Construction Of Compatible Laplace-Beltrami Operator In Mixed Polygon Mesh

With the rise of discrete sampling technology,digital geometric processing and computer graphics are receiving more and more attention.In the processing of polygon meshes,triangular meshes and quadrilateral meshes are the most common.Triangular meshes have relatively good properties and solidity Research history,but for architectural geometry,quadrilateral meshes are often considered to be a more direct and efficient choice.In order to merge the advantages of both,industrial design often also builds a hybrid mesh model that contains both triangles and quadrilaterals.Therefore,it is very necessary to construct a direct processing method for hybrid polygon meshes Constructing discrete Laplace-Beltrami operators has always been a research hotspot for polygon mesh processing.For non-triangular mesh models,the direct approach is to use after the model is triangulated,it is processed according to the triangle mesh,but this method often destroys the structure and properties of the original polygon mesh.In this paper,on the basis of ensuring the original mesh model,it is constructed that it can be applied to Discrete Laplace-Beltrami operator of a polygon mesh with triangles and quadrilateralsIn this paper,the Laplace operator and the Laplace-Beltrami operator are briefly introduced.The basic framework and weight properties of the discrete Laplace-Beltrami operator are given.The uniform weight format and the cotangent weight format in the triangular mesh are introduced.The advantages and disadvantages of the two formats are briefly described.The cotangent weight form is the most widely used in triangle mesh processing,but it lacks its corresponding expression on the quadrilateral mesh.This paper uses a quadrilateral area coordinate representation method,the basic format of the cotangent weight of the discrete Laplace-Beltrami operator based on the quadrilateral mesh is derived.Then this format is normalized to the cotangent weight format of the triangular grid,and a compatible Laplace-Beltrami operator discretization format,which can be directly used in polygon meshes with triangles and quadrilaterals.Then we use algebraic surfaces and three-dimensional mesh models to verify the accuracy,convergence and phase of this format capacitance was verified.At the same time,the grid smoothing experiment was conducted using this discrete format.So far,we have obtained a Laplace-Beltrami operator discrete format with good performance and compatibility for hybrid polygon meshes.