广义重心坐标的递推关系

Recursion on Generalized Barycentric Coordinates

从线性方程组解空间的角度理解广义重心坐标(GBCs),给出平面重心坐标从n边形到n(10)1边形的递推关系式。将构造重心坐标的问题转化为构造函数的问题,不需考虑坐标函数的几何意义,选取满足约束条件的函数即可构造重心坐标。在推导过程中,n(10)1边形(n≥3)可看作n边形与一顶点的组合,将该顶点用n边形的顶点线性表出,可将n(10)1边形上的重心坐标化为n边形上的齐次坐标(homogeneous coordinates)。为第n(10)1个坐标函数施加一定限制条件,即得到n边形上一组重心坐标。

From the view of the solution space of a system of linear equations,the recursion formula is worked out on generalized barycentric coordinates(GBCs)from n-gons to n?1-gons.Unlike the classical way to construct GBCs,which based on geometric meaning of coordinate functions,a new method is provided to construct GBCs for planar n-gons if a coordinate function is chosen which satisfies constraint condition.To get the recursion formula,since a(n?1)-gons(n≥3)can be seen as a n-gons plus one extra vertex,the extra vertex can be represented by affine linear combination of the vertices of the n-gons.Hence the GBCs in(n?1)-gons can be rewritten by homogeneous coordinates in n-gons.Conditions for the(n?1)th coordinate function are presented to satisfy the requirement of GBCs.