On Control Polygons of Planar Sextic Pythagorean Hodograph Curves
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 th...
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| Format: | Article |
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MDPI AG
2023-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/2/383 |
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| author | Yujun Li Lincong Fang Zhihao Zheng Juan Cao |
| author_facet | Yujun Li Lincong Fang Zhihao Zheng Juan Cao |
| author_sort | Yujun Li |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T11:46:30Z |
| format | Article |
| id | doaj.art-698f99f343394381808d7cd64ccf9e0b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:46:30Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-698f99f343394381808d7cd64ccf9e0b2023-11-30T23:21:25ZengMDPI AGMathematics2227-73902023-01-0111238310.3390/math11020383On Control Polygons of Planar Sextic Pythagorean Hodograph CurvesYujun Li0Lincong Fang1Zhihao Zheng2Juan Cao3Information Institute, Zhejiang University of Finance and Economics Dongfang College, Jiaxing 314408, ChinaSchool of Information Management and Artificial Intelligence, Zhejiang University of Finance and Economics, Hangzhou 310018, ChinaSchool of Mathematical Sciences, Zhejiang University, Hangzhou 310058, ChinaSchool of Mathematical Sciences, Xiamen University, Xiamen 361005, ChinaIn 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.https://www.mdpi.com/2227-7390/11/2/383Bézier curvePythagorean hodographcontrol polygongeometric characteristic |
| spellingShingle | Yujun Li Lincong Fang Zhihao Zheng Juan Cao On Control Polygons of Planar Sextic Pythagorean Hodograph Curves Mathematics Bézier curve Pythagorean hodograph control polygon geometric characteristic |
| title | On Control Polygons of Planar Sextic Pythagorean Hodograph Curves |
| title_full | On Control Polygons of Planar Sextic Pythagorean Hodograph Curves |
| title_fullStr | On Control Polygons of Planar Sextic Pythagorean Hodograph Curves |
| title_full_unstemmed | On Control Polygons of Planar Sextic Pythagorean Hodograph Curves |
| title_short | On Control Polygons of Planar Sextic Pythagorean Hodograph Curves |
| title_sort | on control polygons of planar sextic pythagorean hodograph curves |
| topic | Bézier curve Pythagorean hodograph control polygon geometric characteristic |
| url | https://www.mdpi.com/2227-7390/11/2/383 |
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