affine_map_tri.txt   
map a triangle with vertices x0,y0  x1,y1      x2,y2
onto a right regular triangle 0,0   1,0 (in a) 0,1 (in b)
and inverse map

                    x2,y2
                      o**
                     *   ***
                    *       ***o x1,y1
                   *    ****
           x0,y0  o****

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

 x = c0 + c1*a + c2*b
 y = c3 + c4*a + c5*b

 x0 = c0 + c1*0 + c2*0   thus c0 = x0
 y0 = c3 + c4*0 + c5*0   thus c3 = y0

 x1 = x0 + c1*1 + c2*0   thus c1 = x1-x0
 y1 = y0 + c4*1 + c5*0   thus c4 = y1-y0

 x2 = x0 + (x1-x0)*0 + c2*1  thus c2 = x2-x0
 y2 = y0 + (y1-y0)*0 + c5*1  thus c5 = y2-y0

 thus the mapping from  a,b  to  x,y  is:
 x = x0 + (x1-x0)*a + (x2-x0)*b   a in 0 .. 1
 y = y0 + (y1-y0)*a + (y2-y0)*b   b in 0 .. 1-a

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

rewriting the mapping in matrix form:

|(x1-x0)  (x2-x0)| * |a| = |(x-x0)|
|(y1-y0)  (y2-y0)|   |b| = |(y-y0)|

solving the simultaneous equations for a and b

     y *(x2-x0)-x *(y2-y0)+x0*y2-y0*x2
 a = ---------------------------------
     y1*(x2-x0)-x1*(y2-y0)+x0*y2-y0*x2

     x *(y1-y0)-y *(x1-x0)+x1*y0-y1*x0
 b = ---------------------------------
     x2*(y1-y0)-y2*(x1-x0)+x1*y0-y1*x0










map a triangle with vertices  x0,y0   x1,y1       x2,y2
onto a right regular triangle -1,-1   1,-1 (in a) -1,1 (in b)
and inverse map
                    x2,y2
                      o**
                     *   ***
                    *       ***o x1,y1
                   *    ****
           x0,y0  o****
       b
 -1,1  o
       **
       *  *
       *   *
       *     *
 -1,-1 o*******o  a
              1,-1

 x  = c0 + c1*a  + c2*b
 y  = c3 + c4*a  + c5*b

 (x1+x2)/2 = 0 in a
 (y1+y2)/2 = 0 in b  for checking

 x0 = c0 + c1*(-1) + c2*(-1)  |1  -1  -1| |c0| |x0|
 y0 = c3 + c4*(-1) + c5*(-1)  |1   1  -1|*|c1|=|x1|
                              |1  -1   1| |c2| |x2|
 x1 = c0 + c1*1 + c2*(-1)
 y1 = c3 + c4*1 + c5*(-1)
                              |1  -1  -1| |c3| |y0|
 x2 = c0 + c1*(-1) + c2*1     |1   1  -1|*|c4|=|y1|
 y2 = c3 + c4*(-1) + c5*1     |1  -1   1| |c5| |y2|

 c0 = x0 + (x1-x0)/2 + (x2-x0)/2
 c1 = (x1-x0)/2
 c2 = (x2-x0)/2
 c3 = y0 + (y1-y0)/2 + (y2-y0)/2
 c4 = (y1-y0)/2
 c5 = (y2-y0)/2

 thus the mapping from  a,b  to  x,y  is:
 x = x0 + (x1-x0)/2 + (x2-x0)/2 + ((x1-x0)/2)*a + ((x2-x0)/2)*b   a in -1..1
 y = y0 + (y1-y0)/2 + (y2-y0)/2 + ((y1-y0)/2)*a + ((y2-y0)/2)*b   b in -1..-a

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

 a = d0 + d1*x + d2*y
 b = d3 + d4*x + d5*y

-1 = d0 + d1*x0 + d2*y0  |1  x0  y0| |d0| |-1|
-1 = d3 + d4*x0 + d5*y0  |1  x1  y1|*|d1|=| 1|
                         |1  x2  y2| |d2| |-1|
 1 = d0 + d1*x1 + d2*y1
-1 = d3 + d4*x1 + d5*y1
                         |1  x0  y0| |d3| |-1|
-1 = d0 + d1*x2 + d2*y2  |1  x1  y1|*|d4|=|-1|
 1 = d3 + d4*x2 + d5*y2  |1  x2  y2| |d5| | 1|


       y *(x2-x0)-x *(y2-y0)+x0*y2-y0*x2
 a = 2*--------------------------------- - 1
       y1*(x2-x0)-x1*(y2-y0)+x0*y2-y0*x2

       x *(y1-y0)-y *(x1-x0)+x1*y0-y1*x0
 b = 2*--------------------------------- - 1
       x2*(y1-y0)-y2*(x1-x0)+x1*y0-y1*x0