- - For a scalar/vector function \(f\) defined in \(\mathbb{R}^2\) with parameter \((u,v)\): \(\triangle f = \frac{\partial^2 f}{\partial u^2}+\frac{\partial^2 f}{\partial v^2}=div(grad(f))\)
- - Laplacian on a manifold: Called Laplace-Beltrami operator.
- - Differential coordinates \(\delta\): Laplace-Beltrami operator applied on coordinates functions \(f=(x,y,z)\) itself.
- Common approximation in the discrete case
\(\displaystyle \delta_i = \frac{1}{|\mathcal{N}_i|} \sum_{j\in\mathcal{N}_i} (p_i-p_j) = p_i - \frac{1}{|\mathcal{N}_i|} \sum_{j\in\mathcal{N}_i} p_j\)
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- \(\mathcal{N}_i\): one ring neighborhood
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- Note: \(\displaystyle \delta_i \simeq \lim_{|\gamma|\to 0} \frac{1}{|\gamma|}\int_{p\in\gamma} (p_i-p)\,\mathrm{d}l(p) = \mathrm{H}(p_i)\,n_i\)
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- \(\gamma\): small contour around \(p_i\)
- \(\mathrm{H}(p_i)\): Mean curvature at \(p_i\), \(n_i\) normal at \(p_i\)
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- \(\Rightarrow\) Encode an approximation of the mean curvature
