Pythagorean hodographs
Pythagorean hodographs
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DOI：
10.1147/rd.345.0736
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发表时间：
1990-09
期刊：
影响因子：
--
通讯作者：
R. Farouki;T. Sakkalis
中科院分区：
文献类型：
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作者：
R. Farouki;T. Sakkalis
平面参数曲线r（t）=（x（t），y（t））f的hdograph是由该曲线的一阶参数导数r'（t）=（x '（t），y '（t））描述的轨迹。一条多项式参数曲线被称为具有毕达哥拉斯速端图，如果存在一个多项式a（t）使得xt 2（t）+ y“（t）”是毕达哥拉斯三元组。“虽然毕达哥拉斯速端曲线的自由度比相同次数的一般多项式曲线少，但它们在实际应用中表现出非常有吸引力的特性。例如，它们的弧长可表示为参数的多项式函数，并且它们的偏移量是有理曲线。我们提出了一个必不可少的和必要的代数表征的勾股速端特性，分析其几何含义的Bernstein-Bezier形式，并调查它在各种应用中所需要的有用属性。（t），即，（x ′（t），y ′（t），a（t））形成1。介绍曲线和曲面的表示形式服从有效的，系统的计算是计算机辅助设计中的一个基本问题。这些陈述由国际商业机器公司于1990年出版。允许以印刷形式复制供私人使用而无需支付版税，前提是（1）每次复制都未经更改，（2）在第一页上包括Joz参考和IBM版权声明。本文的标题和摘要，但没有其他部分，可以复制或分发版税免费未经进一步许可的计算机和其他信息服务系统。允许重新发布本文的任何其他736部分必须从编辑获得。在实际应用中已获得广泛接受的公式几乎完全是基于（分段）多项式函数的参数公式（参见[ 11]和其中的参考文献）。例如，平面曲线段通常以等价于“x（t）= aktk，k= 0 n y（t）= 1 bktk for t E [0，11. k=O这样的段可以以各种阶的连续性拼接在一起，以形成用于平滑数据插值的样条曲线;它们可以通过均匀递增t并对多项式求值而容易地呈现（1）;并且算法程序可用于计算它们的交点（参见[ 2 ]）。形式（1）的一个直接缺点-它不能容纳抛物线以外的圆锥轨迹[3]-可以通过允许有理形式r（t）=（X（t）/W（t），Y（t）/ W（t））来弥补，其中X（t），Y（t）和W（t）是多项式。…
The hdograph of a plane parametric curve r(t) = (x(t), y(t)f is the locus described by the first parametric derivative r' (t) = (x ' (t), y ' (t)) of that curve. A polynomial parametric curve is said to have a Pythagorean hodograph if there exists a polynomial a(t) such that x t 2 (t) + y " (t) " Pythagorean triple. " Although Pythagorean-hodograph curves have fewer degrees of freedom than general polynomial curves of the same degree, they exhibit remarkably attractive properties for practical use. For example, their arc length is expressible as a polynomial function of the parameter, and their offsets are rational curves. We present a sufficient-and-necessary algebraic characterization of the Pythagorean-hodograph property, analyze its geometric implications in terms of Bernstein-Bezier forms, and survey the useful attributes it entails in various applications. ~ ' (t) , i.e., (x' (t), y ' (t) , a (t)) form a 1. Introduction The representation of curves and surfaces in a form amenable to efficient, systematic computation is a basic issue in computer-aided design. Those representations Topyright 1990 by International Business Machines Corporation. Copying in printed form for private use is permitted without payment of royalty provided that (1) each reproduction is done without alteration and (2) the Joz~rnal reference and IBM copyright notice are included on the first page. The title and abstract, but no other portions, of this paper may be copied or distributed royalty free without further permission by computer-based and other information-service systems. Permission to republish any other 736 portion of this paper must be obtained from the Editor. that have won widespread acceptance in practical use are almost exclusively parametric formulations, based on (piecewise) polynomial functions (see [ 11 and references therein). Plane curve segments, for example, are usually defined in a form equivalent to " x(t) = aktk, k=O n y (t) = 1 bktk for t E [0, 11. k=O Such segments may be pieced together with various orders of continuity to form spline curves for smooth data interpolation; they are easily rendered by uniformly incrementing t and evaluating the polynomials (1); and algorithmic procedures are available for computing their intersections (see [ 2 ]). An immediate shortcoming of the form (1)-its inability to accommodate conic loci other than the parabola [3]-may be remedied by allowing the rational form r(t) = (X(t)/W(t), Y(t)/ W (t)) , where X(t), Y(t), and W(t) are polynomials. …