迭代逼近坐标

ITERATIVE APPROXIMATION COORDINATES

广义重心坐标能把多边形内任意一点表示为其顶点的线性组合,因此广泛应用于计算机图形学等领域.本文用渐进逼近的思想计算广义重心坐标.给定多边形及其内一点,首先将多边形映射到以该点为圆心的单位圆上,依次连接映射到同一圆上的各边中点,形成新的圆内接多边形.然后构造以多边形相邻两个点为顶点,其余点的加权和为另一顶点的三角形,并在该三角形内创建初始迭代点.由三角形顶点及各边中点生成三条有理Bézier曲线.通过曲线调整迭代点的位置,达到逐步缩小其与待求点距离的目的.最后通过回代求出待求点的重心坐标.实例表明,迭代逼近坐标具有非负性和光滑性等良好的性质.

Generalized barycentric coordinates(GBCs)can represent any point in a polygon as a linear combination of its vertices,so they are used widely in many fields,such as computer graphics.GBCs are obtained by the idea of progressive approximation in this paper.Given a polygon and a query point in it,map the polygon onto a unit circle centered at the point,and connect the midpoints of each side mapped onto the same circle in turn to form a new inscribed polygon.Then construct a triangle with two adjacent vertices of the inscribed polygon and the weighted sum of the remaining vertices,in which an initial iteration point is obtained.Generate three rational Bézier curves based on the vertices and the midpoints of the triangle and adjust the position of the iterative point through the curves to gradually reduce the distance to the query point.Finally,GBCs of the query point are obtained by back-substitution.Examples show that iterative approximation coordinates have many good properties,such as non-negativity,smoothness and so on.