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Diffstat (limited to 'doc/bezier-utils-work.txt')
| -rw-r--r-- | doc/bezier-utils-work.txt | 34 |
1 files changed, 34 insertions, 0 deletions
diff --git a/doc/bezier-utils-work.txt b/doc/bezier-utils-work.txt new file mode 100644 index 0000000..8604c12 --- /dev/null +++ b/doc/bezier-utils-work.txt @@ -0,0 +1,34 @@ +min .5 * sum_i lensq(bez_pt(b, u[i]) - d[i]) + +lensq(d)=dot(d, d) = d.x * d.x + d.y * d.y + +sum_i (f(i) + g(i)) = sum_i f(i) + sum_i g(i), so +we can separate into x,y parts. Since they are the same, we write `z' in the below +to mean either x or y. + +.5 * sum_i (bez_pt(b, u[i]) - d[i]).z^2 + += .5 * sum_i (B0(u[i]) * b[0] + + B1(u[i]) * b[1] + + B2(u[i]) * b[2] + + B3(u[i]) * b[3] + - d[i] ).z^2 + += H. + +Suppose that b[0,1,3] are fixed (with b[1] perhaps being calculated +from a prior call to existing generate_bezier). + +d H / d b[2].z = sum_i B2(u[i]) * (bez_pt(b, u[i]) - d[i]).z + +Solve for dH/db[2].z==0: + +-sum_i B2(u[i]) B2(u[i]) * b[2].z = sum_i B2(u[i]) * (B0(u[i]) * b[0] + + B1(u[i]) * b[1] + + B3(u[i]) * b[3] + - d[i] ).z +b[2].z = ((sum_i B2(u[i]) * (B0(u[i]) * b[0] + + B1(u[i]) * b[1] + + B3(u[i]) * b[3] + - d[i] ).z) + / -sum_i (B2(u[i]))^2) |