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diff --git a/buch/chapters/090-pde/membran/membran.pov b/buch/chapters/090-pde/membran/membran.pov
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+//
+// membran.pov
+//
+// (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+#include "math.inc"
+#include "membran.inc"
+
+global_settings {
+        assumed_gamma 1
+}
+
+#declare imagescale = 0.037;
+
+camera {
+	location <-33, 20, 50>
+	look_at <0, -0.005, 0>
+	right x * imagescale
+	up (9/32) * y * imagescale
+}
+
+background { color rgbt <0,0,0,1> }
+
+light_source {
+	<10, 15, 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(<-1.1,0,0>, <1.1,0,0>, 0.01, White)
+//arrow(<0,-1.1,0>, <0,1.1,0>, 0.01, White)
+//arrow(<0,0,-1.1>, <0,0,1.1>, 0.01, White)
+
+mesh {
+	flaeche()
+	pigment {
+		color rgb<0.8,0.8,0.8> * 1.2
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+#declare randr = 0.01;
+
+#macro zylinder(k,n,r)
+	cylinder {
+		< cos((2*k+1)*pi/(2*n)), 0, sin((2*k+1)*pi/(2*n)) >,
+		< -cos((2*k+1)*pi/(2*n)), 0, -sin((2*k+1)*pi/(2*n)) >,
+		r
+	}
+#end
+
+#macro ring(R,r)
+	#declare phisteps = 100;
+	#declare phistep = 2 * pi / 100;
+	#declare phimin = 0;
+	#declare phimax = 2 * pi;
+	#declare phi = phimin;
+	#while (phi < phimax - phistep/2)
+		cylinder {
+			<R*cos(phi), 0, R*sin(phi)>,
+			<R*cos(phi+phistep), 0, R*sin(phi+phistep)>,
+			r
+		}
+		sphere {
+			<R*cos(phi), 0, R*sin(phi)>, r
+		}
+		#declare phi = phi + phistep;
+	#end
+#end
+
+#declare nullstelle = 13.0152;
+
+union {
+	ring(1, randr)
+	pigment {
+		color rgb<1.0,0.8,0.6>
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+union {
+	zylinder(0, 3, 0.5*randr)
+	zylinder(1, 3, 0.5*randr)
+	zylinder(2, 3, 0.5*randr)
+	ring(6.3802/nullstelle, 0.5*randr)
+	ring(9.7610/nullstelle, 0.5*randr)
+	pigment {
+		color rgb<1.0,0.8,0.6>
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+
+#declare r = 0.2*randr;
+union {
+	phigitter()
+	pigment {
+		color rgb<0.2,0.6,1.0>
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}
+union {
+	rgitter()
+	pigment {
+		color rgb<1.0,0.4,0.6>
+	}
+	finish {
+		specular 0.95
+		metallic
+	}
+}