Alec's Web Log

I’m always re-deriving the matrix notation of common triangle mesh energies like the Dirichlet energy, Laplacian energy, or local rigidity energy. Here’s my usual strategy. I’ll assume that if your energy came from a continuous integral you can at least get it to a sum over mesh elements. Here I write my sums over vertices and their neighbors, but surely you could also sum over triangles.

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Uglier plain text, unicode version:

∑i∑j∈N(i) vi  = ∑i * #N(i) * v = VT * D * 1, 
  where D is diagonal with Dii = #N(i)
∑i∑j∈N(i) vj  = VT * A * 1, where A is the adjacency matrix
∑i∑j∈N(i) vi - vj = VT * D * 1 - VT * A * 1 = VT * (D-A) * 1

∑i∑j∈N(i) wij  = 1 * Aw * 1, where Aw adjacency with Awij = wij, if j∈N(i)
∑i∑j∈N(i) wij*vi  = ∑i vi * ∑j∈N(i) wij  = VT * Dw * 1, 
  where Dw diagonal with Dwii = ∑j∈N(i) wij
∑i∑j∈N(i) wij*vj  = VT * Aw * 1

∑i mi * ∑j∈N(i) wij vi = ∑i mi * vi * ∑j∈N(i) wij = VT * M * Dw * 1,
  where M is diagonal with Mii = wi
∑i mi * ∑j∈N(i) wij vj = VT * Aw * M * 1

∑i∑j∈N(i) vi² = VT * D * V
∑i∑j∈N(i) vivj = VT * A * V
∑i∑j∈N(i) vj² = VT * RD * V, where RD is diagonal, RDjj = ∑i 1 if j∈N(i)
  if j∈N(i) implies i∈N(j) then
  D = RD
  so ∑i∑j∈N(i) vj² = ∑i∑j∈N(i) vi² = VT * D * V
∑i∑j∈N(i) (vi - vj)² = ∑i∑j∈N(i) vi² - 2 * ∑i∑j∈N(i) vivj + ∑i∑j∈N(i) vj²
                     = 2 * ∑i∑j∈N(i) vi² - 2 * ∑i∑j∈N(i) vivj
                     = 2 * VT * D * V - 2 * VT * A * V
                     = 2 * VT * (D - A) * V
                     = 2 * VT * L * V

∑i∑j∈N(i) wij*vi²  = ∑i vi² * ∑j∈N(i) wij = VT * Dw * V
∑i∑j∈N(i) wij*vivj = VT * Aw * V
∑i∑j∈N(i) wij*vj² = VT * RDw * V, where DW is diagonal, RDwjj = ∑i wij if j∈N(i)
  if j∈N(i) implies i∈N(j) and wij = wji then 
  Dw = RDw
  so ∑i∑j∈N(i) wij*vj² = ∑i∑j∈N(i) wij*vi² = VT * Dw * V
∑i∑j∈N(i) wij(vi - vj)² = 
  = ∑i∑j∈N(i) wij*vi² - 2 * ∑i∑j∈N(i) wij*vivj + ∑i∑j∈N(i) wij*vj²
  = 2 * ∑i∑j∈N(i) wij*vi² - 2 * ∑i∑j∈N(i) wij*vivj
  = 2 * VT * Dw * V - 2 * VT * Aw * V
  = 2 * VT * (Dw - Aw) * V
  = 2 * VT * Lw * V

∑i mi * ∑j∈N(i) wij vi² = ∑i mi * vi² * ∑j∈N(i) wij = VT * M * Dw * V
∑i mi * ∑j∈N(i) wij vj² = VT * RDw * M * V
∑i mi * ∑j∈N(i) wij vivj = VT * Aw * M * V
∑i mi * ∑j∈N(i) wij(vi - vj)² = 
  = ∑i mi * ∑j∈N(i) wij*vi² 
    - 2 * ∑i mi * ∑j∈N(i) wij*vivj 
    + ∑i mi *∑j∈N(i) wij*vj²
  = 2 * ∑i mi * ∑j∈N(i) wij*vi² - 2 * ∑i mi * ∑j∈N(i) wij*vivj
  = 2 * VT * M * Dw * V - 2 * VT * Aw * M * V
  = 2 * VT * M * (Dw - Aw) * V
  = 2 * VT * M * Lw * V
  Note: let wij = cwij = cot(αij) + cot(βij) / mi and let mi = voronoi area at 
  vi, then the familar cotangent form of the discrete Dirichlet energy is
  EDirichlet = 2 * VT * M * Lcw * V