File:Riemann sphere1.jpg

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Summary

Description	Sphère de Riemann, démonstration de la projection English: rendering of the graph of the Sphere of Rieman
Date
Source	Own work. Merci à Friedrich A. Lohmüller pour sa bibliothèque POV.
Author	Jean-Christophe BENOIST

//------------------------------------------------------------------------
#version 3.6;
global_settings {  assumed_gamma 1 }
//------------------------------------------------------------------------
#include "colors.inc"
#include "textures.inc"
#include "glass.inc"
#include "metals.inc"
#include "golds.inc"
#include "stones.inc"
#include "woods.inc"
#include "shapes.inc"
#include "shapes2.inc"
#include "functions.inc"
#include "math.inc"          
#include "transforms.inc"
//------------------------------------------------------------------------
#declare Camera_0 = camera {                                 // xy-view
                             angle 20
                             location  <8.0 , 8.0 ,-24.0>
                             right     x*image_width/image_height
                             look_at   <1 , 0 , 0>
                           }
 
#declare Camera_1 = camera {                                // diagonal view
                             angle 9
                             location  <15.0 ,15 ,-60.0>
                             right     x*image_width/image_height
                             look_at   <1 , 0 , 0>
                           }
#declare Camera_2 = camera {                               // yz-view
                             angle 20          
                             location  <31.0 , 7.0 ,1.0>
                             right     x*image_width/image_height
                             look_at   <0 , 0 , 0>
                           }
#declare Camera_3 = camera { 
                             angle 20
                             location  < 0.0 , 37.5 ,-0.0001>
                             right     x*image_width/image_height
                             look_at   <0 , 0 , 0>
                           }
 
camera{Camera_0}
 
//------------------------------------------------------------------------
// sun -------------------------------------------------------------------
light_source{<3000,5500,-1000> colour White }
// sky -------------------------------------------------------------------
sky_sphere{ pigment{ gradient <0,1,0>
                     color_map{ [0   colour rgb<1,1,1>         ]//White
                                [0.4 colour rgb<1,1,1>]  //~Navy
                                [0.6 colour rgb<1,1,1>]  //<0.14,0.14,0.56>]//~Navy
                                [1.0 colour rgb<1,1,1>         ]//White
                              }
                     scale 2 }
           } // end of sky_sphere 
//------------------------------------------------------------------------
 
 
 
 
 
 
//------------------------------ the Axes --------------------------------
//------------------------------------------------------------------------
#macro Axis_( AxisLen, Dark_Texture,Light_Texture) 
 union{
    cylinder { <0,-AxisLen-3,0>,<0,AxisLen,0>,0.017
               texture{checker texture{Dark_Texture } 
                               texture{Light_Texture}
                       translate<0.1,0,0.1>}
             }
    cone{<0,AxisLen,0>,0.1,<0,AxisLen+0.3,0>,0
          texture{Dark_Texture}
         }
     } // end of union                   
#end // of macro "Axis()"
//------------------------------------------------------------------------
#macro AxisXYZ( AxisLenX, AxisLenY, AxisLenZ, Tex_Dark, Tex_Light)
//--------------------- drawing of 3 Axes --------------------------------
union{
#if (AxisLenX != 0)
 object { Axis_(AxisLenX, Tex_Dark, Tex_Light)   rotate< 0,0,-90>}// x-Axis
 text   { ttf "arial.ttf",  " ",  0.15,  0  texture{Tex_Dark} 
          scale 0.5 translate <AxisLenX+0.05,0.4,-0.10>}
#end // of #if 
#if (AxisLenY != 0)
 object { Axis_(AxisLenY, Tex_Dark, Tex_Light)   rotate< 0,0,  0>}// y-Axis
 text   { ttf "arial.ttf",  "z",  0.15,  0  texture{Tex_Dark}    
           scale 0.5 translate <-0.55,AxisLenY+0.20,-0.10>}
#end // of #if 
#if (AxisLenZ != 0)
 object { Axis_(AxisLenZ, Tex_Dark, Tex_Light)   rotate<90,0,  0>}// z-Axis
 text   { ttf "arial.ttf",  " ",  0.15,  0  texture{Tex_Dark}
               scale 0.5 translate <-0.55,0.1,AxisLenZ+0.10>}
#end // of #if 
} // end of union
#end// of macro "AxisXYZ( ... )"
//------------------------------------------------------------------------
 
#declare Texture_A_Dark  = texture {
                               pigment{colour rgb<0.5,0.5,0.5>}
                               finish {ambient 0.15 diffuse 0.85 phong 1}
                             }
#declare Texture_A_Light = texture { 
                               pigment{colour rgb<1,1,1>}
                               finish {ambient 0.15 diffuse 0.85 phong 1}
                             }
 
 
 
 
 
 
 
 
 
object{ AxisXYZ( 25.0, 0, 25.0, Texture_A_Dark, Texture_A_Dark)} // <<<<<<<<<<<<<<<<< adapt the axes here !!!
 
 
#declare        TextScale=0.5;
 
//-------------------------------------------------- end of coordinate axes
 
 
// ground -----------------------------------------------------------------
//---------------------------------<<< settings of squered plane dimensions
#declare RasterScale = 2.0/2;
#declare RasterHalfLine  = 0.03/4;  
#declare RasterHalfLineZ = 0.03/2; 
//-------------------------------------------------------------------------
#macro Raster(RScale, HLine) 
       pigment{ gradient x scale RScale
                color_map{[0.000   colour rgbt<1,1,1,1>*0.2]
                          [0+HLine colour rgbt<1,1,1,1>*0.2]
                          [0+HLine colour rgbt<1,1,1,1>]
                          [1-HLine colour rgbt<1,1,1,1>]
                          [1-HLine colour rgbt<1,1,1,1>*0.2]
                          [1.000   colour rgbt<1,1,1,1>*0.2]} }
       finish { ambient 0.15 diffuse 0.85}
 #end// of Raster(RScale, HLine)-macro    
//-------------------------------------------------------------------------
 
 
plane { <0,1,0>, 0    // plane with layered textures
        texture { pigment{colour rgbt<1,1,1,0.7>*1.1}
                  finish {ambient 0.45 diffuse 0.85}}
        texture { Raster(RasterScale,RasterHalfLine ) rotate<0,0,0> }
        texture { Raster(RasterScale,RasterHalfLineZ) rotate<0,90,0>}
        rotate<0,0,0>
        no_shadow
      }
//------------------------------------------------ end of squered plane XZ   
 
 
 
// 
//------------------------------------------------------------------------------------
#include "analytical_g.inc" 
 
 
 
 
 
//--------------------------------------------------------------------------
//---------------------------- objects in scene ----------------------------
//--------------------------------------------------------------------------
#default{ finish {ambient 0.15 diffuse 0.85} } // 
 
 
 
//-----------------------------------------------------------------------------------
#declare Rl  = 0.035; // radius of lines  - Radius der Strecken 
#declare Rp  = 0.06; // radius of points - Radius der Punkte 
 
 
//------------------------------------------------------------------------------------
// ----------------------------------------------------------------------------------- 
 
// ----------------------------------------------------------------------------------- 
// ---------------------  Punti e Vettori  - Punkte und Vektoren -  Points and Vectors
//------------------------------------------------------------------------------------
 
#declare M1 = < 0, 0.0,0.0>;  
#declare R1  = 3;   
 
#declare M2  = < 0.0,0.0, 0.0>;
#declare Rx  = 3.00;
#declare Ry  = 1.00;
#declare Rz  = 2.00;
 
#declare P1   = < 3, 0, 0>;
#declare Pi   = < 0, 0, 3>;
#declare PPinf = < 0, 3, 0>;        
#declare PP0   = < 0, -3, 0>;  
#declare P0   = <0,0,0>;
 
sphere{ P1, Rp  pigment{colour YellowGreen}}
text { ttf "arial.ttf", "1",0.1,0 
       scale TextScale rotate<0,0,0> translate P1+<0.2,0.1,-0.2> pigment{ colour Red } no_shadow}
 
 
sphere{ Pi, Rp  pigment{colour YellowGreen}}
text { ttf "arial.ttf", "i",0.1,0 
       scale TextScale rotate<0,0,0> translate Pi+<-0.1,0.1,0.2> pigment{ colour Red } no_shadow}
 
 
sphere{ PPinf, Rp  pigment{colour YellowGreen}}
text { ttf "Mathematica1.ttf", concat("R(",chr(165),")"),0.1,0 
       scale TextScale rotate<0,0,0> translate PPinf+<-0.1,0.1,0> pigment{ colour Red } no_shadow}             
 
 
sphere{ PP0, Rp  pigment{colour YellowGreen}}
text { ttf "Mathematica1.ttf", "R(0)",-0.1,0 
       scale TextScale rotate<0,0,0> translate PP0+<-0.1,0.1,0> pigment{ colour Red } no_shadow}         
 
sphere{ P0, Rp  pigment{colour YellowGreen}}
text { ttf "Mathematica1.ttf", "0",-0.1,0 
       scale TextScale rotate<0,0,0> translate P0+<-0.1,0.1,0> pigment{ colour Red } no_shadow}           
 
 
//object{ Show_Yxz( M1, Rl/2) pigment{ colour Yellow }}
//object{ Show_Yxz( M2, Rl/2) pigment{ colour Yellow }}
//------------------------------------------------------------------------------------
// --------------------------------------------------------------- Zeichnen ---------- 
// ---------------------------------------------------------------- Drawing ---------- 
 
 
#declare RSphere=
object {
        sphere{ M1, R1                          pigment{ colour Yellow transmit 0.6 } finish {phong 0.3}}
}                               
 
 
#declare PPlan1 = <4.5,0,0>;     
 
object{ Vector (PPinf, PPlan1, Rl)
         pigment{colour Red}}  
 
sphere{ PPlan1, Rp  pigment{colour Green}}          
object{ Show_Yxz( PPlan1, Rl/2)
        pigment{ colour Orange }}
text { ttf "arial.ttf", "A",0.1,0 
       scale TextScale rotate<0,0,0> translate PPlan1+<0.1,0.0,-0.4> pigment{ colour Red } no_shadow}        
 
// point of intersection 
#declare Hit_the_Object =  trace ( RSphere, PPinf, PPlan1 - PPinf );                      
 
sphere{ Hit_the_Object, Rp  pigment{colour Green}}
text { ttf "Mathematica1.ttf", "a=R(A)",0.1,0 
       scale TextScale rotate<0,0,0> translate Hit_the_Object+<0.2,0.0,-0.2> pigment{ colour Red } no_shadow}  
 
 
 
 
 
 
 
#declare PPlan2 = <-1.5,0,0>;     
 
 
 
sphere{ PPlan2, Rp  pigment{colour Green}}          
object{ Show_Yxz( PPlan2, Rl/2)
        pigment{ colour Orange }}
text { ttf "arial.ttf", "B",0.1,0 
       scale TextScale rotate<0,0,0> translate PPlan2+<0.3,0.0,-0.4> pigment{ colour Red } no_shadow}        
 
// point of intersection 
#declare Hit_the_Object =  trace ( RSphere, PPinf, PPlan2 - PPinf );   
 
object{ Vector (PPinf, Hit_the_Object, Rl)
         pigment{colour Red}}                     
 
sphere{ Hit_the_Object, Rp  pigment{colour Green}}
text { ttf "Mathematica1.ttf", "b=R(B)",0.1,0 
       scale TextScale rotate<0,0,0> translate Hit_the_Object+<0.2,0.0,-0.2> pigment{ colour Red } no_shadow}  
 
 
 
 
 
 
object {RSphere}

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The following pages on Schools Wikipedia link to this image (list may be incomplete):

Riemann surface

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it. If the file has been modified from its original state, some details may not fully reflect the modified file.

Orientation	Normal
Horizontal resolution	72 dpi
Vertical resolution	72 dpi
Software used	Adobe Photoshop CS3 Windows
File change date and time	18:09, 4 June 2007
Colour space	sRGB

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