平面三次Pythagorean-Hodograph Hyperbolic曲线

Planar Cubic Pythagorean-Hodograph Hyperbolic Curves

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

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.