Polygon Laplacian Made Simple. (13th July 2020)

• Record Type: Journal Article
• Authors: Bunge, Astrid; Herholz, Philipp; Kazhdan, Misha; Botsch, Mario
• Is Part Of: Computer graphics forum. Volume 39:Number 2(2020)
• Page Start: 303
• Page End: 313
• Publication Date: 2020-07-13
• Subjects: CCS Concepts -- Computing methodologies → Mesh geometry models -- Theory of computation → Computational geometry; Computer graphics -- Periodicals; 006.605
• DOI: 10.1111/cgf.13931
• Languages: English
• ISSNs: 0167-7055

Abstract

Abstract: The discrete Laplace‐Beltrami operator for surface meshes is a fundamental building block for many (if not most) geometry processing algorithms. While Laplacians on triangle meshes have been researched intensively, yielding the cotangent discretization as the de‐facto standard, the case of general polygon meshes has received much less attention. We present a discretization of the Laplace operator which is consistent with its expression as the composition of divergence and gradient operators, and is applicable to general polygon meshes, including meshes with non‐convex, and even non‐planar, faces. By virtually inserting a carefully placed point we implicitly refine each polygon into a triangle fan, but then hide the refinement within the matrix assembly. The resulting operator generalizes the cotangent Laplacian, inherits its advantages, and is empirically shown to be on par or even better than the recent polygon Laplacian of Alexa and Wardetzky [AW11] — while being simpler to compute.