Bilde:Euclidian and non euclidian geometry.png
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Euclidian_and_non_euclidian_geometry.png (790 × 310 punkter, filstørrelse: 47 KB, MIME-type: image/png)
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Shows three "blocks" with varying surfaces, whereon a point and som lines are "drawn" to demonstrate
- euclidian
- elliptical and
- hyperbolic geometry
To be exact, the hyperbolic surface was approximated using a chunk of the "inside" of a torus - this may not be perfectly accurate, but apparently it "looks right" in this illustration. :-)
Rendered using POV-Ray (see http://www.povray.org) and the scene description "code" below, then cropped and "numbered" using a graphics software package.
/* ==================================================================== Surfaces demonstrating euclidian, elliptical and hyperbolic geometry -------------------------------------------------------------------- Created by Søren Peo Pedersen - see my user page at http://da.wikipedia.org/wiki/Bruger:Peo ==================================================================== */ // Macro for texture with background pattern, lines, and point "marker" #macro SurfaceDrawing ( Point, // Position vector: Location of white point RefLineMove, // Transformation: Where to place the yellow reference line Lines // Object that renders the blue line(s) ) #local BasePattern=pigment { // The underlying, drak grey checkerboard pattern checker color rgb 0.4 color rgb 0.6 scale <1,1000,1> } #local SP1 = pigment { // Checkerboard pattern with yellow "reference line" added object {box {<-.07,-10,-10>,<.07,10,10> transform{RefLineMove}} pigment {BasePattern} pigment {color rgb <1,.8,.3>} } } #local SP2 = pigment { // Checkerborad + yellow line with user-defined (blue) lines added object {Lines pigment {SP1} pigment {color rgb <.6,.8,1>} } } #local SP3 = pigment { // Checkerboard + lines with whit point "marker" added object { sphere { Point, .2 } pigment {SP2} pigment {color rgb 1} } } pigment { // Definition of pigment, confined to a box slightly smaller than the "blocks" boxed // to make the lines and point only appear on the top of the blocks pigment_map { [0.0000 BasePattern scale .5 translate <0,-1,0>] [0.0001 BasePattern scale .5 translate <0,-1,0>] [0.0001 SP3 scale .5 translate <0,-1,0>] [1.0000 SP3 scale .5 translate <0,-1,0>] } scale <2,2,2> translate <0,2,0> } finish {ambient .7 diffuse .3} #end // End of macro plane {<0,1,0>,0 // White "tabletop" to receive shadows of the blocks (helps visualisation) pigment {color rgb 1} finish {ambient .8 diffuse .5} } box {<-2,0,-2>,<2,2,2> // Demonstrating euclidean geometry texture { SurfaceDrawing(<-.3,2,0>, transform {rotate <0,35,0> translate <.3,0,0>}, box {<-.07,-10,-10>,<.07,10,10> rotate <0,35,0> translate <-.3,0,0>} ) } rotate <0,-20,0> translate <-6,0,0> } intersection { // Demonstrating elliptical, or "Riemann", geometry box {<-2,0,-2>,<2,5,2>} sphere {<0,0,0>,2.8285} texture { SurfaceDrawing(<.65,2.45,1.25>, transform {rotate <0,5,0> rotate <-45,35,0>}, box {<-.07,-10,-10>,<.07,10,10> rotate <0,-4,0> rotate <-45,35,0>} ) } rotate <0,-20,0> translate <0,0,0> } intersection { // Demonstrating hyperbolic geometry box {<-2,0,-2>,<2,4,2>} torus {7.03,5 rotate <90,90,0> translate <0,4,0>} texture { SurfaceDrawing(<0,2.3,1>, transform {rotate <0,90,0> rotate <-40,0,0> translate <0,4,0>}, merge { box {<-.07,-10,-10>,<.07,10,10> rotate <0,85,0> rotate <-30,0,0> translate <0,4,0> } box {<-.07,-10,-10>,<.07,10,10> rotate <0,95,0> rotate <-30,0,0> translate <0,4,0> } } ) } rotate <0,-20,0> translate <6,0,0> } camera { // Viewpoint location <0,2100,-800> look_at <.5,0,0> angle .47 } light_source {<-50,200,-100> color rgb 1.5} // Illumination
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Dato/klokkeslett | Dimensjoner | Bruker | Kommentar | |
---|---|---|---|---|
nåværende | 8. apr 2005 kl. 01:21 | 790×310 (47 KB) | Peo | (Shows three "blocks" with varying surfaces, whereon a point and som lines are "drawn" to demonstrate #euclidian #elliptical and #hyperbolic geometry Rendered using POV-Ray (see http://www.povray.org) and the scene description "code" below, then cropped an) |
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