The Construction And Properties Of Generalized Barycentric Coordinates Of Planar Polygon

As a new method of interpolation and deformation,the barycentric coordinates has the advantages of simplicity,intuition,detail preserving and controllable computing complexity.It has been widely used in the improvement of rendering,image interpolation and graph morphing.This paper first introduces the concept of the barycentric coordinates,as well as several common barycentric coordinates,mainly about Wachspress coordinates and mean value coordinates and related properties.Next,the change rule of the mean value coordinate of the planar polygon is introduced as the polygon changes the num of its vertices.When the planar polygon increases or decreases a vertex,except for the changing vertice and the adjacent vertices,the other coordinates corresponding to other vertices are a constant time of the original.And through the numerical examples of convex or concave decagon,it can be seen that,whether the point is in polygon affects the comparison between the constant and 1 and there is no obvious correlation between polygon concavity and convexity with the constant.In addition,through the comparison of the efficiency,the calculation efficiency of the modified barycentric coordinates is obviously higher than that of the original method.Then,the problem of the parameter limit of the single parameter quadrilateral barycentric coordinates is theoretically proved.That is,when the parameter tends to infinity or infinitesimal,the distribution of the barycentric coordinates of the single parameter quadrilateral is consistent with the distribution of the barycentric coordinates of the two individual triangles divided by the diagonal line.In the case of different parameters,the distribution of the barycentric coordinates is given.After that,an improved algorithm of mean value coordinates is proposed.In order to solve the problem of negative value of mean value coordinates in concave polygon,this paper uses the theory of non negativity of mean value coordinates in quadrilateral to transform the problem of the mean value in the concave polygon into a linear combination of the mean value coordinates in several quadrilateral.The algorithm can make the barycentric coordinates in any polygon is non negative,and the numerical examples show that the improved algorithm can guarantee thesmoothness of the barycentric coordinates if the number of edges is small.However,the smoothness of the barycentric coordinates is weaker than that of the mean value coordinates' when the number of edges is large.The problem may be solved by modifying the corresponding parameters of the algorithm.This problem can be used as a follow-up study.