Approximate symmetries of planar algebraic curves with inexact input

Bizzarri, Michal; Lávička, Miroslav; Vršek, Jan

doi:10.1016/j.cagd.2019.101794

Cited by 10 publications

(2 citation statements)

References 15 publications

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“…Projective equivalence of special algebraic varieties, including projective (and other) equivalences of rational curves were studied and presented in [11]. A simple algorithm for an approximate reconstruction of an inexact planar curve which is assumed to be a perturbation of some unknown planar curve with symmetry was formulated in [10]. Exact and approximate similarities of non-necessarily rational planar, parameterized curves, using centers of gravity and inertia tensors were studied in [8].…”

Section: Introductionmentioning

confidence: 99%

Symmetries of discrete curves and point clouds via trigonometric interpolation

Bizzarri¹,

Lávička²,

Vršek³

2021

Preprint

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We formulate a simple algorithm for computing global exact symmetries of closed discrete curves in plane. The method is based on a suitable trigonometric interpolation of vertices of the given polyline and consequent computation of the symmetry group of the obtained trigonometric curve. The algorithm exploits the fact that the introduced unique assigning of the trigonometric curve to each closed discrete curve commutes with isometries. For understandable reasons, an essential part of the paper is devoted to determining rotational and axial symmetries of trigonometric curves. We also show that the formulated approach can be easily applied on nonorganized clouds of points. A functionality of the designed detection method is presented on several examples.

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Section: Introductionmentioning

confidence: 99%

Symmetries of discrete curves and point clouds via trigonometric interpolation

Bizzarri¹,

Lávička²,

Vršek³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…Recently Alcazar (2014); Alcazar et al (2014bAlcazar et al ( ,a, 2015; Alcázar et al (2019a,b); Bizzarri et al (2020aBizzarri et al ( , 2021 published a series of papers. In these papers, the authors investigate the problem of detecting symmetries and similarities of the rational plane and space curves.…”

Section: Introductionmentioning

confidence: 99%

A new method to detect projective equivalences and symmetries of rational $3D$ curves

Gözütok¹,

Çoban²,

Sağıroğlu³

et al. 2021

Preprint

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We present a new approach to detecting projective equivalences and symmetries of rational parametric 3D curves. To detect projective equivalences, we first derive two projective differential invariants that are also invariant with respect to the change of parameters called Möbius transformations. Given two rational curves, we form a system consists of two homogeneous polynomials in four variables using the projective differential invariants. The solution of the system yields the Möbius transformations, each of which corresponds to a projective equivalence. If the input curves are the same, then our method detects the projective symmetries of the input curve. Our method is substantially faster than methods addressing a similar problem and provides solutions even for the curves with degree up to 24 and coefficients up to 78 digits.

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Detecting Affine Equivalences Between Implicit Planar Algebraic Curves

Alcázar

Gözütok²,

Çoban³

et al. 2022

Acta Appl Math

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Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In this paper we generalize previous results by the authors to provide an algorithm for computing the affine equivalences between two parametric curves of certain types, in any dimension. In more detail, the algorithm is valid for rational curves, and for parametric curves with non-rational but meromorphic components admitting a rational inverse. Unlike other algorithms already known for rational curves, the algorithm completely avoids polynomial system solving, and uses bivariate factoring, instead, as a fundamental tool. The algorithm has been implemented in the computer algebra system Maple, and can be freely downloaded and used. Introduction.We say that two curves are affinely equivalent if one of them is the image of the other curve by means of an affine mapping. If the affine mapping preserves angles, then the two curves are similar, i.e. both correspond to the same shape and differ only in position and/or scaling. If the affine mapping preserves distances, the curves are congruent, so they differ only in position. Finally, if the two curves coincide, finding the self-congruences of the curve is equivalent to computing its symmetries.Because of the nature of the problem, it has received some attention in the applied fields of Computer Aided Geometric Design, Pattern Recognition and Computer Vision. In the last years, the problem has been also addressed in the Computer Algebra field, and this paper follows this trend. Examples of papers where this question has been studied are [2,4,7]; for other related papers, the interested reader can check the bibliographies of [2,4,7]. These papers address the problem for rational curves, i.e. parametric curves whose components are quotients of polynomial functions, and aim to the more general question of checking projective, and not just affine, equivalence. While [2,7] provide solutions for this problem, with different strategies, for curves in any dimension and using as a fundamental tool polynomial system solving, the paper [4] addresses the question only for space rational curves, but employing bivariate factoring as an alternative to solving polynomial systems; this leads to better timings and performance. To do this, in [4] two rational invariants, i.e. two functions rationally depending on the parametrizations to be studied which stay invariant under projective transformations, are found and used.In this paper we generalize the ideas of [4] in three different ways. First, while the development of the invariants in [4] was more of an "art" than of a "craft", here we provide a complete algorithm to generate such invariants. Second, the technique is valid for curves in any dimension, and not just space curves, which was the case addressed in [4]: we provide an algorithm, that can be downloaded from [5], to generate the corresponding inv...

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Approximate symmetries of planar algebraic curves with inexact input

Cited by 10 publications

References 15 publications

Symmetries of discrete curves and point clouds via trigonometric interpolation

Symmetries of discrete curves and point clouds via trigonometric interpolation

A new method to detect projective equivalences and symmetries of rational $3D$ curves

Detecting Affine Equivalences Between Implicit Planar Algebraic Curves

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