基于双参数的几何细分法

提出一种基于两个参数的几何细分方法。首先，借助于标准型的二次有理Bézier 曲 线公式，以相邻的两个初始控制点及其切向量所在直线的交点作为该二次有理Bézier 曲线的控制 顶点；同时，选取分点参数值t ? 0.5，并以该曲线的权因子作为控制顶点的参数λ，计算新增控 制顶点。其次，定义每个顶点的临时切向量，以每点及其相邻两点确定该点的圆切向；引入切向 量的控制参数?，从而确定该顶点新切向量的计算公式。然后，从理论上证明了该方法的保凸性 与收敛性。取定切向量参数?=0，重新定义每步的权因子参数λ，其极限曲线是C1连续的分段二 次有理Bézier 曲线；令?=1，在每一步骤中采用不同的权因子参数λ 求新增点，具有保圆性。最 后，通过一些实例说明了该方法的有效性。

Double-Parameter Geometric Subdivision Method

A geometric subdivision method based on double parameters is proposed in this paper. Firstly, the new control points are determined by the original control points and their tangents: using the quadratic rational Bézier curves formula in which the parameter t is 0.5, let two adjacent points and the intersecting point of their tangents be the control points of Bézier curves, and take its weight as the first parameter ? to calculate new points. Then we calculate new tangent vectors of all points: after define provisional tangent vectors, the circle-tangent of this point is computed by the point and its two adjacent points; whereafter define the formula of new tangents for all points by introducing the second parameter ? related to tangent vectors. Theoretical analyses show its convexity preserving and convergence. If the second parameter ?=0, and next step we define a new factor by the initial parameter ?, its limit curve is a piecewise rational quadratic C1 curve. The circle preserving of this scheme can be obtained by computing new points with different parameters ? in every step under ?=1. The effectiveness of this approach is verified by some numerical examples.