CGAL 4.9 - Triangulated Surface Mesh Deformation

Here, I would like to derive the formula (10)

For triangle vjvivmv_jv_iv_m, we have the following three equation which is related to viv_i

wij∥(v′i−v′j)−Ri(vi−vj)∥2w_{ij}\|(v'_i-v'_j)-R_i(v_i-v_j)\|^2
wij∥(v′i−v′j)−Rj(vi−vj)∥2w_{ij}\|(v'_i-v'_j)-R_j(v_i-v_j)\|^2
wij∥(v′i−v′j)−Rm(vi−vj)∥2w_{ij}\|(v'_i-v'_j)-R_m(v_i-v_j)\|^2

The same as for for triangle vivjvnv_iv_jv_n we have:

wji∥(v′j−v′i)−Ri(vi−vj)∥2w_{ji}\|(v'_j-v'_i)-R_i(v_i-v_j)\|^2
wji∥(v′j−v′i)−Rj(vi−vj)∥2w_{ji}\|(v'_j-v'_i)-R_j(v_i-v_j)\|^2
wji∥(v′j−v′i)−Rn(vi−vj)∥2w_{ji}\|(v'_j-v'_i)-R_n(v_i-v_j)\|^2

Taking derivative of those equations w.r.t viv_i and sum them up yields formula (10)

Then, let us look into alec jacobson’s matlab code

 function K = spokes_and_rims_linear_block(V,F,d)% Computes a matrix K such that K * R computes%  鈭�   -2*(cot(aij) + cot(bij) * (V(i,d)-V(j,d)) * (Ri + Rj) +%       -2*cot(aij) * (V(i,d)-V(j,d)) * Raij +%       -2*cot(bij) * (V(i,d)-V(j,d)) * Rbij% j鈭圢(i)% % where:          vj%              /  |  \%             /   |   \%            /    |    \%           /     |     \%          aij    |    bij%           \     |     /%            \    |    /%             \fij|gij/%              \  |  /%                 vi% % Inputs:%   V  #V by dim list of coordinates%   F  #F by 3 list of triangle indices into V%   d  index into columns of V% Output:%   K  #V by #F matrix%if simplex_size == 3% trianglesC = cotangent(V,F);i1 = F(:,1); i2 = F(:,2); i3 = F(:,3);I = [i1;i2;i2;i3;i3;i1;i1;i2;i3];J = [i2;i1;i3;i2;i1;i3;i1;i2;i3];v = [ ...C(:,3).*(V(i1,d)-V(i2,d)) + C(:,2).*(V(i1,d)-V(i3,d)); ... -C(:,3).*(V(i1,d)-V(i2,d)) + C(:,1).*(V(i2,d)-V(i3,d)); ... C(:,1).*(V(i2,d)-V(i3,d)) + C(:,3).*(V(i2,d)-V(i1,d)); ... -C(:,1).*(V(i2,d)-V(i3,d)) + C(:,2).*(V(i3,d)-V(i1,d)); ... C(:,2).*(V(i3,d)-V(i1,d)) + C(:,1).*(V(i3,d)-V(i2,d)); ... -C(:,2).*(V(i3,d)-V(i1,d)) + C(:,3).*(V(i1,d)-V(i2,d)); ... ... % diagonalC(:,3).*(V(i1,d)-V(i2,d)) - C(:,2).*(V(i3,d)-V(i1,d)); ... C(:,1).*(V(i2,d)-V(i3,d)) - C(:,3).*(V(i1,d)-V(i2,d)); ... C(:,2).*(V(i3,d)-V(i1,d)) - C(:,1).*(V(i2,d)-V(i3,d)); ... ];% construct and divide by 3 so laplacian can be used as isK = sparse(I,J,v,n,n)/3;elseif simplex_size == 4% tetrahedraassert(false)endend

The column dimension of matrix k goes through each vertex that edges connect to. Elements on each row of matrix k are the same for linearizing the rotation matrix summation.
For each vertex on each triangle, we sum cot* edge up through edges connected to it.

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