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[Maximilian Weiherer <>]

Dear cgal-community,

 

I have a rather theoretical question concerning the spokes-and-rims as-rigid-as-possible (ARAP) surface mesh deformation, see https://doc.cgal.org/latest/Surface_mesh_deformation/index.html, chapter 5 "Deformation Techniques, Energies and Weighting Schemes", section 5.3 "Spokes and Rims Version".

 

I am trying to re-calculate the partial derivative w.r.t. new positions p'_i as stated in equation 10. However, I want to use symmetric cotangent weights, thus w_ij = w_ji. Taking partial derivative of equation 8 w.r.t. p'_i, only the vertices i, j, m, and n remain since they are directly connected to p'_i (see Fig. 69.9). All the other derivatives vanish. Thus, I would expect the rhs of equation 10 to be something like: sum_j [w_ij * ((R_i + R_j + R_m + R_n) / 4) * (p_i - p_j)]. However, using the fact that w_ij = w_ji equation 10 becomes clearly not the same as what I obtained: sum_j [w_ij * ((2R_i + 2R_j + R_m + R_n) / 6) * (p_i - p_j)]. I do not understand why R_i and R_j have double influcence.

 

Any hints are greatly appreciated.

 

Max