Discrete Laplacians on general polygonal meshes

Abstract

While the theory and applications of discrete Laplacians on triangulated surfaces are well developed, far less is known about the general polygonal case. We present here a principled approach for constructing geometric discrete Laplacians on surfaces with arbitrary polygonal faces, encompassing non-planar and non-convex polygons. Our construction is guided by closely mimicking structural properties of the smooth Laplace-Beltrami operator. Among other features, our construction leads to an extension of the widely employed cotan formula from triangles to polygons. Besides carefully laying out theoretical aspects, we demonstrate the versatility of our approach for a variety of geometry processing applications, embarking on situations that would have been more difficult to achieve based on geometric Laplacians for simplicial meshes or purely combinatorial Laplacians for general meshes.CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Curve, surface, solid, and object representations, Geometric algorithms, languages, and systems.Keywords: Discrete Laplace operator, generalized cotan formula, geometry processing with polygonal meshes. MotivationAmong the geometric atomic building blocks of graphics, triangles have by far attracted the most attention-perhaps because triangles are the simplest geometric figures that are able to represent twodimensional shapes, or perhaps even due to Plato's foreshadowing work Timaeus, where he records that "every solid must necessarily be contained in planes; and every planar rectilinear figure is composed of triangles". Yet, as favorable as the simplicity of triangles might appear from a purist's perspective, exclusively restricting to triangles largely impedes artistic freedom. Ornaments, tilings, kaleidoscope pattern, cubist art, architecture, or design would be paltry without quadrilaterals, pentagons, hexagons, and in fact arbitrary polygonal shapes.Likewise, in geometry processing, consider, for example, the clipping, trimming, and intersection of meshes, the insertion of meshes into spatial data structures, such as kd-or BSP-trees, or the reconstruction of meshes using marching cubes or Voronoi tessellations. Consider modeling and animating meshes with mixed quad-triangle control nets, such as they commonly appear in geometric design. Or consider the isolines of a surface parameterization, nets of curvature * e-mail: marc.alexa@tu-berlin.de † e-mail: wardetzky@math.uni-goettingen.de Figure 1: The non-planar faces of a polygonal mesh (left) can be planarized (middle) using a gradient flow of an energy that is based on our Laplacian. Likewise, the mesh can be conformally mapped to the plane with automatic boundary placement (right).lines, or Morse complexes on surfaces. All of these innately lead to tessellations by general, non-triangular polygons.Scrutinizing a parallel development, it seems fair to point out that the majority of contemporary geometry processing tools rely on, if only in the background, discrete Laplace operators, with t...

Perfect Laplacians for Polygon Meshes

“…with H i being the mean curvature and n i being the normal at vertex i . The mean curvature normal may be computed using the cotan‐Laplacian or the area gradient [AW11]. Condition (9) is consistent with the planar case, since discrete mean curvature vanishes for planar vertices.…”

“…The situation is far less developed for general polygon meshes, i.e., meshes with face degrees larger than three. Alexa and Wardetzky [AW11] provide a construction that reduces to the cotan‐Laplace for triangle meshes, but it is unclear for which meshes this operator is perfect. More importantly, there is no obvious generalization for weighted meshes, and therefore the question of existence of perfect Laplace operators for polygon meshes is open.…”

Variational Tangent Plane Intersection for Planar Polygonal Meshing

“…Here we briefly demonstrate the results of this approach and compare them to the recently proposed planarizing flow of [Alexa and Wardetzky 2011]. Figure 4 shows the results of both methods on the FERTILITY and FELINE quad mesh.…”

“…VTPI tends to concentrate these rotations to single points (see bottom figure on the right) yielding "triangles" (degenerate quads), whereas the method of [Alexa and Wardetzky 2011] tends to preserve the quad structure better at the expense of introducing ripples and plateaus (cf. the marked regions in Figure 4).…”

Error diffusion on meshes

“…To this end, we can associate to each vertex a mass m i that is deduced from the geometry. This is typically the area associated to a vertex (e.g., the Voronoi area [21], or the incident face areas based on vertex degrees [22]) but the term mass is common in the related literature on finite element methods. In some application it might be convenient and sufficient to assume that all masses are equal, i.e., m i ¼ 1.…”