Quadrics / distance from planes

gistfile1.txt

Okay, to make this simpler I'll write:

  p = (x, y, z)^T
  q_i = (x_i, y_i, z_i)^T
  g_i = (grad f)(q_i)

And I'll also assume that f(x,y,z) = c not w (since I'll be using w for homogeneous coords!)

then

  T_i(p)
  = dot(g_i, p - q_i)

and

  (c - T_i(p))^2
  = (c - dot(g_i, p - q_i))^2

Okay. We can go on in that form, but it's nicer if we go to homogeneous first:

  p' = (x, y, z, 1)^T
  q_i' = (x_i, y_i, z_i, 0)^T
  

  g_i' = [-g_i] (stack the "c" below "-g_i").
         [  c ]

Then

  (c - T_i(p))^2
  = (c - dot(g_i, p - q_i))^2
  = (dot(g_i', p' - q_i'))^2
  = (dot(p' - q_i', g_i'))^2
  = (dot(p', g_i') - dot(q_i', g_i'))^2

and that later term is just a constant offset we can bake into the constant offset term we already have in g_i', yielding:

  g_i'' = [        -g_i         ]
          [ c - dot(q_i', g_i') ]

which means

  (c - T_i(p))^2
  = (dot(p', g_i''))^2
  = (dot(p', g_i'')) * (dot(g_i'', p'))
  = (p'^T * g_i'') * (g_i''^T * p')
  = p'^T * (g_i'' * g_i''^T) * p'

which is a quadratic form in standard form (the factor in parentheses is A). Summing over several of these just sums their entries for A.