
+    \,dx
     +
     \frac{1}{s(s-1)}.
 \end{equation}
@@ -206,12 +303,12 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s
     \int_0^{1}
     x^{\frac{s}{2}-1}
     \psi(x)
-    dx
+    \,dx
     +
     \int_1^{\infty}
     x^{\frac{s}{2}-1}
     \psi(x)
-    dx
+    \,dx
     \nonumber
     \\
     &=
@@ -220,12 +317,12 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s
     \int_{1}^{\infty}
     x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)}
     \psi(x)
-    dx
+    \,dx
     +
     \int_1^{\infty}
     x^{\frac{s}{2}-1}
     \psi(x)
-    dx
+    \,dx
     \\
     &=
     \frac{1}{s(s-1)}
@@ -237,7 +334,7 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s
     x^{\frac{s}{2}-1}
     \right)
     \psi(x)
-    dx
+    \,dx
     \\
     &=
     \frac{-1}{s(1-s)}
@@ -249,7 +346,7 @@ Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um s
     x^{\frac{s}{2}}
     \right)
     \frac{\psi(x)}{x}
-    dx,
+    \,dx,
 \end{align}
 zu erhalten.
 Wenn wir dieses Resultat genau anschauen, erkennen wir dass sich nichts verändert wenn $s$ mit $1-s$ ersetzt wird.
@@ -261,4 +358,120 @@ Somit haben wir die analytische Fortsetzung gefunden als
     \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}}
     \zeta(1-s).
 \end{equation}
+%TODO Definitionen und Gleichungen klarer unterscheiden
+
+\subsection{Poissonsche Summenformel} \label{zeta:subsec:poisson_summation}
+
+Der Beweis für Gleichung \ref{zeta:equation:psi} folgt direkt durch die poissonsche Summenformel.
+Um diese zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta Funktion.
+
+\begin{lemma}
+    Die Fourierreihe der periodischen Dirac Delta Funktion $\sum \delta(x - 2\pi k)$ ist
+    \begin{equation} \label{zeta:equation:fourier_dirac}
+        \sum_{k=-\infty}^{\infty}
+        \delta(x - 2\pi k)
+        =
+        \frac{1}{2\pi}
+        \sum_{n=-\infty}^{\infty}
+        e^{i n x}.
+    \end{equation}
+\end{lemma}
+
+\begin{proof}[Beweis]
+    Eine Fourierreihe einer beliebigen periodischen Funktion $f(x)$ berechnet sich als
+    \begin{align}
+        f(x)
+        &=
+        \sum_{n=-\infty}^{\infty}
+        c_n
+        e^{i n x} \\
+        c_n
+        &=
+        \frac{1}{2\pi}
+        \int_{-\pi}^{\pi}
+        f(x)
+        e^{-i n x}
+        \, dx.
+    \end{align}
+    Wenn $f(x)=\delta(x)$ eingesetz wird ergeben sich konstante Koeffizienten
+    \begin{equation}
+        c_n
+        =
+        \frac{1}{2\pi}
+        \int_{-\pi}^{\pi}
+        \delta(x)
+        e^{-i n x}
+        \, dx
+        =
+        \frac{1}{2\pi},
+    \end{equation}
+    womit die sehr einfache Fourierreihe der Dirac Delta Funktion berechnet wäre.
+\end{proof}
+
+\begin{satz}[Poissonsche Summernformel]
+    Die Summe einer Funktion $f(n)$ über alle ganzen Zahlen $n$ ist äquivalent zur Summe ihrer Fouriertransformation $F(k)$ über alle ganzen Zahlen $k$
+    \begin{equation}
+        \sum_{n=-\infty}^{\infty}
+        f(n)
+        =
+        \sum_{k=-\infty}^{\infty}
+        F(k).
+    \end{equation}
+\end{satz}
 
+\begin{proof}[Beweis]
+    Wir schreiben die Summe über die Fouriertransformation aus
+    \begin{align}
+        \sum_{k=-\infty}^{\infty}
+        F(k)
+        &=
+        \sum_{k=-\infty}^{\infty}
+        \int_{-\infty}^{\infty}
+        f(x)
+        e^{-i 2\pi x k}
+        \, dx
+        \\
+        &=
+        \int_{-\infty}^{\infty}
+        f(x)
+        \underbrace{
+        \sum_{k=-\infty}^{\infty}
+        e^{-i 2\pi x k}
+        }_{\text{\eqref{zeta:equation:fourier_dirac}}}
+        \, dx,
+    \end{align}
+    und verwenden die Fouriertransformation der Dirac Funktion aus \eqref{zeta:equation:fourier_dirac}
+    \begin{align}
+        \sum_{k=-\infty}^{\infty}
+        e^{-i 2\pi x k}
+        &=
+        2 \pi
+        \sum_{k=-\infty}^{\infty}
+        \delta(-2\pi x - 2\pi k)
+        \\
+        &=
+        \frac{2 \pi}{2 \pi}
+        \sum_{k=-\infty}^{\infty}
+        \delta(x + k).
+    \end{align}
+    Wenn wir dies einsetzen und erhalten wir den gesuchten Beweis für die poissonsche Summenformel
+    \begin{equation}
+        \sum_{k=-\infty}^{\infty}
+        F(k)
+        =
+        \int_{-\infty}^{\infty}
+        f(x)
+        \sum_{k=-\infty}^{\infty}
+        \delta(x + k)
+        \, dx
+        =
+        \sum_{k=-\infty}^{\infty}
+        \int_{-\infty}^{\infty}
+        f(x)
+        \delta(x + k)
+        \, dx
+        =
+        \sum_{k=-\infty}^{\infty}
+        f(k).
+    \end{equation}
+\end{proof}
diff --git a/buch/papers/zeta/continuation_overview.tikz.tex b/buch/papers/zeta/continuation_overview.tikz.tex
new file mode 100644
index 0000000..836ab1d
--- /dev/null
+++ b/buch/papers/zeta/continuation_overview.tikz.tex
@@ -0,0 +1,18 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=0.9cm, scale=2,
+    dot/.style={fill, circle, inner sep=0, minimum size=0.1cm}]
+
+    \draw[->] (-2,0) -- (-1,0) node[dot]{} node[anchor=north]{$-1$} -- (0,0) node[anchor=north west]{$0$} -- (0.5,0) node[anchor=north west]{$0.5$}-- (1,0) node[anchor=north west]{$1$} -- (2,0) node[anchor=west]{$\Re(s)$};
+
+    \draw[->] (0,-1.2) -- (0,1.2) node[anchor=south]{$\Im(s)$};
+    \begin{scope}[yscale=0.1]
+        \draw[] (1,-1) -- (1,1);
+    \end{scope}
+    \draw[dotted] (0.5,-1) -- (0.5,1);
+
+    \begin{scope}[]
+        \fill[opacity=0.2, red] (-1.8,1) rectangle (0, -1);
+        \fill[opacity=0.2, blue] (0,1) rectangle (1, -1);
+        \fill[opacity=0.2, green] (1,1) rectangle (1.8, -1);
+    \end{scope}
+
+\end{tikzpicture}
diff --git a/buch/papers/zeta/euler_product.tex b/buch/papers/zeta/euler_product.tex
new file mode 100644
index 0000000..a6ed512
--- /dev/null
+++ b/buch/papers/zeta/euler_product.tex
@@ -0,0 +1,85 @@
+\section{Eulerprodukt} \label{zeta:section:eulerprodukt}
+\rhead{Eulerprodukt}
+
+Das Eulerprodukt stellt die Verbindung der Zetafunktion und der Primzahlen her.
+Diese Verbindung ist sehr wichtig, da durch sie eine Aussage zur Primzahlverteilung gemacht werden kann.
+Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche eines der grössten ungelösten Probleme der Mathematik ist.
+
+\begin{satz}
+    Für alle Zahlen $s$ mit $\Re(s) > 1$ ist die Zetafunktion identisch mit dem unendlichen Eulerprodukt
+    \begin{equation}\label{zeta:eq:eulerprodukt}
+        \zeta(s)
+        =
+        \sum_{n=1}^\infty
+        \frac{1}{n^s}
+        =
+        \prod_{p \in P}
+        \frac{1}{1-p^{-s}}
+    \end{equation}
+    wobei $P$ die Menge aller Primzahlen darstellt.
+\end{satz}
+
+\begin{proof}[Beweis]
+    Der Beweis startet mit dem Eulerprodukt und stellt dieses so um, dass die Zetafunktion erscheint.
+    Als erstes ersetzen wir die Faktoren durch geometrische Reihen
+    \begin{equation}
+        \prod_{i=1}^{\infty}
+        \frac{1}{1-p^{-s}}
+        =
+        \prod_{p \in P}
+        \sum_{k_i=0}^{\infty}
+        \left(
+        \frac{1}{p_i^s}
+        \right)^{k_i}
+        =
+        \prod_{p \in P}
+        \sum_{k_i=0}^{\infty}
+        \frac{1}{p_i^{s k_i}},
+    \end{equation}
+    dabei iteriert der Index $i$ über alle Primzahlen $p_i$.
+    Durch Ausschreiben der Multiplikation und Ausklammern der Summen erhalten wir
+    \begin{align}
+        \prod_{p \in P}
+        \sum_{k_i=0}^{\infty}
+        \frac{1}{p_i^{s k_i}}
+        &=
+        \sum_{k_1=0}^{\infty}
+        \frac{1}{p_1^{s k_1}}
+        \sum_{k_2=0}^{\infty}
+        \frac{1}{p_2^{s k_2}}
+        \ldots
+        \nonumber \\
+        &=
+        \sum_{k_1=0}^{\infty}
+        \sum_{k_2=0}^{\infty}
+        \ldots
+        \left(
+        \frac{1}{p_1^{k_1}}
+        \frac{1}{p_2^{k_2}}
+        \ldots
+        \right)^s.
+        \label{zeta:equation:eulerprodukt2}
+    \end{align}
+    Der Fundamentalsatz der Arithmetik (Primfaktorzerlegung) besagt, dass jede beliebige Zahl $n \in \mathbb{N}$ durch eine eindeutige Primfaktorzerlegung beschrieben werden kann
+    \begin{equation}
+        n = \prod_i p_i^{k_i} \quad \forall \quad n \in \mathbb{N}.
+    \end{equation}
+    Jeder Summand der Summen in \eqref{zeta:equation:eulerprodukt2} ist somit eine Zahl $n$.
+    Da die Summen alle möglichen Kombinationen von Exponenten und Primzahlen in \eqref{zeta:equation:eulerprodukt2} enthält haben wir
+    \begin{equation}
+        \sum_{k_1=0}^{\infty}
+        \sum_{k_2=0}^{\infty}
+        \ldots
+        \left(
+        \frac{1}{p_1^{k_1}}
+        \frac{1}{p_2^{k_2}}
+        \ldots
+        \right)^s
+        =
+        \sum_{n=1}^\infty
+        \frac{1}{n^s}
+        =
+        \zeta(s)
+    \end{equation}
+\end{proof}
+
diff --git a/buch/papers/zeta/main.tex b/buch/papers/zeta/main.tex
index e0ea8e1..caddace 100644
--- a/buch/papers/zeta/main.tex
+++ b/buch/papers/zeta/main.tex
@@ -11,6 +11,7 @@
 %TODO Einleitung
 
 \input{papers/zeta/einleitung.tex}
+\input{papers/zeta/euler_product.tex}
 \input{papers/zeta/zeta_gamma.tex}
 \input{papers/zeta/analytic_continuation.tex}
 
diff --git a/buch/papers/zeta/zeta_gamma.tex b/buch/papers/zeta/zeta_gamma.tex
index 59c8744..db41676 100644
--- a/buch/papers/zeta/zeta_gamma.tex
+++ b/buch/papers/zeta/zeta_gamma.tex
@@ -1,38 +1,46 @@
-\section{Zusammenhang mit Gammafunktion} \label{zeta:section:zusammenhang_mit_gammafunktion}
-\rhead{Zusammenhang mit Gammafunktion}
+\section{Zusammenhang mit der Gammafunktion} \label{zeta:section:zusammenhang_mit_gammafunktion}
+\rhead{Zusammenhang mit der Gammafunktion}
 
-Dieser Abschnitt stellt die Verbindung zwischen der Gamma- und der Zetafunktion her.
+In diesem Abschnitt wird gezeigt, wie sich die Zetafunktion durch die Gammafunktion $\Gamma(s)$ ausdrücken lässt.
+Dieser Zusammenhang der Art $\zeta(s) = f(\Gamma(s))$ ist nicht nur interessant, er wird später auch für die Herleitung der analytischen Fortsetzung gebraucht.
 
-%TODO ref Gamma
-Wenn in der Gammafunkion die Integrationsvariable $t$ substituieren mit $t = nu$ und $dt = n du$, dann können wir die Gleichung umstellen und erhalten den Zusammenhang mit der Zetafunktion
-\begin{align}
+Wir erinnern uns an die Definition der Gammafunktion in \eqref{buch:rekursion:gamma:integralbeweis}
+\begin{equation*}
+    \Gamma(s)
+    =
+    \int_0^{\infty} t^{s-1} e^{-t} \,dt,
+\end{equation*}
+wobei die Notation an die Zetafunktion angepasst ist.
+Durch die Substitution von $t$ mit $t = nu$ und $dt = n\,du$ wird daraus
+\begin{align*}
     \Gamma(s)
     &=
-    \int_0^{\infty} t^{s-1} e^{-t} dt
-    \\
+    \int_0^{\infty} n^{s-1}u^{s-1} e^{-nu} n \,du \\
     &=
-    \int_0^{\infty} n^{s\cancel{-1}}u^{s-1} e^{-nu} \cancel{n}du
-    &&
-    \text{Division durch }n^s
-    \\
+    \int_0^{\infty} n^s u^{s-1} e^{-nu} \,du.
+\end{align*}
+Durch Division mit durch $n^s$ ergibt sich die Quotienten
+\begin{equation*}
     \frac{\Gamma(s)}{n^s}
-    &=
-    \int_0^{\infty} u^{s-1} e^{-nu}du
-    &&
-    \text{Zeta durch Summenbildung } \sum_{n=1}^{\infty}
-    \\
+    =
+    \int_0^{\infty} u^{s-1} e^{-nu} \,du,
+\end{equation*}
+welche sich zur Zetafunktion summieren
+\begin{equation}
+    \sum_{n=1}^{\infty} \frac{\Gamma(s)}{n^s}
+    =
     \Gamma(s) \zeta(s)
-    &=
+    =
     \int_0^{\infty} u^{s-1}
     \sum_{n=1}^{\infty}e^{-nu}
-    du.
+    \,du.
     \label{zeta:equation:zeta_gamma1}
-\end{align}
+\end{equation}
 Die Summe über $e^{-nu}$ können wir als geometrische Reihe schreiben und erhalten
 \begin{align}
-    \sum_{n=1}^{\infty}e^{-u^n}
+    \sum_{n=1}^{\infty}\left(e^{-u}\right)^n
     &=
-    \sum_{n=0}^{\infty}e^{-u^n}
+    \sum_{n=0}^{\infty}\left(e^{-u}\right)^n
     -
     1
     \\
@@ -42,12 +50,12 @@ Die Summe über $e^{-nu}$ können wir als geometrische Reihe schreiben und erhal
     &=
     \frac{1}{e^u - 1}.
 \end{align}
-Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir
+Wenn wir dieses Resultat einsetzen in \eqref{zeta:equation:zeta_gamma1} und durch $\Gamma(s)$ teilen, erhalten wir den gewünschten Zusammenhang
 \begin{equation}\label{zeta:equation:zeta_gamma_final}
     \zeta(s)
     =
     \frac{1}{\Gamma(s)}
     \int_0^{\infty}
     \frac{u^{s-1}}{e^u -1}
-    du.
+    du \qed
 \end{equation}
author	JODBaer <55744603+JODBaer@users.noreply.github.com>	2022-06-13 09:18:25 +0200
committer	GitHub <noreply@github.com>	2022-06-13 09:18:25 +0200
commit	3010b2b87e56a8e2fbc2476b9971d9ef886f17a0 (patch)
tree	9de92825e4293741d7d617d40e661fb5863bb8b9 /buch/papers
parent	Merge branch 'AndreasFMueller:master' into master (diff)
parent	flow (diff)
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