Characteristics of planar sextic indirect-PH curves

Yujun Li; Yujun Li

doi:10.3934/math.2024110

AIMS Mathematics

2024, Volume 9, Issue 1: 2215-2231. doi: 10.3934/math.2024110

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Research article

Characteristics of planar sextic indirect-PH curves

Yujun Li ^,

School of Information, Zhejiang University of Finance and Economics Dongfang College, 310044 Jiaxing, China

Received: 16 September 2023 Revised: 08 December 2023 Accepted: 11 December 2023 Published: 21 December 2023
MSC : 65D17

By a fractional quadratic transformation, an indirect-PH curve can have rational offsets. In this paper, I study properties of planar sextic indirect-PH curves, in terms of their Bézier control polygon legs. With our results, sextic Bézier curves can be efficiently tested whether they are indirect-PH curves. The main strategy to achieve our results is using complex representation of planar parametric curve. Sextic indirect-PH curves can be classified into three classes according to different factorizations of their hodographs. Necessary and sufficient conditions for all classes of sextic indirect-PH curves can be described by non-linear complex systems. By analyzing these non-linear systems, algebraic conditions for a sextic Bézier curve to be an indirect-PH curve are first discussed, then geometric characteristics in terms of legs of its control polygon are revealed.
- geometric design,
- characteristics,
- Pythagorean hodograph,
- control polygon
Citation: Yujun Li. Characteristics of planar sextic indirect-PH curves[J]. AIMS Mathematics, 2024, 9(1): 2215-2231. doi: 10.3934/math.2024110

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Abstract

By a fractional quadratic transformation, an indirect-PH curve can have rational offsets. In this paper, I study properties of planar sextic indirect-PH curves, in terms of their Bézier control polygon legs. With our results, sextic Bézier curves can be efficiently tested whether they are indirect-PH curves. The main strategy to achieve our results is using complex representation of planar parametric curve. Sextic indirect-PH curves can be classified into three classes according to different factorizations of their hodographs. Necessary and sufficient conditions for all classes of sextic indirect-PH curves can be described by non-linear complex systems. By analyzing these non-linear systems, algebraic conditions for a sextic Bézier curve to be an indirect-PH curve are first discussed, then geometric characteristics in terms of legs of its control polygon are revealed.

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