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authorAndreas Müller <andreas.mueller@ost.ch>2022-06-15 20:25:27 +0200
committerAndreas Müller <andreas.mueller@ost.ch>2022-06-15 20:25:27 +0200
commit864b17ee949de5c14ebc3bbf50a90178b4b804f3 (patch)
treebf43dfb50e5874e6b0bd083d71c77171517554a6 /buch/chapters/110-elliptisch
parentMerge pull request #19 from enezerdem/master (diff)
downloadSeminarSpezielleFunktionen-864b17ee949de5c14ebc3bbf50a90178b4b804f3.tar.gz
SeminarSpezielleFunktionen-864b17ee949de5c14ebc3bbf50a90178b4b804f3.zip
fix some minor issues
Diffstat (limited to 'buch/chapters/110-elliptisch')
-rw-r--r--buch/chapters/110-elliptisch/images/Makefile7
-rw-r--r--buch/chapters/110-elliptisch/images/jacobiplots.pdfbin56975 -> 56975 bytes
-rw-r--r--buch/chapters/110-elliptisch/images/kegelpara.pdfbin139626 -> 202828 bytes
-rw-r--r--buch/chapters/110-elliptisch/images/kegelpara.pov122
-rw-r--r--buch/chapters/110-elliptisch/images/kegelpara.tex6
-rw-r--r--buch/chapters/110-elliptisch/images/torusschnitt.pdfbin137159 -> 301677 bytes
-rw-r--r--buch/chapters/110-elliptisch/images/torusschnitt.pov88
-rw-r--r--buch/chapters/110-elliptisch/lemniskate.tex12
8 files changed, 211 insertions, 24 deletions
diff --git a/buch/chapters/110-elliptisch/images/Makefile b/buch/chapters/110-elliptisch/images/Makefile
index 30642fe..c8f98cb 100644
--- a/buch/chapters/110-elliptisch/images/Makefile
+++ b/buch/chapters/110-elliptisch/images/Makefile
@@ -79,11 +79,14 @@ slcldata.tex: slcl
slcl.pdf: slcl.tex slcldata.tex
pdflatex slcl.tex
+KEGELSIZE = -W256 -H256
+KEGELSIZE = -W128 -H128
+KEGELSIZE = -W1080 -H1080
kegelpara.png: kegelpara.pov
- povray +A0.1 -W1080 -H1080 -Okegelpara.png kegelpara.pov
+ povray +A0.1 $(KEGELSIZE) -Okegelpara.png kegelpara.pov
kegelpara.jpg: kegelpara.png Makefile
- convert -extract 1080x1000+0+40 kegelpara.png \
+ convert -extract 1080x1040+0+0 kegelpara.png \
-density 300 -units PixelsPerInch kegelpara.jpg
kegelpara.pdf: kegelpara.tex kegelpara.jpg
diff --git a/buch/chapters/110-elliptisch/images/jacobiplots.pdf b/buch/chapters/110-elliptisch/images/jacobiplots.pdf
index 4c74a5c..c11affc 100644
--- a/buch/chapters/110-elliptisch/images/jacobiplots.pdf
+++ b/buch/chapters/110-elliptisch/images/jacobiplots.pdf
Binary files differ
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.pdf b/buch/chapters/110-elliptisch/images/kegelpara.pdf
index 6683709..2f76593 100644
--- a/buch/chapters/110-elliptisch/images/kegelpara.pdf
+++ b/buch/chapters/110-elliptisch/images/kegelpara.pdf
Binary files differ
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.pov b/buch/chapters/110-elliptisch/images/kegelpara.pov
index 5d85eed..13b66cc 100644
--- a/buch/chapters/110-elliptisch/images/kegelpara.pov
+++ b/buch/chapters/110-elliptisch/images/kegelpara.pov
@@ -67,11 +67,11 @@ union {
}
#end
-arrow(<-2,0,0>,<2,0,0>,0.02,White)
-arrow(<0,-2,0>,<0,2,0>,0.02,White)
-arrow(<0,0,-2>,<0,0,2>,0.02,White)
+arrow(<-2.6,0,0>,<2.5,0,0>,0.02,White)
+arrow(<0,-2,0>,<0,2.3,0>,0.02,White)
+arrow(<0,0,-3.2>,<0,0,3.7>,0.02,White)
-#declare epsilon = 0.001;
+#declare epsilon = 0.0001;
#declare l = 1.5;
#macro Kegel(farbe)
@@ -94,6 +94,54 @@ union {
}
#end
+#macro Kegelpunkt(xx, phi)
+ < xx, xx * sin(phi), xx * cos(phi) >
+#end
+
+#macro Kegelgitter(farbe, r)
+union {
+ #declare s = 0;
+ #declare smax = 2 * pi;
+ #declare sstep = pi / 6;
+ #while (s < smax - sstep/2)
+ cylinder { Kegelpunkt(l, s), Kegelpunkt(-l, s), r }
+ #declare s = s + sstep;
+ #end
+ #declare phimax = 2 * pi;
+ #declare phisteps = 100;
+ #declare phistep = phimax / phisteps;
+ #declare xxstep = 0.5;
+ #declare xxmax = 2;
+ #declare xx = xxstep;
+ #while (xx < xxmax - xxstep/2)
+ #declare phi = 0;
+ #while (phi < phimax - phistep/2)
+ cylinder {
+ Kegelpunkt(xx, phi),
+ Kegelpunkt(xx, phi + phistep),
+ r
+ }
+ sphere { Kegelpunkt(xx, phi), r }
+ cylinder {
+ Kegelpunkt(-xx, phi),
+ Kegelpunkt(-xx, phi + phistep),
+ r
+ }
+ sphere { Kegelpunkt(-xx, phi), r }
+ #declare phi = phi + phistep;
+ #end
+ #declare xx = xx + xxstep;
+ #end
+ pigment {
+ color farbe
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
#macro F(w, r)
<r * cos(w), r * r/sqrt(2), r * sin(w) >
#end
@@ -139,6 +187,50 @@ mesh {
}
#end
+#macro Paraboloidgitter(farbe, gr)
+union {
+ #declare phi = 0;
+ #declare phimax = 2 * pi;
+ #declare phistep = pi / 6;
+
+ #declare rmax = 1.5;
+ #declare rsteps = 100;
+ #declare rstep = rmax / rsteps;
+
+ #while (phi < phimax - phistep/2)
+ #declare r = rstep;
+ #while (r < rmax - rstep/2)
+ cylinder { F(phi, r), F(phi, r + rstep), gr }
+ sphere { F(phi, r), gr }
+ #declare r = r + rstep;
+ #end
+ #declare phi = phi + phistep;
+ #end
+
+ #declare rstep = 0.2;
+ #declare r = rstep;
+
+ #declare phisteps = 100;
+ #declare phistep = phimax / phisteps;
+ #while (r < rmax)
+ #declare phi = 0;
+ #while (phi < phimax - phistep/2)
+ cylinder { F(phi, r), F(phi + phistep, r), gr }
+ sphere { F(phi, r), gr }
+ #declare phi = phi + phistep;
+ #end
+ #declare r = r + rstep;
+ #end
+ pigment {
+ color farbe
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
#declare a = sqrt(2);
#macro G(phi,sg)
< a*sg*sqrt(cos(2*phi))*cos(phi), a*cos(2*phi), a*sqrt(cos(2*phi))*sin(phi)>
@@ -215,11 +307,23 @@ union {
}
#end
-#declare kegelfarbe = rgbt<0.2,0.6,0.2,0.2>;
-#declare paraboloidfarbe = rgbt<0.2,0.6,1.0,0.2>;
+#declare kegelfarbe = rgbf<0.2,0.6,0.2,0.2>;
+#declare kegelgitterfarbe = rgb<0.2,0.8,0.2>;
+#declare paraboloidfarbe = rgbf<0.2,0.6,1.0,0.2>;
+#declare paraboloidgitterfarbe = rgb<0.4,1,1>;
+
+//intersection {
+// union {
+ Paraboloid(paraboloidfarbe)
+ Paraboloidgitter(paraboloidgitterfarbe, 0.004)
+
+ Kegel(kegelfarbe)
+ Kegelgitter(kegelgitterfarbe, 0.004)
+// }
+// plane { <0, 0, -1>, 0.6 }
+//}
+
-Paraboloid(paraboloidfarbe)
-Kegel(kegelfarbe)
-Lemniskate3D(0.02, Blue)
+Lemniskate3D(0.02, rgb<0.8,0.0,0.8>)
Lemniskate(0.02, Red)
Projektion(0.01, Yellow)
diff --git a/buch/chapters/110-elliptisch/images/kegelpara.tex b/buch/chapters/110-elliptisch/images/kegelpara.tex
index 5a724ee..8fcefbf 100644
--- a/buch/chapters/110-elliptisch/images/kegelpara.tex
+++ b/buch/chapters/110-elliptisch/images/kegelpara.tex
@@ -31,9 +31,9 @@
\fill (0,0) circle[radius=0.05];
}{}
-\node at (3.3,-1.0) {$X$};
-\node at (0.2,3.4) {$Z$};
-\node at (-2.5,-1.4) {$-Y$};
+\node at (4.1,-1.4) {$X$};
+\node at (0.2,3.8) {$Z$};
+\node at (4.0,1.8) {$Y$};
\end{tikzpicture}
diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.pdf b/buch/chapters/110-elliptisch/images/torusschnitt.pdf
index 2f8c204..11bd353 100644
--- a/buch/chapters/110-elliptisch/images/torusschnitt.pdf
+++ b/buch/chapters/110-elliptisch/images/torusschnitt.pdf
Binary files differ
diff --git a/buch/chapters/110-elliptisch/images/torusschnitt.pov b/buch/chapters/110-elliptisch/images/torusschnitt.pov
index 94190be..43d50c6 100644
--- a/buch/chapters/110-elliptisch/images/torusschnitt.pov
+++ b/buch/chapters/110-elliptisch/images/torusschnitt.pov
@@ -123,9 +123,34 @@ union {
}
#end
-#macro Ebene(farbe)
-box {
- <-1.8, 0, -1.4>, <1.8, 0.001, 1.4>
+#macro Ebene(l, b, farbe)
+mesh {
+ triangle { <-l, 0, -b>, < l, 0, -b>, < l, 0, b> }
+ triangle { <-l, 0, -b>, < l, 0, b>, <-l, 0, b> }
+ pigment {
+ color farbe
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
+#macro Ebenengitter(l, b, s, r, farbe)
+union {
+ #declare lmax = floor(l / s);
+ #declare ll = -lmax;
+ #while (ll <= lmax)
+ cylinder { <ll * s, 0, -b>, <ll * s, 0, b>, r }
+ #declare ll = ll + 1;
+ #end
+ #declare bmax = floor(b / s);
+ #declare bb = -bmax;
+ #while (bb <= bmax)
+ cylinder { <-l, 0, bb * s>, <l, 0, bb * s>, r }
+ #declare bb = bb + 1;
+ #end
pigment {
color farbe
}
@@ -141,6 +166,59 @@ box {
b * < (2 + cos(theta)) * cos(phi), (2 + cos(theta)) * sin(phi) + 1, sin(theta) >
#end
+#macro breitenkreis(theta, r)
+ #declare phi = 0;
+ #declare phimax = 2 * pi;
+ #declare phisteps = 200;
+ #declare phistep = phimax / phisteps;
+ #while (phi < phimax - phistep/2)
+ cylinder { T(phi, theta), T(phi + phistep, theta), r }
+ sphere { T(phi, theta), r }
+ #declare phi = phi + phistep;
+ #end
+#end
+
+#macro laengenkreis(phi, r)
+ #declare theta = 0;
+ #declare thetamax = 2 * pi;
+ #declare thetasteps = 200;
+ #declare thetastep = thetamax / thetasteps;
+ #while (theta < thetamax - thetastep/2)
+ cylinder { T(phi, theta), T(phi, theta + thetastep), r }
+ sphere { T(phi, theta), r }
+ #declare theta = theta + thetastep;
+ #end
+#end
+
+#macro Torusgitter(farbe, r)
+union {
+ #declare phi = 0;
+ #declare phimax = 2 * pi;
+ #declare phistep = pi / 6;
+ #while (phi < phimax - phistep/2)
+ laengenkreis(phi, r)
+ #declare phi = phi + phistep;
+ #end
+ #declare thetamax = pi;
+ #declare thetastep = pi / 6;
+ #declare theta = thetastep;
+ #while (theta < thetamax - thetastep/2)
+ breitenkreis(theta, r)
+ breitenkreis(thetamax + theta, r)
+ #declare theta = theta + thetastep;
+ #end
+ breitenkreis(0, 1.5 * r)
+ breitenkreis(pi, 1.5 * r)
+ pigment {
+ color farbe
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
#macro Torus(farbe)
mesh {
#declare phi = 0;
@@ -181,5 +259,7 @@ mesh {
#declare ebenenfarbe = rgbt<0.2,0.6,1.0,0.2>;
Lemniskate(0.02, Red)
-Ebene(ebenenfarbe)
+Ebene(1.8, 1.4, ebenenfarbe)
+Ebenengitter(1.8, 1.4, 0.5, 0.005, rgb<0.4,1,1>)
Torus(torusfarbe)
+Torusgitter(Yellow, 0.005)
diff --git a/buch/chapters/110-elliptisch/lemniskate.tex b/buch/chapters/110-elliptisch/lemniskate.tex
index f81f0e2..fceaadf 100644
--- a/buch/chapters/110-elliptisch/lemniskate.tex
+++ b/buch/chapters/110-elliptisch/lemniskate.tex
@@ -81,7 +81,7 @@ Blatt der Lemniskate.
\begin{figure}
\center
\includegraphics{chapters/110-elliptisch/images/kegelpara.pdf}
-\caption{Leminiskate (rot) als Projektion (gelb) der Schnittkurve (blau)
+\caption{Leminiskate (rot) als Projektion (gelb) der Schnittkurve (pink)
eines geraden
Kreiskegels (grün) mit einem Rotationsparaboloid (hellblau).
\label{buch:elliptisch:lemniskate:kegelpara}}
@@ -126,7 +126,7 @@ kann mit der Parametrisierung
\begin{pmatrix}
(2+\cos s) \cos t \\
\sin s \\
-(2+\cos s) \sin t
+(2+\cos s) \sin t + 1
\end{pmatrix}
\]
beschrieben werden.
@@ -134,9 +134,9 @@ Die Gleichung $z=1$ beschreibt eine
achsparallele Ebene, die den inneren Äquator berührt.
Die Schnittkurve erfüllt daher
\[
-(2+\cos s)\sin t = 1,
+(2+\cos s)\sin t + 1 = 0,
\]
-was wir auch als $2 +\cos s = 1/\sin t$ schreiben können.
+was wir auch als $2 +\cos s = -1/\sin t$ schreiben können.
Wir müssen nachprüfen dass die Koordinaten
$X=(2+\cos s)\cos t$ und $Y=\sin s$ die Gleichung einer Lemniskate
erfüllen.
@@ -147,9 +147,9 @@ X
=
(2+\cos s) \cos t
=
-\frac{1}{\sin t}\cos t
+-\frac{1}{\sin t}\cos t
=
-\frac{\cos t}{\sin t}
+-\frac{\cos t}{\sin t}
\qquad\Rightarrow\qquad
X^2
=