郝永霞,廖莲星.平面三次Pythagorean-Hodograph Hyperbolic曲线[J].数学研究及应用,2022,42(2):206~220 |
平面三次Pythagorean-Hodograph Hyperbolic曲线 |
Planar Cubic Pythagorean-Hodograph Hyperbolic Curves |
投稿时间:2021-03-02 修订日期:2021-09-04 |
DOI:10.3770/j.issn:2095-2651.2022.02.011 |
中文关键词: Pythagorean-hodograph曲线 代数双曲B\'ezier曲线 $G^1$ Hermite插值 |
英文关键词:Pythagorean-hodograph curve algebraic hyperbolic B\'{e}zier curve $G^{1}$ Hermite interpolation |
基金项目:国家自然科学基金(Grant No.11801225),江苏省教育厅高校自然科学项目(Grant No.18KJB11005),江苏大学高级人才科研启动项目(Grant No.14JDG034). |
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中文摘要: |
本文基于Pythagorean-hodograph (PH)曲线和代数双曲线的良好几何特性,构造了Pythagorean-Hodograph Hyperbolic (PH-H)曲线,并给出了PH-H曲线的定义以及相应性质.同时,分别利用Hyperbolic基函数和Algebraic Hyperbolic (AH) B\'ezier基函数,得到了平面三次AH B\'ezier曲线为PH曲线的两个不同的充要条件.此外,三次PH-H曲线也被用于求解具有确定解的$G^1$ Hermite插值问题.文中给出了具体实例来说明我们的方法. |
英文摘要: |
The purpose of this paper is to develop a method to construct the Pythagorean-hodograph hyperbolic (PH-H) curves based on the good geometric properties of PH curves and algebraic hyperbolic curves. The definition of Pythagorean-hodograph hyperbolic curves is given and their properties are examined. By using hyperbolic basis functions and algebraic B\'{e}zier basis functions respectively, two different necessary and sufficient conditions for a planar cubic algebraic hyperbolic B\'{e}zier curve to be a PH curve are obtained. Moreover, cubic PH-H curves are applied in the problem of $G^{1}$ Hermite interpolation with determined closed form solutions. Several examples serve to illustrate our method. |
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