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The Associated Discrete Laplacian in $\mathbb{R}^3$ and Mean Curvature with Higher order Approximations

Wei-Hung Liao

TL;DR

This work investigates discrete Laplacians on tetrahedral meshes, showing that primal and dual constructions yield distinct operators in $\mathbb{R}^3$. It develops an associated dual (cotangent-like) Laplacian, proves it satisfies the Euler–Lagrange equation for the Dirichlet energy and establishes its optimality over the primal construction in $\mathbb{R}^3$ via a weak-* convergence argument. The authors introduce an associated mean curvature on edges and propose a higher-order angular-variation framework to improve approximation accuracy, providing explicit formulas and convergence insight as mesh size tends to zero. These contributions advance discrete geometric PDEs by offering a principled, higher-order and mesh-sensitive discretization that better captures extrinsic curvature and the Laplacian in 3D geometry processing tasks.

Abstract

In $\mathbb{R}^3$, the primal and dual constructions yield completely different discrete Laplacians for tetrahedral meshes.In this article, we prove that the discrete Laplacian satisfies the Euler-Lagrange equation of the Dirichlet energy in terms of the associated discrete Laplacian corresponding to the dual construction. Specifically, for a three simplex immersed in $\mathbb{R}^3$, the associated discrete Laplacian on the tetrahedron can be expressed as the discrete Laplacian of the faces of the tetrahedron and the associated discrete mean curvature term given by the ambient space $\mathbb{R}^3$. Based on geometric foundations, we provide a mathematical proof showing that the dual construction gives a optimal Laplacian in $\mathbb{R}^3$ compared to the primal construction. Moreover, we show that the associated discrete mean curvature is more sensitive to the initial mesh than other state-of-the-art discrete mean curvatures when the angle changes instantaneously. Instead of improving the angular transient accuracy through mesh subdivision, we can improve the accuracy by providing a higher order approximation of the instantaneous change in angle to reduce the solution error.

The Associated Discrete Laplacian in $\mathbb{R}^3$ and Mean Curvature with Higher order Approximations

TL;DR

This work investigates discrete Laplacians on tetrahedral meshes, showing that primal and dual constructions yield distinct operators in . It develops an associated dual (cotangent-like) Laplacian, proves it satisfies the Euler–Lagrange equation for the Dirichlet energy and establishes its optimality over the primal construction in via a weak-* convergence argument. The authors introduce an associated mean curvature on edges and propose a higher-order angular-variation framework to improve approximation accuracy, providing explicit formulas and convergence insight as mesh size tends to zero. These contributions advance discrete geometric PDEs by offering a principled, higher-order and mesh-sensitive discretization that better captures extrinsic curvature and the Laplacian in 3D geometry processing tasks.

Abstract

In , the primal and dual constructions yield completely different discrete Laplacians for tetrahedral meshes.In this article, we prove that the discrete Laplacian satisfies the Euler-Lagrange equation of the Dirichlet energy in terms of the associated discrete Laplacian corresponding to the dual construction. Specifically, for a three simplex immersed in , the associated discrete Laplacian on the tetrahedron can be expressed as the discrete Laplacian of the faces of the tetrahedron and the associated discrete mean curvature term given by the ambient space . Based on geometric foundations, we provide a mathematical proof showing that the dual construction gives a optimal Laplacian in compared to the primal construction. Moreover, we show that the associated discrete mean curvature is more sensitive to the initial mesh than other state-of-the-art discrete mean curvatures when the angle changes instantaneously. Instead of improving the angular transient accuracy through mesh subdivision, we can improve the accuracy by providing a higher order approximation of the instantaneous change in angle to reduce the solution error.
Paper Structure (19 sections, 3 theorems, 53 equations, 6 figures, 2 tables)

This paper contains 19 sections, 3 theorems, 53 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary and overview
  3. Euler-Lagrange equation for the Dirichlet energy
  4. Discrete approximation to the Dirichlet energy
  5. Primal topology and dual topology
  6. Primal and dual construction of the discrete Laplacian
  7. Primal discrete Laplacian
  8. Dual discrete Laplacian
  9. Discrete Construction of Mean curvature
  10. Discrete mean curvature on vertex
  11. Discrete mean curvature on edge
  12. Geometric aspects of the dual construction
  13. Theoretical foundation
  14. Associative Laplacian for the dual construction
  15. Geometric perspective of $\vec{H}_{\mathbf{e}}^{\mathsf{c}}$
  16. ...and 4 more sections

Key Result

Lemma 4.1

Let $\mathbf{v}$ be an interior vertex of $\mathcal{M}$ and $\mathcal{M}_\mathbf{v}$ be the triangles forming a one--ring neighborhood of $\mathbf{v}$. Then the gradient of $\mathrm{Area}(\mathcal{M}_{\mathbf{v}})$ with respect to variation of vertices can be written in the cotangent formula where $\alpha_i$ and $\beta_i$ are the opposite angles relative to the edge $[\mathbf{v}_i, \mathbf{v}]$.

Figures (6)

  • Figure 1: (a) The one-ring neighborhood $\mathcal{M}_{\mathbf{v}_i}$ and dual cell $*\mathbf{v}_i$ of the vertex $\mathbf{v}_i$. (b) The Hodge star $*$ maps the vertex $\mathbf{v}_i$ to its dual $*\mathbf{v}_i$
  • Figure 3: (a) The dihedral angle $\theta_{kl}^{ij}$ in the primal construction of the weight $w_{ijkl}$ to the edge $[\mathbf{v}_i, \mathbf{v}_j]$. (b) The dihedral angle $\theta_{ij}^{kl}$ and triangular angles $\alpha_{ij}$, $\beta_{ij}$ in the dual construction of the weight $w_{ijkl}$ to the edge $[\mathbf{v}_i, \mathbf{v}_j]$.
  • Figure 4: (a) The discrete mean curvature vector $\vec{H}_{\mathbf{e}}$ along the edge $\mathbf{e}:=[\mathbf{v}_i, \mathbf{v}_j]$. (b) The reciprocal discrete mean curvature vector $\vec{H}_{\mathbf{e}}^{\mathsf{c}}$ along the edge $\mathbf{e}:=[\mathbf{v}_i, \mathbf{v}_j]$.
  • Figure 5: Select the coordinates on the plane $*\mathbf{e}$ and redefine the coordinates of $\mathbf{c}_{ij}$, $\mathbf{c}_{ijk}$, $\mathbf{c}_{ijkl}$ and $\mathbf{c}_{ijl}$.
  • Figure 6: (a) Discrete mean curvatures as angle varies from $0$ to $\pi$ (b) The reciprocal discrete mean curvature vector $\vec{H}_{\mathbf{e}}^{\mathsf{c}}$ along the edge $\mathbf{e}:=[\mathbf{v}_i, \mathbf{v}_j]$.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 1.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Remark 3.1
  • Lemma 4.1
  • Remark 4.2
  • Definition 4.3
  • Definition 4.4
  • ...and 9 more