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authorAlain <mceagle117@gmail.com>2021-05-26 17:41:38 +0200
committerAlain <mceagle117@gmail.com>2021-05-26 17:41:38 +0200
commita1a45cd5bd0e487cb69916f8c3e636a5e326c935 (patch)
treefe363da909509dfea40eb934ad0c89ea78fb3d5c /buch/papers/ifs
parentiniteur (diff)
downloadSeminarMatrizen-a1a45cd5bd0e487cb69916f8c3e636a5e326c935.tar.gz
SeminarMatrizen-a1a45cd5bd0e487cb69916f8c3e636a5e326c935.zip
Fraktale Kapitel Fertig
Diffstat (limited to 'buch/papers/ifs')
-rw-r--r--buch/papers/ifs/images/koch0.eps1004
-rw-r--r--buch/papers/ifs/images/koch1.eps1073
-rw-r--r--buch/papers/ifs/images/koch2.eps1085
-rw-r--r--buch/papers/ifs/images/koch8.eps26780
-rw-r--r--buch/papers/ifs/teil0.tex2
-rw-r--r--buch/papers/ifs/teil1.tex116
6 files changed, 30023 insertions, 37 deletions
diff --git a/buch/papers/ifs/images/koch0.eps b/buch/papers/ifs/images/koch0.eps
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+S
+GR
+%%Trailer
+%%Pages: 1
+%%EOF
diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex
index 7e3d344..d61c013 100644
--- a/buch/papers/ifs/teil0.tex
+++ b/buch/papers/ifs/teil0.tex
@@ -9,6 +9,4 @@ Mit der Hilfe von Iterierten Funktionsschemata mit nur wenigen Funktionen, kompl
In der Regel sind diese Bilder Fraktale.
Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir im folgenden Kapitel untersuchen.
-\subsection{Metrische Räume}
-
diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex
index 76bc828..327a082 100644
--- a/buch/papers/ifs/teil1.tex
+++ b/buch/papers/ifs/teil1.tex
@@ -8,43 +8,89 @@
\rhead{Problemstellung}
Bevor wir die IFS genauer ansehen, schauen wir uns Fraktale genauer an.
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{ifs: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.
+\subsection{Was sind Fraktale?
+\label{ifs:subsection:finibus}}
+Über die genaue Definition von Fraktalen sind sich die Mathematiker noch nicht einig.
+In diesem Kapitel orientieren wir uns an den Eigneschaften welche Kenneth Flaconer in seinem Buch Fractal Geometry beschreibt.
+Von einem Fraktal $F$ können wir folgende Eigneschaften erwarten:
+\begin{enumerate}
+ \item $F$ hat eine unendlich feine Struktur
+ \item $F$ kann nicht mit der klassischen Geometrie beschrieben werden.
+ \item Oftmals haf $F$ eine Form von Selbstähnlichkeit.
+ \item Die 'fraktale Dimension' ist grösser als die Topologische Dimension
+ \item Viele Fraktale lassen sich einfach beschrieben
+\end{enumerate}
+\subsection{Koch Kurve
+ \label{ifs:subsection:lilkoch}}
+Diese Eigenschaften möchten wir nun anhand der Koch Kurve näher anschauen.
+In \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Wie man schon erahnen kann, besteht die aus lauter kleineren Kopien von sich selber.
+Den Konstruktionvorgang sehen wir in \ref{ifs:kochconst}.
+Gestartet wird mit einer einzelnen Strecke der Länge $a$.
+Diese wird in ersten Schritt mit vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt.
+In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtilich.
+Dieser Schritt wird nun für jeden der resultierten Streckenabschnitten wiederholt.
+Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist.
-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{ifs: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}.
+\begin{figure}
+ \label{ifs:kochkurve8}
+ \centering
+ \includegraphics{papers/ifs/images/koch8}
+ \caption{Koch Kurve}
+\end{figure}
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{ifs: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{ifs: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.
+\begin{figure}
+ \label{ifs:kochconst}
+ \centering
+ \subfigure[]{
+ \label{ifs:kochconsta}
+ \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch0}}
+ \subfigure[]{
+ \label{ifs:kochconstb}
+ \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch1}}
+ \subfigure[]{
+ \label{kochconstc}
+ \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch2}}
+ \caption{(a) Start (b) 1. Iteration (c) 2. Iteration}
+ \label{fig:foobar}
+\end{figure}
+Die resultierende Kurve hat ein paar interessante Eigenschaften.
+Die Länge der Kurve lasst sich einfach berechnen.
+\begin{align*}
+ l_0 = a ,\quad l_1 = a \frac{4}{3} ,\quad l_2 = a \left( \frac{4}{3}\right)^2 , \quad ... , \quad
+ l_n = a * \left( \frac{4}{3}\right)^n \quad
+ \Rightarrow \quad
+ \lim_{n\to\infty} a \left( \frac{4}{3}\right)^n = \infty
+\end{align*}
+In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verglängert. Somit divergiert die Länge gegen Unendlich.
+Die Fläche unter der Kurve lässt sich folgendermassen berechnen
+\begin{align*}
+ A_0 = 0 , \quad A_1 = \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\
+ A_2 = A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\
+ A_3 = A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1
+\end{align*}
+Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist.
+Daraus resultiert eine konvergierende Geometrische Rheie.
+\begin{align*}
+ A_n = A_1 \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n \\
+ \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2
+\end{align*}
+Wie wir sehen ist die Kochkurve ein Konstrukt mit endlicher Fläche, aber unendlichem Umfang.
+Zu guter letzt bestimmen wir die Dimension der Kurve.
+Es gibt viele verschidene Arten die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern.
+Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur viele verschiedene Arten.
+In diesem Beispiel werden wir die Ähnlichkeits-Dimension.
+\begin{align*}
+ D = - \frac{log(N)}{log(\epsilon)}
+\end{align*}
+Mit ihr kann man einfach die Dimension selbstähnlicher Mengen bestimmen.
+Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge.
+Somit hat das Dreieck die Dimension $D = 2$.
+Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon = 1/3$.
+\begin{align*}
+ D = - \frac{log(N)}{log(\epsilon)} = - \frac{log(4)}{log(1/3)} \approx 1.2619
+\end{align*}
+Wie wir nun sehen besitzt die Kochkurve alle oben beschriebenen Eigenschaften von Fraktalen.
+Dies muss jedoch nicht bei allen Fraktalen der Fall. Sonst wäre die Frage nach einer 'richtigen' Definition einfach zu beantworten.