Construction Of Laplace Beltrami Operators On Polygonal Meshes Based On Generalized Barycenter Coordinates

With the continuous development and improvement of digital technology,digital geometry processing and computer graphics are more and more concerned by people.In digital geometry processing,discrete representation with polygonal mesh is a routine operation.Triangular mesh is widely used in mesh processing because of its good properties and rich research history.However,in practical industrial design,mixed polygonal mesh model is often produced,and for processing mixed mesh model,it is difficult to solve the problem,The simplest way is to triangulate the original mesh model,which obviously destroys the structure and properties of the original polygon mesh.The idea of this paper is to propose a discrete Laplace Beltrami operator on polygonal meshes based on quadrilateral generalized barycenter coordinates without destroying the original mesh model.This paper first introduces the research background of generalized barycenter coordinates and Laplace Beltrami operator,then briefly introduces the discrete Laplace Beltrami operator on triangular mesh,analyzes and compares these schemes,especially the cotangent weight scheme of discrete Laplace Beltrami operator on triangular mesh,and makes a complete derivation.Then,the recurrence formula of generalized barycenter coordinate and its proof are given.Then,two kinds of generalized barycenter coordinates based on the upper and lower bounds of recursive constraints are constructed for quadrilateral meshes.According to the constructed generalized barycenter coordinates,the cotangent weight scheme of discrete Laplace Beltrami operator for quadrilateral meshes is derived,and this scheme is normalized with the cotangent weight scheme of triangular meshes,In this paper,a discrete scheme of Laplace Beltrami operator for triangular and quadrilateral meshes is obtained.Compared with the existing schemes based on quadrilateral bilinear interpolation and quadrilateral area coordinates,this scheme is not only similar to the scheme of triangle cotangent weight,but also has the advantages of high efficiency and high efficiency,Moreover,the good properties of the generalized barycenter coordinates also provide sufficient theoretical support for the calculation and derivation.Finally,numerical experiments are carried out to verify the convergence of the scheme,and the mesh smoothing operation is carried out by using the discrete scheme.