Positive Anisotropic Barycentric Coordinate On Arbitrary Polygons

The concept of barycentric coordinates was first proposed by Mobobius in 1827,that is,the coordinates of any point in a triangle can be represented by a linear combination of its three vertices.Subsequently,this concept was generalized to polygons with more than three vertices,which we call the generalized barycen-tric coordinates.In recent years,barycentric coordinates has gradually become the most common mathematical tool in computational geometry.It has been widely used to solve the problems of PDE equation,surface reconstruction,im-age graphics deformation,surface parameterization,etc.In addition to the most basic normality and precision,the barycentric coordinates have several other important properties,such as positive and smoothness.This article is for this purpose,a series of studies on the anisotropy of barycentric coordinates.The main content of this paper proposes an anisotropic barycentric coordinate construction method that maintains positive on an arbitrary polygon.For any polygon,our barycentric coordinates can integrate the anisotropy well with pos-itive and continuous guarantees.Compared with most classical barycentric co-ordinates construction methods,we provide a flexible barycentric coordinates paradigm,which allows users to select appropriate anisotropic parameters ac-cording to their actual problems and obtain different anisotropic barycentric co-ordinates.We present some specific application methods and results in the experi-mental results,showing some of the advantages of our proposed center of barycen-tric coordinates in practical applications.However,in practical applications,we find that the above properties sometimes do not meet the needs well.For exam-ple,when we approximate some objective functions with obvious anisotropy,it is necessary to study the anisotropy of the basis functions.And the results show some of the advantages of our proposed center of gravity coordinates in practical applications.At the same time,this paper also introduces some existing classical gravity center coordinate construction methods and some extensive applications of gravity center coordinates,and discusses some problems and difficulties that will occur in the construction of barycentric coordinates.