ncts

Discrete Laplace operators are fundamental tools in geometry processing, simulation, and modeling. For applications requiring robustness and accuracy, it is crucial that the discrete operators preserve essential properties of their continuous counterparts. In this work, we investigate discrete Laplacians on tetrahedral meshes embedded in $\mathbb{R}^3$, focusing on the structural differences between primal and dual constructions. We demonstrate that the discrete Laplacian derived from the dual construction satisfies the Euler-Lagrange equation for the Dirichlet energy, and can be decomposed into contributions from the tetrahedron's faces and a discrete mean curvature term related to the surrounding space. Through mathematical analysis, we show that the dual construction yields a more optimal and structurally faithful Laplacian compared to the primal approach. Furthermore, our results reveal that the dual mean curvature is more responsive to mesh variations, particularly in regions with high curvature changes, outperforming several existing discrete curvature models.