15 research outputs found

    A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

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    AbstractAlgorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G1 Hermite data; however, one could also obtain higher order algorithms

    Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines

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    open3siProvided that they are in appropriate configurations (tight data), given planar G1 Hermite data generate a unique cubic Pythagorean hodograph (PH) spline curve interpolant. On a given associated knot-vector, the corresponding spline function cannot be C1, save for exceptional cases. By contrast, we show that replacing cubic spaces by cubic-like sparse spaces makes it possible to produce infinitely many C1 PH spline functions interpolating any given tight G1 Hermite data. Such cubic-like sparse spaces involve the constants and monomials of consecutive degrees, and they have long been used for design purposes. Only lately they were investigated in view of producing PH curves and associated G1 PH spline interpolants with some flexibility. The present work strongly relies on these recent results.embargoed_20220415Ait-Haddou R.; Beccari C.V.; Mazure M.-L.Ait-Haddou R.; Beccari C.V.; Mazure M.-L

    Path planning with PH G2 splines in R2

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    International audienceIn this article, we justify the use of parametric planar Pythagorean Hodograph spline curves in path planning. The elegant properties of such splines enable us to design an efficient interpolator algorithm, more precise than the classical Taylor interpolators and faster than an interpolator based on arc length computations

    Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable

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    Geometric Hermite interpolation by rational curves of constant width

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    A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve.Funding for the other authors not affiliated with BCAM: Grant PID2021-124577NBI00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. Project PID2019-104927GB-C21 funded by MCIN/AEI/10.13039/501100011033. Project UJI-B2022-19 funded by Universitat Jaume I. Project CIAICO/2021/180 funded by Generalitat Valenciana

    Nonlinear Geometric Models

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