A Hands-On Introduction to Discrete Differential Operators on Polygon Meshes
Robust Laplacians on Polygon Meshes
Sven D. Wagner
TU Dortmund
Astrid Bunge
AutoForm Engineering
Mario Botsch
TU Dortmund
Problems with the Centroid
Centroid is simple, but not good 🫤
How Should we Choose the Virtual Vertex?
- Minimize sum of squared areas
\[\begin{align*} \mathbf w &= \operatorname{argmin}_{\mathbf w} \sum_i \operatorname{area}\left(\mathbf x_i, \mathbf x_{i+1}, \sum_j w_j \mathbf x_j \right)^2\\ &\text{s.t.} \, \sum_i w_i = 1 \end{align*}\]
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- Solve simple linear system
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- No flipped triangles
What else can we do? 🤔
Making it More Robust
- Minimize harmonic index
- Can be related to triangle quality and stiffness matrix conditioning
\[\begin{align*} \mathbf x_f &= \operatorname{argmin}_{\mathbf x_f} \operatorname{tr}\left(\mat S^{\mathrm{tri}}\right)\\ &\text{and}\\ \mathbf w &= \operatorname{argmin}_{\mathbf w} \operatorname{tr}\left(\mat S\right) \end{align*}\]
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- Small iterative optimization
Still problems with near-
degenerate polygon edges 🫤