src/display/bezier-utils-work.txt

   1 min .5 * sum_i lensq(bez_pt(b, u[i]) - d[i])
   2
   3 lensq(d)=dot(d, d) = d.x * d.x + d.y * d.y
   4
   5 sum_i (f(i) + g(i)) = sum_i f(i) + sum_i g(i), so
   6 we can separate into x,y parts.  Since they are the same, we write `z' in the below
   7 to mean either x or y.
   8
   9 .5 * sum_i (bez_pt(b, u[i]) - d[i]).z^2
  10
  11 = .5 * sum_i (B0(u[i]) * b[0] +
  12               B1(u[i]) * b[1] +
  13               B2(u[i]) * b[2] +
  14               B3(u[i]) * b[3]
  15               - d[i]           ).z^2
  16
  17 = H.
  18
  19 Suppose that b[0,1,3] are fixed (with b[1] perhaps being calculated
  20 from a prior call to existing generate_bezier).
  21
  22 d H / d b[2].z = sum_i B2(u[i]) * (bez_pt(b, u[i]) - d[i]).z
  23
  24 Solve for dH/db[2].z==0:
  25
  26 -sum_i B2(u[i]) B2(u[i]) * b[2].z = sum_i B2(u[i]) * (B0(u[i]) * b[0] +
  27                                                       B1(u[i]) * b[1] +
  28                                                       B3(u[i]) * b[3]
  29                                                       - d[i]           ).z
  30 b[2].z = ((sum_i B2(u[i]) * (B0(u[i]) * b[0] +
  31                              B1(u[i]) * b[1] +
  32                              B3(u[i]) * b[3]
  33                              - d[i]           ).z)
  34           / -sum_i (B2(u[i]))^2)