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| View Original: | Polyhedron with no vertex visible from center.png (1280x1024) | |||
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| Keywords: Polyhedron with no vertex visible from center.png This simple non-convex polyhedron is constructed so that none of its 24 vertices marked with black balls can be seen from a region around its center This counterexample shows that placing observers at each vertex is not a solution to the 3-dimensional generalization of the art gallery problem The blue faces are also not visible from the center Self-made using POV-Ray 3 6 for Windows 2007-11-06 Ilmari Karonen File // File 3d-art-gallery pov // Vers 3 6 // Desc A simple polyhedron with none of the vertices visible from the center // Date 2007-11-06 // Auth Ilmari Karonen // parameters controlling the width and depth of hidden gaps declare a 1 - 1/3 ; declare b 1 + 2/3 ; if b < 1/a warning B too small for a vertices will be visible from center end // textures for various surfaces declare T1 texture pigment color rgb<1 0 0 8 0 8> finish ambient 0 3 diffuse 0 7 declare T2 texture pigment color rgb<1 0 1 0 0 8> finish ambient 0 3 diffuse 0 7 declare T3 texture pigment color rgb<0 8 0 9 1 0> //pigment color rgb 0 transmit 1 finish ambient 0 3 diffuse 0 7 declare T4 texture pigment color rgb<0 8 0 8 0 8> finish diffuse 0 3 specular 1 0 // the polyhedron itself mesh // corner triangles triangle < 1 a b> < a b 1> < b 1 a> texture T1 triangle < 1 a -b> < a b -1> < b 1 -a> texture T1 triangle < 1 -a b> < a -b 1> < b -1 a> texture T1 triangle < 1 -a -b> < a -b -1> < b -1 -a> texture T1 triangle <-1 a b> <-a b 1> <-b 1 a> texture T1 triangle <-1 a -b> <-a b -1> <-b 1 -a> texture T1 triangle <-1 -a b> <-a -b 1> <-b -1 a> texture T1 triangle <-1 -a -b> <-a -b -1> <-b -1 -a> texture T1 // visible sides triangle < 1 a b> < 1 -a b> < 1 a -b> texture T2 triangle < 1 -a -b> < 1 -a b> < 1 a -b> texture T2 triangle <-1 a b> <-1 -a b> <-1 a -b> texture T2 triangle <-1 -a -b> <-1 -a b> <-1 a -b> texture T2 triangle < a b 1> <-a b 1> < a -b 1> texture T2 triangle <-a -b 1> <-a b 1> < a -b 1> texture T2 triangle < a b -1> <-a b -1> < a -b -1> texture T2 triangle <-a -b -1> <-a b -1> < a -b -1> texture T2 triangle < b 1 a> < b 1 -a> <-b 1 a> texture T2 triangle <-b 1 -a> < b 1 -a> <-b 1 a> texture T2 triangle < b -1 a> < b -1 -a> <-b -1 a> texture T2 triangle <-b -1 -a> < b -1 -a> <-b -1 a> texture T2 // hidden sides triangle < 1 a b> < b 1 a> < 1 a -b> texture T3 triangle < b 1 -a> < b 1 a> < 1 a -b> texture T3 triangle < 1 -a b> < b -1 a> < 1 -a -b> texture T3 triangle < b -1 -a> < b -1 a> < 1 -a -b> texture T3 triangle <-1 a b> <-b 1 a> <-1 a -b> texture T3 triangle <-b 1 -a> <-b 1 a> <-1 a -b> texture T3 triangle <-1 -a b> <-b -1 a> <-1 -a -b> texture T3 triangle <-b -1 -a> <-b -1 a> <-1 -a -b> texture T3 triangle < a b 1> < 1 a b> < a -b 1> texture T3 triangle < 1 -a b> < 1 a b> < a -b 1> texture T3 triangle <-a b 1> <-1 a b> <-a -b 1> texture T3 triangle <-1 -a b> <-1 a b> <-a -b 1> texture T3 triangle < a b -1> < 1 a -b> < a -b -1> texture T3 triangle < 1 -a -b> < 1 a -b> < a -b -1> texture T3 triangle <-a b -1> <-1 a -b> <-a -b -1> texture T3 triangle <-1 -a -b> <-1 a -b> <-a -b -1> texture T3 triangle < b 1 a> < a b 1> <-b 1 a> texture T3 triangle <-a b 1> < a b 1> <-b 1 a> texture T3 triangle < b 1 -a> < a b -1> <-b 1 -a> texture T3 triangle <-a b -1> < a b -1> <-b 1 -a> texture T3 triangle < b -1 a> < a -b 1> <-b -1 a> texture T3 triangle <-a -b 1> < a -b 1> <-b -1 a> texture T3 triangle < b -1 -a> < a -b -1> <-b -1 -a> texture T3 triangle <-a -b -1> < a -b -1> <-b -1 -a> texture T3 // mark vertices with black spheres sphere < 1 a b> 0 03 texture T4 sphere < a b 1> 0 03 texture T4 sphere < b 1 a> 0 03 texture T4 sphere < 1 a -b> 0 03 texture T4 sphere < a b -1> 0 03 texture T4 sphere < b 1 -a> 0 03 texture T4 sphere < 1 -a b> 0 03 texture T4 sphere < a -b 1> 0 03 texture T4 sphere < b -1 a> 0 03 texture T4 sphere < 1 -a -b> 0 03 texture T4 sphere < a -b -1> 0 03 texture T4 sphere < b -1 -a> 0 03 texture T4 sphere <-1 a b> 0 03 texture T4 sphere <-a b 1> 0 03 texture T4 sphere <-b 1 a> 0 03 texture T4 sphere <-1 a -b> 0 03 texture T4 sphere <-a b -1> 0 03 texture T4 sphere <-b 1 -a> 0 03 texture T4 sphere <-1 -a b> 0 03 texture T4 sphere <-a -b 1> 0 03 texture T4 sphere <-b -1 a> 0 03 texture T4 sphere <-1 -a -b> 0 03 texture T4 sphere <-a -b -1> 0 03 texture T4 sphere <-b -1 -a> 0 03 texture T4 sky_sphere pigment color rgb 1 // outside lights light_source <50 80 0> color rgb 0 2 area_light 5 x 5 y 9 9 adaptive 0 jitter circular orient parallel point_at <0 0 0> rotate 15 y light_source <50 -10 0> color rgb 0 1 area_light 5 x 5 y 9 9 adaptive 0 jitter circular orient parallel point_at <0 0 0> rotate 45 y light_source <50 20 0> color rgb 0 3 area_light 5 x 5 y 9 9 adaptive 1 jitter circular orient parallel point_at <0 0 0> rotate 75 y light_source <50 80 0> color rgb 0 2 area_light 5 x 5 y 9 9 adaptive 0 jitter circular orient parallel point_at <0 0 0> rotate 135 y light_source <50 -10 0> color rgb 0 1 area_light 5 x 5 y 9 9 adaptive 0 jitter circular orient parallel point_at <0 0 0> rotate 165 y light_source <50 20 0> color rgb 0 3 area_light 5 x 5 y 9 9 adaptive 1 jitter circular orient parallel point_at <0 0 0> rotate 195 y light_source <50 80 0> color rgb 0 2 area_light 5 x 5 y 9 9 adaptive 0 jitter circular orient parallel point_at <0 0 0> rotate 255 y light_source <50 -10 0> color rgb 0 1 area_light 5 x 5 y 9 9 adaptive 0 jitter circular orient parallel point_at <0 0 0> rotate 285 y light_source <50 20 0> color rgb 0 3 area_light 5 x 5 y 9 9 adaptive 1 jitter circular orient parallel point_at <0 0 0> rotate 315 y // inside lights light_source <0 0 0> color rgb 0 5 area_light 0 6 x 0 6 z 9 9 adaptive 1 jitter circular orient light_source <0 0 0> color rgb 0 5 area_light 0 4 x 0 4 z 9 9 adaptive 1 jitter circular orient // cameras camera location <8 0 5 0 -5 0> angle 30 look_at <0 0 0 0 0 0> right x image_width/image_height // outside //camera panoramic location <0 0 0> look_at <1 0 1> angle 180 // inside 180 degree panorama //camera location <0 0 0> look_at <1 1 1> angle 70 // inside looking at corner </pre> Polyhedra Created with Persistence of Vision | ||||
