G1约束下基于三次内点插值的曲线逼近方法  

Cubic Inner Point Interpolation Based Method for Approximating Offset Curves under G1 Constraint

作　　者：金佳培 赵伟华[1] 曾宇[1] 陈小雕[1] 

机构地区：[1]杭州电子科技大学计算机学院,浙江杭州310018

出　　处：《杭州电子科技大学学报（自然科学版）》2017年第5期38-43,共6页Journal of Hangzhou Dianzi University：Natural Sciences

基　　金：国家自然科学基金资助项目(61672009)

摘　　要：讨论了平面曲线的逼近问题,并提出了基于三次Bézier曲线插值的逼近算法.首先给出了插值三点三切向的三次Bézier曲线的计算公式,其本质上等价于一元三次方程的求解问题,并讨论了相关插值曲线的存在性.该插值曲线具有高达6次的逼近阶,可期望获取更好的逼近效果.然后,已满足误差的部分保持不变,针对不满足误差的部分,事先估算参数区间的划分段数,并计算每一小区间对应的逼近曲线.多段插值Bézier曲线自动具有G1连续性,可进一步合并成C2连续的三次B样条曲线.该方法只需修改不满足误差的局部曲线段,具有修改的局部性.数值实例证明了该方法具有更好的逼近效果和计算效率.Offset curves have wide applications in computer-aided design （CAD） and robot path planning（RPP）. Interpolation method needs no information of the control polygon and seems to be more flexible. This paper discusses approximating offset curves and proposes a method based on cubic Bezier inner point interpolation. It derives the formulae of cubic Bezier curves which interpolate three points and three of their directional tangent vectors, which is turned into a univariate cubic polynomial equation; it also discusses the existence of the interpolation curve. It can achieve approximation order of 6, which means to possibly obtain better approximation effect. The cubic interpolation Bezier curves are G1 continuous, which can be merged into a C2 continuous B-spline curve. The proposed method is a local one, which means that only the segments not satisfying the given tolerance need to be subdivided into several sub-segments, and the number of the sub-segments can be pre-estimated. Numerical examples show that the proposed method can achieve much better approximation effect than previous methods.

关 键 词：等距 逼近 内点插值法 三次BÉZIER曲线 

分 类 号：TP391.41[自动化与计算机技术—计算机应用技术]