The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining G2 2 smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all x-coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the G2 2 interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels.

Publication:

Computer Aided Geometric Design Volume 113, September 2024

https://doi.org/10.1016/j.cagd.2024.102374

Author:

Xin-Yu Wang

School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049, Beijing, China

Email: wangxinyu205@mails.ucas.ac.cn

Li-Yong Shen

School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049, Beijing, China

Email: lyshen@ucas.ac.cn

Chun-Ming Yuan

School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049, Beijing, China

KLMM, Academy of Mathematics and Systems Sciences, CAS, 100190, Beijing, China

Email: cmyuan@mmrc.iss.ac.cn

Sonia Pérez-Díaz

Universidad de Alcalá, Departamento de Física y Matemáticas, E-28871 Madrid, Spain

Email: sonia.perez@uah.es