fix some minor issues

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
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
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
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
 =