Here's the mathematically correct version:

// Persistence of Vision Ray Tracer Scene Description File
// File: PigFunc.pov
// Vers: 3.5+
// Desc: Pigment function demo
// Date: 2005.02.19
// Auth: PM 2Ring
//
//  -D +A0.01 +AM2 +R4 +W640 +H640
//  -F -A0.4 +AM2 +R1
//

global_settings {assumed_gamma 1}

//---Color Maps--------------------------------------------

//Make a color_map from a0 to a1
#macro CMRange(a0, a1)color_map{[0 a0][1 a1]}#end

//Make a color_map from Black to a
#macro CMFull(a)CMRange(rgb 0, a)#end

//---Stereographically projected
pigment!-----------------------------------------

#declare R = 3;   //Radius of projection
#declare R2 = R*R;
#declare f_proj = function(x,z){1/sqrt(1 - (x*x+z*z)/R2)}

#declare TestPig = pigment{hexagon rgb<1 .9 .5>, rgb <.85 .5 .6>, rgb <0 0
...8> scale .5}

#declare f_Pig = function{pigment{TestPig}}
#declare Brite = 3;
#declare PigRGB =
pigment{
  average
  pigment_map {
    [1 function{f_Pig(x*f_proj(x,z), y, z*f_proj(x,z)).red} CMFull(red
Brite)]
    [1 function{f_Pig(x*f_proj(x,z), y, z*f_proj(x,z)).green} CMFull(green
Brite)]
    [1 function{f_Pig(x*f_proj(x,z), y, z*f_proj(x,z)).blue} CMFull(blue
Brite)]
  }
}

//---Scene--------------------------------

cylinder{
  -.1*y, .1*y, R

  pigment{
    PigRGB
    //TestPig
  }
  finish{ambient 1 diffuse 0}
}

camera {
  orthographic
  location y * R * 4
  look_at  0
  right x*image_width/image_height up y
  direction z

  angle 30
}

background{rgb .5}

//-------------------------------------------------------

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