update to current state of book

6 files changed, 385 insertions, 385 deletions

diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile
index cc67c8a..9de1c34 100644
--- a/vorlesungen/slides/7/images/Makefile
+++ b/vorlesungen/slides/7/images/Makefile
@@ -1,19 +1,19 @@
-#
-# Makefile -- Illustrationen zu 
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-# 
-all:	rodriguez.jpg 
-
-rodriguez.png:	rodriguez.pov
-	povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
-
-rodriguez.jpg:	rodriguez.png 
-	convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg
-
-commutator:	commutator.ini commutator.pov common.inc
-	povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini
-jpg:
-	for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done
-
-
+#

+# Makefile -- Illustrationen zu 

+#

+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+# 

+all:	rodriguez.jpg 

+

+rodriguez.png:	rodriguez.pov

+	povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov

+

+rodriguez.jpg:	rodriguez.png 

+	convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg

+

+commutator:	commutator.ini commutator.pov common.inc

+	povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini

+jpg:

+	for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done

+

+

diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc
index 0e27c9a..b028956 100644
--- a/vorlesungen/slides/7/images/common.inc
+++ b/vorlesungen/slides/7/images/common.inc
@@ -1,70 +1,70 @@
-//
-// common.inc
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-
-global_settings {
-	assumed_gamma 1
-}
-
-#declare imagescale = 0.025;
-#declare O = <0, 0, 0>;
-#declare at = 0.015;
-
-camera {
-	location <18, 15, -50>
-	look_at <0.0, 0.5, 0>
-	right 16/9 * x * imagescale
-	up y * imagescale
-}
-
-light_source {
-	<-40, 30, -50> color White
-	area_light <1,0,0> <0,0,1>, 10, 10
-	adaptive 1
-	jitter
-}
-
-sky_sphere {
-	pigment {
-		color rgb<1,1,1>
-	}
-}
-
-#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 l = 1.2;
-
-arrow(< -l,  0,  0 >, < l, 0, 0 >, at, White)
-arrow(<  0,  0, -l >, < 0, 0, l >, at, White)
-arrow(<  0, -l,  0 >, < 0, l, 0 >, at, White)
-
+//

+// common.inc

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#version 3.7;

+#include "colors.inc"

+

+global_settings {

+	assumed_gamma 1

+}

+

+#declare imagescale = 0.025;

+#declare O = <0, 0, 0>;

+#declare at = 0.015;

+

+camera {

+	location <18, 15, -50>

+	look_at <0.0, 0.5, 0>

+	right 16/9 * x * imagescale

+	up y * imagescale

+}

+

+light_source {

+	<-40, 30, -50> color White

+	area_light <1,0,0> <0,0,1>, 10, 10

+	adaptive 1

+	jitter

+}

+

+sky_sphere {

+	pigment {

+		color rgb<1,1,1>

+	}

+}

+

+#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 l = 1.2;

+

+arrow(< -l,  0,  0 >, < l, 0, 0 >, at, White)

+arrow(<  0,  0, -l >, < 0, 0, l >, at, White)

+arrow(<  0, -l,  0 >, < 0, l, 0 >, at, White)

+

diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini
index 8c2211e..44a5ac5 100644
--- a/vorlesungen/slides/7/images/commutator.ini
+++ b/vorlesungen/slides/7/images/commutator.ini
@@ -1,8 +1,8 @@
-Input_File_Name=commutator.pov
-Initial_Frame=1
-Final_Frame=60
-Initial_Clock=1
-Final_Clock=60
-Cyclic_Animation=off
-Pause_when_Done=off
-
+Input_File_Name=commutator.pov

+Initial_Frame=1

+Final_Frame=60

+Initial_Clock=1

+Final_Clock=60

+Cyclic_Animation=off

+Pause_when_Done=off

+

diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m
index 5a448db..3f5ea17 100644
--- a/vorlesungen/slides/7/images/commutator.m
+++ b/vorlesungen/slides/7/images/commutator.m
@@ -1,111 +1,111 @@
-#
-# commutator.m
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-
-X = [
-	0, 0,  0;
-	0, 0, -1;
-	0, 1,  0
-];
-
-Y = [
-	 0, 0, 1;
-	 0, 0, 0;
-	-1, 0, 0
-];
-
-Z = [
-	0, -1, 0;
-	1,  0, 0;
-	0,  0, 0
-];
-
-function retval = Dx(alpha)
-	retval = [
-		1, 0,           0         ;
-		0, cos(alpha), -sin(alpha);
-		0, sin(alpha),  cos(alpha)
-	];
-end
-
-function retval = Dy(beta)
-	retval = [
-		 cos(beta), 0, sin(beta);
-		 0,         1, 0        ;
-		-sin(beta), 0, cos(beta)
-	];
-end
-
-t = 0.9;
-P = Dx(t) * Dy(t)
-Q = Dy(t) * Dx(t)
-P - Q
-(P - Q) * [0;0;1]
-
-function	retval = kurven(filename, t)
-	retval = -1;
-	N = 20;
-	fn = fopen(filename, "w");
-	fprintf(fn, "//\n");
-	fprintf(fn, "// %s\n", filename);
-	fprintf(fn, "//\n");
-	fprintf(fn, "#macro XYkurve()\n");
-	for i = (0:N-1)
-		v1 = Dx(t * i / N) * [0;0;1];
-		v2 = Dx(t * (i+1) / N) * [0;0;1];
-		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
-			v1(1,1), v1(3,1), v1(2,1));
-		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
-			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
-	end
-	for i = (0:N-1)
-		v1 = Dx(t) * Dy(t * i / N) * [0;0;1];
-		v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1];
-		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
-			v1(1,1), v1(3,1), v1(2,1));
-		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
-			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
-	end
-	fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
-		v2(1,1), v2(3,1), v2(2,1));
-	fprintf(fn, "#end\n");
-	fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n",
-		v2(1,1), v2(3,1), v2(2,1));
-	fprintf(fn, "#macro YXkurve()\n");
-	for i = (0:N-1)
-		v1 = Dy(t * i / N) * [0;0;1];
-		v2 = Dy(t * (i+1) / N) * [0;0;1];
-		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
-			v1(1,1), v1(3,1), v1(2,1));
-		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
-			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
-	end
-	for i = (0:N-1)
-		v1 = Dy(t) * Dx(t * i / N) * [0;0;1];
-		v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1];
-		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
-			v1(1,1), v1(3,1), v1(2,1));
-		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
-			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
-	end
-	fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
-		v2(1,1), v2(3,1), v2(2,1));
-	fprintf(fn, "#end\n");
-	fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n",
-		v2(1,1), v2(3,1), v2(2,1));
-
-	fclose(fn);
-	retval = 0;
-end
-
-function	retval = kurve(i)
-	n = pi / 180;
-	filename = sprintf("f/%04d.inc", i);
-	kurven(filename, n * i);
-end
-
-for i = (1:60)
-	kurve(i);
-end
+#

+# commutator.m

+#

+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+#

+

+X = [

+	0, 0,  0;

+	0, 0, -1;

+	0, 1,  0

+];

+

+Y = [

+	 0, 0, 1;

+	 0, 0, 0;

+	-1, 0, 0

+];

+

+Z = [

+	0, -1, 0;

+	1,  0, 0;

+	0,  0, 0

+];

+

+function retval = Dx(alpha)

+	retval = [

+		1, 0,           0         ;

+		0, cos(alpha), -sin(alpha);

+		0, sin(alpha),  cos(alpha)

+	];

+end

+

+function retval = Dy(beta)

+	retval = [

+		 cos(beta), 0, sin(beta);

+		 0,         1, 0        ;

+		-sin(beta), 0, cos(beta)

+	];

+end

+

+t = 0.9;

+P = Dx(t) * Dy(t)

+Q = Dy(t) * Dx(t)

+P - Q

+(P - Q) * [0;0;1]

+

+function	retval = kurven(filename, t)

+	retval = -1;

+	N = 20;

+	fn = fopen(filename, "w");

+	fprintf(fn, "//\n");

+	fprintf(fn, "// %s\n", filename);

+	fprintf(fn, "//\n");

+	fprintf(fn, "#macro XYkurve()\n");

+	for i = (0:N-1)

+		v1 = Dx(t * i / N) * [0;0;1];

+		v2 = Dx(t * (i+1) / N) * [0;0;1];

+		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",

+			v1(1,1), v1(3,1), v1(2,1));

+		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",

+			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));

+	end

+	for i = (0:N-1)

+		v1 = Dx(t) * Dy(t * i / N) * [0;0;1];

+		v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1];

+		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",

+			v1(1,1), v1(3,1), v1(2,1));

+		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",

+			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));

+	end

+	fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",

+		v2(1,1), v2(3,1), v2(2,1));

+	fprintf(fn, "#end\n");

+	fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n",

+		v2(1,1), v2(3,1), v2(2,1));

+	fprintf(fn, "#macro YXkurve()\n");

+	for i = (0:N-1)

+		v1 = Dy(t * i / N) * [0;0;1];

+		v2 = Dy(t * (i+1) / N) * [0;0;1];

+		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",

+			v1(1,1), v1(3,1), v1(2,1));

+		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",

+			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));

+	end

+	for i = (0:N-1)

+		v1 = Dy(t) * Dx(t * i / N) * [0;0;1];

+		v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1];

+		fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",

+			v1(1,1), v1(3,1), v1(2,1));

+		fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",

+			v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));

+	end

+	fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",

+		v2(1,1), v2(3,1), v2(2,1));

+	fprintf(fn, "#end\n");

+	fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n",

+		v2(1,1), v2(3,1), v2(2,1));

+

+	fclose(fn);

+	retval = 0;

+end

+

+function	retval = kurve(i)

+	n = pi / 180;

+	filename = sprintf("f/%04d.inc", i);

+	kurven(filename, n * i);

+end

+

+for i = (1:60)

+	kurve(i);

+end

diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov
index 9ae11b9..8229a06 100644
--- a/vorlesungen/slides/7/images/commutator.pov
+++ b/vorlesungen/slides/7/images/commutator.pov
@@ -1,59 +1,59 @@
-//
-// commutator.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-sphere { O, 0.99
-	pigment {
-		color rgbt<1,1,1,0.5>
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
-#declare filename = concat("f/", str(clock, -4, 0), ".inc");
-
-#include filename
-
-#declare n1 = vcross(<0,1,0>, finalXY);
-#declare n2 = vcross(<0,1,0>, finalYX);
-
-intersection {
-	sphere { O, 1 }
-	plane { -n1, 0 }
-	plane { n2, 0 }
-	pigment {
-		color rgb<0,0.4,0.1>
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
-union {
-        XYkurve()
-        pigment {
-                color Red
-        }
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
-union {
-        YXkurve()
-        pigment {
-                color Blue
-        }
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
+//

+// commutator.pov

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#include "common.inc"

+

+sphere { O, 0.99

+	pigment {

+		color rgbt<1,1,1,0.5>

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

+#declare filename = concat("f/", str(clock, -4, 0), ".inc");

+

+#include filename

+

+#declare n1 = vcross(<0,1,0>, finalXY);

+#declare n2 = vcross(<0,1,0>, finalYX);

+

+intersection {

+	sphere { O, 1 }

+	plane { -n1, 0 }

+	plane { n2, 0 }

+	pigment {

+		color rgb<0,0.4,0.1>

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

+union {

+        XYkurve()

+        pigment {

+                color Red

+        }

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

+union {

+        YXkurve()

+        pigment {

+                color Blue

+        }

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov
index 07aec19..62306f8 100644
--- a/vorlesungen/slides/7/images/rodriguez.pov
+++ b/vorlesungen/slides/7/images/rodriguez.pov
@@ -1,118 +1,118 @@
-//
-// rodriguez.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-
-global_settings {
-	assumed_gamma 1
-}
-
-#declare imagescale = 0.020;
-#declare O = <0, 0, 0>;
-#declare at = 0.015;
-
-camera {
-	location <8, 15, -50>
-	look_at <0.1, 0.475, 0>
-	right 16/9 * x * imagescale
-	up y * imagescale
-}
-
-light_source {
-	<-4, 20, -50> color White
-	area_light <1,0,0> <0,0,1>, 10, 10
-	adaptive 1
-	jitter
-}
-
-sky_sphere {
-	pigment {
-		color rgb<1,1,1>
-	}
-}
-
-#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 K = vnormalize(<0.2,1,0.1>);
-#declare X = vnormalize(<1.1,1,-1.2>);
-#declare O = <0,0,0>;
-
-#declare r = vlength(vcross(K, X)) / vlength(K);
-
-#declare l = 1.0;
-
-arrow(< -l,  0,  0 >, < l, 0, 0 >, at, White)
-arrow(<  0,  0, -l >, < 0, 0, l >, at, White)
-arrow(<  0, -l,  0 >, < 0, l, 0 >, at, White)
-
-arrow(O, X, at, Red)
-arrow(O, K, at, Blue)
-
-#macro punkt(H,phi)
-	((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K)
-#end
-
-cone { vdot(K, X) * K, r, O, 0
-	pigment {
-		color rgbt<0.6,0.6,0.6,0.5>
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
-
-union {
-	#declare phistep = pi / 100;
-	#declare phi = 0;
-	#while (phi < 2 * pi - phistep/2) 
-		sphere { punkt(K, phi), at/2 }
-		cylinder {
-			punkt(X, phi),
-			punkt(X, phi + phistep),
-			at/2
-		}
-		#declare phi = phi + phistep;
-	#end
-	pigment {
-		color Orange
-	}
-	finish {
-		specular 0.9
-		metallic
-	}
-}
-
-arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow)
-#declare Darkgreen = rgb<0,0.5,0>;
-arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen)
+//

+// rodriguez.pov

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#version 3.7;

+#include "colors.inc"

+

+global_settings {

+	assumed_gamma 1

+}

+

+#declare imagescale = 0.020;

+#declare O = <0, 0, 0>;

+#declare at = 0.015;

+

+camera {

+	location <8, 15, -50>

+	look_at <0.1, 0.475, 0>

+	right 16/9 * x * imagescale

+	up y * imagescale

+}

+

+light_source {

+	<-4, 20, -50> color White

+	area_light <1,0,0> <0,0,1>, 10, 10

+	adaptive 1

+	jitter

+}

+

+sky_sphere {

+	pigment {

+		color rgb<1,1,1>

+	}

+}

+

+#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 K = vnormalize(<0.2,1,0.1>);

+#declare X = vnormalize(<1.1,1,-1.2>);

+#declare O = <0,0,0>;

+

+#declare r = vlength(vcross(K, X)) / vlength(K);

+

+#declare l = 1.0;

+

+arrow(< -l,  0,  0 >, < l, 0, 0 >, at, White)

+arrow(<  0,  0, -l >, < 0, 0, l >, at, White)

+arrow(<  0, -l,  0 >, < 0, l, 0 >, at, White)

+

+arrow(O, X, at, Red)

+arrow(O, K, at, Blue)

+

+#macro punkt(H,phi)

+	((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K)

+#end

+

+cone { vdot(K, X) * K, r, O, 0

+	pigment {

+		color rgbt<0.6,0.6,0.6,0.5>

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

+

+union {

+	#declare phistep = pi / 100;

+	#declare phi = 0;

+	#while (phi < 2 * pi - phistep/2) 

+		sphere { punkt(K, phi), at/2 }

+		cylinder {

+			punkt(X, phi),

+			punkt(X, phi + phistep),

+			at/2

+		}

+		#declare phi = phi + phistep;

+	#end

+	pigment {

+		color Orange

+	}

+	finish {

+		specular 0.9

+		metallic

+	}

+}

+

+arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow)

+#declare Darkgreen = rgb<0,0.5,0>;

+arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen)