I tried to use an isosurface with a moderately complex formula but got very
weird results, so I started with a working example with the object of adding the
complexity bit by bit.

I started with the code below and found I was getting black features on the
surface. There are five versions of the function line, the first renders ok but
the rest give increasingly worse results.

#version 3.7;
global_settings { assumed_gamma 1.0 }
#default{ finish{ ambient 0.1 diffuse 0.9 }}

#include "colors.inc"
#include "functions.inc"

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

#declare CamLoc = < 0, 0,-10>;

camera {
 location CamLoc
        right     x*image_width/image_height
 look_at < 0, 0, 0>
 angle 42
}

light_source {CamLoc color White*0.35}
light_source {<-50, 150,-75> color White}

background { color rgb<1,1,1>*0.35 }


/*
box {<-4,-3, 1>, < 4, 3, 2>
 texture {
  pigment {checker color rgb < 0.05, 0, 0.35>, color White}
  finish {ambient 0 diffuse 1}
 }
}
*/
//-------------------------------------------
#local Yo = 0.08;
#local Zo = .635;
#local Xa = 0.75;
#local Yb = 1.6;
#local Rr = 2.560;
#local Ko = 0.23;


#declare IsoFinish =
finish {
 ambient 0 diffuse 1
 specular 1 roughness 0.02
 brilliance 2
}

#declare  Shape =
isosurface {
// function {sqrt(x*x + y*y + z*z) - 1}
// function {sqrt(pow((x/Xa),2) + pow((y),2) + z*z) - 1}
// function {sqrt(pow((x/Xa),2) + pow((y/Yb),2) + z*z) - 1}
// function {sqrt(pow((x/Xa),2) + pow((y/Yb),2) + pow((z/Zo),2)) - 1}
  function {sqrt(pow((x/Xa),2) + pow((y/Yb),2) + (z/Zo)*(z/Zo)) - 1}
 threshold 0
 contained_by {box {<-1,-1,-1>, < 1, 1, 1>}}
 texture {
  pigment {color  rgb<0.8,0,0>}
  finish {IsoFinish}
         }
        }
 object {Shape rotate <0, 0, 45> translate <-2, 2, 0>}
 object {Shape rotate <0, 45, 0> translate <-2, 0, 0>}
 object {Shape rotate <0, 45, 45> translate <-2, -2, 0>}

 object {Shape rotate <-45, 0, 0> translate <0, 2, 0>}
 object {Shape rotate <0, 0, 0> translate <0, 0, 0>}
 object {Shape rotate <45, 0, 0> translate <0, -2, 0>}

 object {Shape rotate <45, 45, 45> translate <2, 2, 0>}
 object {Shape rotate <0, -45, 0> translate <2, 0, 0>}
 object {Shape rotate <45, 45, 0> translate <2, -2, 0>}

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