affine_map_tetra.txt   
map a tetrahedron with vertices 0,0,0  1,0,0  0,1,0  0,0,1
onto a tetrahedron with vertices x1,y1,z1  x2,y2,z2  x3,y3,z3  x4,y4,z4

map a,b,c  to  x,y,z

                   x2,y2,z2
                      o
                     *   
                    *       ***o x3,y3,z3
                   *    ****
         x1,y2,z1  o****
                     **
                       **
                         **o x4,y4,z5


         c
  0,0,1  o
         *             0,1,0
         *           b
         *        **
         *      **
         *   **
         * **    
  0,0,0  o***********o  a
                   1,0,0

 x = c0 + c1*a + c2 *b + c3 *c
 y = c4 + c5*a + c6 *b + c7 *c
 z = c8 + c9*a + c10*b + c11*c 

 x1 = c0 + c1*0 + c2 *0 + c3 *0   thus c0 = x1
 y1 = c4 + c5*0 + c6 *0 + c7 *0   thus c3 = y1
 z1 = c8 + c9*0 + c10*0 + c11*0   thus c8 = z1

 x2 = x1 + c1*1 + c2 *0 + c3 *0   thus c1 = x2-x1
 y2 = y1 + c5*1 + c6 *0 + c7 *0   thus c5 = y2-y1
 z2 = z1 + c9*1 + c10*0 + c11*0   thus c9 = z2-z1

 x3 = x1 + (x2-x1)*0 + c2 *1 + c3 *0   thus c2  = x3-x1
 y3 = y1 + (y2-y1)*0 + c6 *1 + c7 *0   thus c6  = y3-y1
 z3 = z1 + (z2-z1)*0 + c10*1 + c11*0   thus c10 = z3-z1

 x4 = x1 + (x2-x1)*0 + (x3-x1)*0 + c3 *1   thus c3  = x4-x1
 y4 = y1 + (y2-y1)*0 + (y3-y1)*0 + c7 *1   thus c7  = y4-y1
 z4 = z1 + (z2-z1)*0 + (z3-z1)*0 + c11*1   thus c11 = z4-z1


 thus the mapping from  a,b,c  to  x,y,z  is:

 x = x1 + (x2-x1)*a + (x3-x1)*a + (x4-x1)*a
 y = y1 + (y2-y1)*b + (y3-y1)*b + (y4-y1)*b
 z = z1 + (z2-z1)*c + (z3-z1)*c + (z4-z1)*c


This mapping: does not preserve area, does not preserve angle,
              is a full covering

rewriting the mapping in matrix form:

|(x2-x1)  (x3-x1)  (x4-x1)| * |a| = |(x-x1)|
|(y2-y1)  (y3-y1)  (y4-y1)| * |b| = |(y-y1)|
|(z2-z1)  (z3-z1)  (z4-z1)| * |c| = |(z-z1)|


solving the simultaneous equations for a,b,c
gives the inverse mapping.