Curvatures of Smooth and Discrete Surfaces

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation.

💡 Research Summary

The paper “Curvatures of Smooth and Discrete Surfaces” provides a thorough comparison between the classical differential‑geometric notions of curvature on smooth surfaces and their discrete counterparts on polyhedral (triangular) meshes. It begins by recalling that for a smooth oriented surface (M\subset\mathbb{R}^3) the Gauss curvature (K) is the product of the principal curvatures (\kappa_1\kappa_2) and the mean curvature (H) is their arithmetic mean ((\kappa_1+\kappa_2)/2). These quantities are not merely pointwise invariants; they satisfy two fundamental integral relations. The Gauss–Bonnet theorem states that (\int_M K,dA = 2\pi\chi(M)), linking curvature to the Euler characteristic, while the mean curvature appears as the first variation of the area functional, leading to the force‑balance identity (\int_{\partial\Omega} H,\mathbf{n},ds = \int_{\Omega} \Delta\mathbf{x},dA). Both relations are global and therefore provide a natural target for discretization: a good discrete curvature should preserve the integral formulas as the mesh is refined.

The authors then turn to polyhedral surfaces (S) composed of flat triangles. For Gauss curvature they adopt the classic “angle‑deficit” definition: at each vertex (v) the discrete curvature is (\theta_v = 2\pi - \sum_i \alpha_i), where the (\alpha_i) are the interior angles of the incident triangles. This definition is purely combinatorial, yet it satisfies a discrete Gauss–Bonnet theorem (\sum_v \theta_v = 2\pi\chi(S)) exactly, regardless of mesh quality. Moreover, as the mesh size tends to zero, the normalized deficit (\theta_v / A_v) converges in the weak sense to the smooth Gauss curvature, where (A_v) denotes an appropriate vertex area (often the Voronoi or mixed area around (v)).

For mean curvature the paper presents the cotangent formula, originally derived from a finite‑element discretization of the Laplace–Beltrami operator. For an interior edge (e=(i,j)) shared by two triangles with opposite angles (\alpha_{ij}) and (\beta_{ij}), the weight \