"Luc H." <nomail@nomail> wrote:
> Hello
>
Luc,

This must be a solved problem, since it amounts to forming a tetrahedron given
the edge lengths, however...

What follows puts the initial triangle onto the x-y plane to make the geometry
easier. If it is something like what you want, I'll put it into vector form for
a general triangle orientation,

Jim

global_settings { ambient_light 4 }
//
#include "colors.inc"
//
#macro draw_axes (aT,aRad,xMin,xMax,yMin,yMax,zMin,zMax)
  object{
    union{
    cylinder{ <xMin,0,0>, <xMax,0,0>, aRad }
    cone    { <xMax,0,0>, 3*aRad, <xMax+8*aRad,0,0>, 0 }
    cylinder{ <0,yMin,0>, <0,yMax,0>, aRad }
    cone    { <0,yMax,0>, 3*aRad, <0,yMax+8*aRad,0>, 0 }
    cylinder{ <0,0,zMin>, <0,0,zMax>, aRad }
    cone    { <0,0,zMax>, 3*aRad, <0,0,zMax+8*aRad>, 0 }
    }
    texture{ aT }
  }
#end
//
//Draw the axes
//
#declare xMin = -5;
#declare xMax = 5;
#declare yMin = -5;
#declare yMax = 5;
#declare zMin = -5;
#declare zMax = 5;
#declare aT   = texture{ pigment { Black } }
#declare aRad = 0.05;
//
draw_axes (aT,aRad,xMin,xMax,yMin,yMax,zMin,zMax)
//
#macro tetrahedron(cRad,srad,p1,p2,p3,p4,d1,d2,d3)
//
union{
// Find the side lengths of the initial triangle
  #local d12 = vlength(p2-p1);
  #local d23 = vlength(p3-p2);
  #local d31 = vlength(p1-p3);
// Set the trial triangle on the x-y plane
  #local pp1 = <0,0,0>;
  #local pp2 = <d12,0,0>;
  #local x3  = (d12*d12+d31*d31-d23*d23)/(2*d12);
  #local y3  = sqrt(d31*d31-x3*x3);
  #local pp3 = <x3,y3,0>;
// Trial triangle vertices
  sphere{pp1 srad texture{pigment{Red}}}
  sphere{pp2 srad texture{pigment{Red}}}
  sphere{pp3 srad texture{pigment{Red}}}
// Joining trial triangle vertices
  cylinder{pp1 pp2 cRad texture{pigment{Blue}}}
  cylinder{pp2 pp3 cRad texture{pigment{Blue}}}
  cylinder{pp3 pp1 cRad texture{pigment{Blue}}}
// Trial triangle
  triangle{ pp1 pp2 pp3 texture{hT} }
// Calculating fourth vertex
// Start off working on the p1-p2-p4 triangle placed on the x-y plane
  #local       d     = (d12*d12+d1*d1-d2*d2)/(2*d12);
  #if(d>0)
     sphere{<0,0,-5> 0.5 pigment{Green}}
    #local     h     = d1*d1-d*d;
    #if(h>0)
      sphere{<0,0,-4> 0.5 pigment{Green}}
      #local   h     = sqrt(h);
      #local   costh = (d3*d3 - (x3-d)*(x3-d) - y3*y3 - h*h)/(2*y3*h);
      #local   sinth = 1 - costh*costh;
      #if(sinth > 0)
        sphere{<0,0,-3> 0.5 pigment{Green}}
        #local sinth = sqrt(sinth);
        #local pp4   = <d,-h*costh,h*sinth>;
      #else
        sphere{<0,0,-3> 0.5 pigment{Red}}
        #local pp4   = <0,0,0>;
      #end
    #else
      sphere{<0,0,-4> 0.5 pigment{Red}}
      #local   pp4   = <0,0,0>;
    #end
  #else
    sphere{<0,0,-5> 0.5 pigment{Red}}
    #local     pp4   = <0,0,0>;
  #end
//
  #if(abs(vlength(pp1-pp4)-d1)<(d1/10000000))
    sphere{  pp4     srad texture{pigment{Red}}}
    cylinder{pp1 pp4 cRad texture{pigment{Blue}}}
    cylinder{pp2 pp4 cRad texture{pigment{Blue}}}
    cylinder{pp3 pp4 cRad texture{pigment{Blue}}}
    triangle{pp1 pp2 pp4  texture{hT}}
    triangle{pp2 pp3 pp4  texture{hT}}
    triangle{pp3 pp1 pp4  texture{hT}}
  #end
// End of main union
     }
//
#end
//
// Joining line and vertex radii
//
#declare cRad = 0.05;
#declare sRad = 0.08;
//
// Joining line, convex hull and spline textures.
//
#declare cT   = texture{ pigment {  rgb<0,0,1> } }     //Blue
#declare hT   = texture{ pigment { rgbf<1,1,0,0.9> } } //Transparent yellow
#declare sT   = texture{ pigment {  rgb<0,1,0> } }     //Green
//
// Vertex definitions
//
#declare p1 = <0, 0  ,0>;
#declare p2 = <4, 0  ,0>;
#declare p3 = <2,3.46,0>;
#declare p4 = <0, 0  ,0>;
//
// Side lengths
//
#declare d1 = 4;
#declare d2 = 4;
#declare d3 = 4;
//
tetrahedron(cRad,sRad,p1,p2,p3,p4,d1,d2,d3)
//
background {color White}
light_source { <20,20,20> White shadowless}
camera{orthographic location <20,6,4> look_at <0,0,0> }

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