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.

Farouki, R.t., Pelosi, F., Sampoli, M.l. (2021). Approximation of monotone clothoid segments by degree 7 Pythagorean-hodograph. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 382 [10.1016/j.cam.2020.113110].

### Approximation of monotone clothoid segments by degree 7 Pythagorean-hodograph

#### Abstract

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.
##### Scheda breve Scheda completa Scheda completa (DC)
15-gen-2021
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/08
English
Con Impact Factor ISI
clothoid; cornu spiral; Fresnel integrals; Arc length; Pythagorean-hodograph curves; geometric Hermite interpolation
This work was partially supported by the MIUR Excellence Department Project, Department of Mathematics, University of Rome ‘‘Tor Vergata’’ (CUP E83C1000100006F), Italy and by INdAM–GNCS, Gruppo Nazionale per il Calcolo Scientifico, which F. Pelosi and M. L. Sampoli are members of.
doi.org/10.1016/j.cam.2020
Farouki, R.t., Pelosi, F., Sampoli, M.l. (2021). Approximation of monotone clothoid segments by degree 7 Pythagorean-hodograph. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 382 [10.1016/j.cam.2020.113110].
Farouki, Rt; Pelosi, F; Sampoli, Ml
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2108/261685`
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