A general framework for the optimal approximation of circular arcs by parametric polynomial curves

Vavpetič, Aleš; Žagar, Emil

doi:10.1016/j.cam.2018.06.020

Cited by 11 publications

(10 citation statements)

References 20 publications

(53 reference statements)

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“…But even more important is the analysis of some low degree cases for which n − k > 2. The first step in this direction was done in [16] and in [15] but only for the simplified radial error. It is clear that the problem of finding the best geometric interpolant in the case of the radial error is much more challenging issue.…”

Section: Discussionmentioning

confidence: 99%

“…In the latter paper the Hausdorff distance has been considered but only for the case where interpolation of the boundary points is not required. A general framework for the approximation of circular arcs by parametric polynomials was given in [16] and some particular cases following this approach can be found in [15]. The optimal approximants of maximal geometric smoothness are characterized in [9].…”

Section: Preliminariesmentioning

confidence: 99%

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On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

Vavpetič,

Žagar

2020

Preprint

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The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and have not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic G 0 , cubic G 1 , quartic G 2 and quintic G 3 interpolation. Several numerical examples are presented which confirm theoretical results.

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“…Most of the approximation techniques namely consider the interpolation of local geometric data only and use the remaining parameters to minimize the distance between the interpolant and the circular arc, to minimize the deviation of the curvature, etc. The results of this type can be found in Dokken et al (1990), Goldapp (1991), Lyche and Mørken (1994), Mørken (1995), Ahn and Kim (1997), Kim and Ahn (2007), Jaklič et al (2007), Jaklič et al (2013), Kovač and Žagar (2016), Jaklič (2016), Jaklič and Kozak (2018), Knez and Žagar (2018), Vavpetič and Žagar (2019), Ahn (2019), Vavpetič (2020) and Vavpetič and Žagar (2021), if we mention just the most important and recent ones. Although the proposed algorithms provide good approximations of circular arcs if the Hausdorff distance is considered as a measure of the error, they do not include an arc length in interpolation data.…”

Section: Introductionmentioning

confidence: 85%

Arc length preserving approximation of circular arcs by Pythagorean-hodograph curves of degree seven

Žagar¹

2022

Preprint

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In this paper interpolation of two planar points, corresponding tangent directions and curvatures with Pythagorean-hodograph (PH) curves of degree seven preserving an arc length is considered. A general approach using complex representation of PH curves is presented and a detailed analysis of the problem for data arising from a circular arc is provided. In the case of several solutions some criteria for the selection of the most appropriate one are described and an asymptotic analysis is given. Several numerical examples are included which confirm theoretical results.

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“…This is due to the fact that a rotational cylinder cannot carry a polynomial curve transversal to the rulings as it would project onto a circle. While a circle does not have an exact polynomial parameterization, it is possible to achieve good approximations with cubics (see [33] and the references therein). This is sufficient for our purposes.…”

Section: Bicubic Patches On Cones and Cylinders A Cone With Vertex Vmentioning

confidence: 99%

Optimizing B-spline surfaces for developability and paneling architectural freeform surfaces

Gavriil

Schiftner²,

Pottmann

2019

Computer-Aided Design

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Motivated by applications in architecture and design, we present a novel method for increasing the developability of a B-spline surface. We use the property that the Gauss image of a developable surface is 1-dimensional and can be locally well approximated by circles. This is cast into an algorithm for thinning the Gauss image by increasing the planarity of the Gauss images of appropriate neighborhoods. A variation of the main method allows us to tackle the problem of paneling a freeform architectural surface with developable panels, in particular enforcing rotational cylindrical, rotational conical and planar panels, which are the main preferred types of developable panels in architecture due to the reduced cost of manufacturing.

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A general framework for the optimal approximation of circular arcs by parametric polynomial curves

Cited by 11 publications

References 20 publications

On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

Arc length preserving approximation of circular arcs by Pythagorean-hodograph curves of degree seven

Optimizing B-spline surfaces for developability and paneling architectural freeform surfaces

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