On Control Polygons of Planar Sextic Pythagorean Hodograph Curves

Yujun Li (), Lincong Fang, Zhihao Zheng and Juan Cao
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Yujun Li: Information Institute, Zhejiang University of Finance and Economics Dongfang College, Jiaxing 314408, China
Lincong Fang: School of Information Management and Artificial Intelligence, Zhejiang University of Finance and Economics, Hangzhou 310018, China
Zhihao Zheng: School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China
Juan Cao: School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Mathematics, 2023, vol. 11, issue 2, 1-15

Abstract: In this paper, we analyze planar parametric sextic curves to determine conditions for Pythagorean hodograph (PH) curves. By expressing the curves to be analyzed in the complex form, the analysis is conducted in algebraic form. Since sextic PH curves can be classified into two classes according to the degrees of their derivatives’ factors, we introduce auxiliary control points to reconstruct the internal algebraic structure for both classes. We prove that a sextic curve is completely characterized by the lengths of legs and angles formed by the legs of their Bézier control polygons. As such conditions are invariant under rotations and translations, we call them the geometric characteristics of sextic PH curves. We demonstrate that the geometric characteristics form the basis for an easy and intuitive method for identifying sextic PH curves. Benefiting from our results, the computations of the parameters of cusps and/or inflection points can also be simplified.

Keywords: Bézier curve; Pythagorean hodograph; control polygon; geometric characteristic (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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