Laplacian of the coordinates: Approximation
- Simplest approximation of the laplacian (Graph-Laplacian): \(\displaystyle \delta_i = \frac{1}{|\mathcal{N}_i|} \sum_{j\in\mathcal{N}_i}
(p_i-p_j)\)
-
- (-) Depends on the connectivity, assume homogeneous and regular geometrical sampling
- Other possible approximations: \(\displaystyle \delta_i = \sum_{j\in\mathcal{N}_i} \omega_{ij} \, (p_i-p_j) \)
- Ex. Cotangent weights ("Mesh Laplacian")
-
- \(\displaystyle \delta_i = \frac{1}{|\Omega_i|} \sum_{j\in\mathcal{N}_i} \frac{1}{2} (\mathrm{cot} \alpha_{ij}+\mathrm{cot} \beta_{ij})\, (p_i-p_j)
\)
-
- Better geometrical properties: independant of the connectivity
- (-) Can become negative
- Or using mean-value coordinates
- No perfect approximation (satisfying all properties of the continuous Laplacian operator)
