五次间接PH曲线的几何特征  

Geometric characteristics of quintic indirect-PH curves

作　　者：李毓君 方林聪[2] Yujun LI;Lincong FANG(Zhejiang University of Finance and Economics Dongfang College,Raining 314408,China;School of Information Management and Artificial Intelligence,Zhejiang University of Finance and Economics,Hangzhou 310018,China)

机构地区：[1]浙江财经大学东方学院,海宁314408 [2]浙江财经大学信息管理与人工智能学院,杭州310018

出　　处：《中国科学：信息科学》2021年第5期808-821,共14页Scientia Sinica(Informationis)

基　　金：浙江省自然科学基金(批准号:LY18F020023);浙江省一流学科A类(浙江财经大学统计学)资助;浙江财经大学东方学院院级重点课题(批准号:2020dfy007)资助。

摘　　要：针对五次间接PH曲线的判别问题,本文结合高斯消元法与几何方法给出Bézier控制多边形满足的充分必要条件.间接PH曲线通过一个二次有理参数变换后,其等距线是有理形式的.间接PH曲线的代数充分必要条件本质是其一阶导数的因式分解满足特定条件,是一种积的形式.考虑到Bézier曲线的表示是Bernstein多项式形式,是一种和的形式.通过这两种形式的相容性引出待求解的非线性方程组并讨论求解问题,最后将所得结果应用在控制多边形上,得到五次间接PH曲线的几何特征.This paper studies the problem of identification of quintic indirect-PH curves. By employing Gaussian elimination and geometric approaches, we give necessary and sufficient conditions for a planar parametric curve to be an indirect-PH curve. Indirect-PH curves own rational offsets after being reparameterized by a fractional quadratic transformation. Algebraic conditions for curves to have rational offsets are constraints on their first derivatives, which are the product of polynomials. However, Bézier curves are represented as a sum of Bernstein polynomials. By considering the equivalence of the product and the sum, we derive non-linear systems and discuss the solutions. Finally, the results are applied to study the Bézier control polygon, thus we get geometric characteristics of quintic indirect-PH curves.

关 键 词：BÉZIER曲线 等距曲线 几何特征 有理参数化 

分 类 号：TP391.7[自动化与计算机技术—计算机应用技术]