On interpolation by Planar cubic $G^2$ pythagorean-hodograph spline curves

Jaklič, Gašper; Kozak, Jernej; Krajnc, Matjaž; Vitrih, Vito; Žagar, Emil

doi:10.1090/s0025-5718-09-02298-4

Cited by 31 publications

(16 citation statements)

References 15 publications

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“…Several authors [22,27] have considered the approximation order of PH curve interpolants to discrete data, based on rather complicated asymptotic analyses. Such an analysis may be possible in the present context, although the presence of an integral constraint is an additional complication.…”

Section: Remarkmentioning

confidence: 99%

Construction of G 1 planar Hermite interpolants with prescribed arc lengths

Farouki

2016

Computer Aided Geometric Design

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The problem of constructing a plane polynomial curve with given end points and end tangents, and a specified arc length, is addressed. The solution employs planar quintic Pythagorean-hodograph (PH) curves with equal-magnitude end derivatives. By reduction to canonical form it is shown that, in this context, the problem can be expressed in terms of finding the real solutions to a system of three quadratic equations in three variables. This system admits further reduction to just a single univariate biquadratic equation, which always has positive roots. It is found that this construction of G 1 Hermite interpolants of specified arc length admits two formal solutions -of which one has attractive shape properties, and the other must be discarded due to undesired looping behavior. The algorithm developed herein offers a simple and efficient closed-form solution to a fundamental constructive geometry problem that avoids the need for iterative numerical methods.

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Section: Remarkmentioning

confidence: 99%

Construction of G 1 planar Hermite interpolants with prescribed arc lengths

Farouki

2016

Computer Aided Geometric Design

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“…Pythagorean Hodograph PH curves form a special subclass of the planar polynomial parametric curves that have expression of the arc-length using polynomial function of parameter (Jakliè et al, 2010;Alves Neto et al, 2010;Farouki and Sakkalis, 1990). The representations of the curves are exclusively parametric formulations based on the cubic polynomial function in the Bernstein-Bezier form as (11) where Bernstein polynomial given by (12) where b k are the control points for k = 0,…,3 (Farouki and Sakkalis, 1990).…”

Section: Smoothing Based On Pythagorean Hodographmentioning

confidence: 99%

“…The PH cubic curves are used to make shortcut on the primitive path by connecting an internal state of the primitive path to the initial state, which makes it possible to enhance the path quality related to kinematically feasible, smooth, and collision-free characteristics. The PH cubic curve provides the piecewise G 1 continuity and arc-length re-parameterization of the curve in a closed form (Jakliè et al, 2010). These properties of PH cubic curve simplify the formulations of the maneuvering for obstacle avoidance with smooth motion in terms of implementation.…”

Section: Introductionmentioning

confidence: 99%

Real-time path planning of autonomous vehicles for unstructured road navigation

Chu

Kim

et al. 2015

Int.J Automot. Technol.

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This paper presents a path planning algorithm for autonomous vehicles to generate feasible and smooth maneuvers in unstructured environments. Our approach uses a graph-based search algorithm and a discrete kinematic vehicle model to find a primitive path that is non-holonomic, collision-free, and safe. The algorithm then improves the quality and smoothness of the identified path by making local shortcuts. This shortcut for smoothing the path is determined by connecting the internal state of the primitive path to the initial state based on the Pythagorean Hodograph (PH) cubic curve. The formulation of the PH cubic curves in the smoothing path provides arc-length parameterization of the curve in a closed form.We present examples and experimental results for the real time path planning in unstructured road from an implementation on an autonomous vehicle.

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“…These results were later generalized in [4] to G 2 interpolation by the same objects, and in [5], where a thorough analysis of the number of solutions and their properties was done. For quintic planar PH curves, several results on first and second order continuous spline interpolation are given in [6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

C1Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

Bastl

Bizzarri

Krajnc

et al. 2014

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper the C 1 Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described. Each PH cubic segment interpolates C 1 data at one point and they are then joined together with a C 1 continuity at some unknown common point sharing some unknown tangent vector. Biarcs are expressed in a closed form with three shape parameters. Two of them are selected based on asymptotic approximation order, while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic bending energy. The final interpolating spline curve is globally C 1 continuous, it can be constructed locally and it exists for arbitrary Hermite data configurations.

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On interpolation by Planar cubic $G^2$ pythagorean-hodograph spline curves

Cited by 31 publications

References 15 publications

Construction of G 1 planar Hermite interpolants with prescribed arc lengths

Construction of G 1 planar Hermite interpolants with prescribed arc lengths

Real-time path planning of autonomous vehicles for unstructured road navigation

C1Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

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