广义重心坐标的递推关系

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 ?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 ?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 ?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 ?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。

Recursion on Generalized Barycentric Coordinates

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.