期刊
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
卷 382, 期 -, 页码 -出版社
ELSEVIER
DOI: 10.1016/j.cam.2020.113110
关键词
Clothoid; Cornu spiral; Fresnel integrals; Arc length; Pythagorean-hodograph curves; Geometric Hermite interpolation
资金
- MIUR Excellence Department Project, Department of Mathematics, University of Rome Tor Vergata'', Italy [CUP E83C1000100006F]
- INdAM-GNCS, Gruppo Nazionale per il Calcolo Scientifico
The study investigates the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, with precise matching of arc length.
The clothoid is a planar curve with the intuitive geometrical property of a linear variation of the curvature with arc length, a feature that is important in many geometric design applications. However, the exact parameterization of the clothoid is defined in terms of the irreducible Fresnel integrals, which are computationally expensive to evaluate and incompatible with the polynomial/rational representations employed in computer aided geometric design. Consequently, applications that seek to exploit the simple curvature variation of the clothoid must rely on approximations that satisfy a prescribed tolerance. In the present study, we investigate the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to monotone clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, and precise matching of the clothoid segment arc length. The construction, employing PH curves of degree 7, involves iterative solution of a system of five algebraic equations in five real unknowns. This is achieved by exploiting a closed-form solution to the problem of interpolating the specified data (except the curvatures) using quintic PH curves, to determine starting values that ensure rapid and accurate convergence to the desired solution. (C) 2020 Elsevier B.V. All rights reserved.
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