基于衍生多边形的混合坐标

将多边形三角化，利用三角网格将三角形衍生为点多边形、边多边形和面多边形，再根据已有 的重心坐标提出基于衍生多边形的混合坐标。首先在三角网格内根据初始多边形内部一点所在的三角形得到衍 生多边形，然后使用调和坐标、局部重心坐标、迭代坐标中任意一种计算衍生多边形的顶点关于初始多边形顶 点的重心坐标，再使用迭代坐标计算初始多边形内部一点关于衍生多边形顶点的重心坐标，最后使用合适的混 合函数计算该点关于初始多边形顶点的重心坐标。通过实例证明，这种新的重心坐标具有良好的局部性、非负 性和光滑性。

Blended coordinates based on derived polygons

Blended barycentric coordinates based on derived polygons were proposed based on the existing barycentric coordinates by triangulating the initial polygon, and deriving the point polygons, the edge polygons, and the face polygons of the triangle using a triangular mesh. First, the derived polygons were obtained using a triangular mesh. Then, the derived polygons’ vertices, which are about the initial polygons’ vertices, could be calculated by referring to harmonic coordinates, local barycentric coordinates, or iterative coordinates. Next, a point as an affine combination of the derived polygons’ vertices could be represented using iterative coordinates. The most important step was to represent the point as an affine combination of the initial polygon’s vertices using a suited mixing coefficient. Taking all the instances into consideration, the conclusion can be naturally drawn that the proposed construction of barycentric coordinates are local, non-negative, and smooth.