G1 Hermite interpolation method for spatial PH curves with PH planar projections

• Published in: Computer aided geometric design Vol. 97; p. 102132
• Main Authors: Song, Yoonae, Kim, Soo Hyun, Moon, Hwan Pyo
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2022
• Subjects: [formula omitted] Hermite interpolation; Monte-Carlo simulation; PH curve; PHoPH curve; Quaternion representation; PHoPH curve; PH curve; Quaternion representation; G1 Hermite interpolation; Monte-Carlo simulation
• ISSN: 0167-8396; 1879-2332
• DOI: 10.1016/j.cagd.2022.102132

The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or PHoPH curve. We investigate the algebraic structure of PHoPH curves and show that their hodographs are obtained by applying two squaring maps successively to quaternion generator polynomials. The simplest nontrivial PHoPH curves generated from linear quaternion generators are quintic curves, which have adequate degrees of freedom to solve the G1 Hermite interpolation problem. From the algebraic structure, we can derive a system of nonlinear equation for G1 interpolation, which is addressable by numerical methods. We also suggest the choice of initial values for the numerical method. The solvability is not guaranteed for arbitrary G1 data in general, however, we show the feasibility of the system for the G1 data taken from a small segment of reference curves without inflection points using extensive Monte-Carlo simulation. We also present a few illustrative examples of PHoPH spline curves that approximate the given reference curves. •Spatial Pythagorean hodograph curves over planer Pythagorean hodograph (PHoPH) are introduced.•The algebraic structure of PHoPH curves is characterized by two squaring maps applied to quaternion generator polynomials.•G1 Hermite interpolation method using quintic PHoPH curves is developed.•The feasibility analysis for the G1 Hermite interpolation problem was conducted using extensive Monte-Carlo simulation.