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Recognition of projectively transformed planar figures. XV. Methods for searching for axes and centers of ovals with symmetries, using a set of dual pairs of cevian triads

© 2021 P. P. Nikolaev

Institute for Information Transmission Problems “Kharkevich Institute” RAS, 127994 Moscow, Bolshoy Karetny per., 19, Russia
Smart Engines Service LLC, 117312 Moscow, Prospect 60-Letiya Oktyabrya, 9, Russia

Received 02 Nov 2020

A number of new enumeration-type procedures for searching symmetry elements of oval curves (their axes or centers) are suggested and modelled, which are based on alternative methods involving one of the two projectively invariant structures on the figure field: a set of so-called dual pairs (DP) or a cevian triad (CT) having the property of their intersection at the common interior point c of the oval (o). In this paper, each of the two DPs is specified on a Plucker polar curve determined by its external pole P, for the position of which conditions are formulated that are satisfied in the enumeration scheme, according to the properties of P to belong either the symmetry axis of o or a chord intersecting the desired center of o (for the cases of radial or rotational symmetry of an odd index). Similarly, CTs with the node c are used when searching for the positions of projectively symmetric pairs of points of the contour o, which, when iterating over the vertices, satisfy certain relations that are valid for the central (two kinds) or axial structure of o.

Key words: oval, center and axis of symmetry, Plucker pole and polar curve, dual pairs, harmonic wurf, wurf function, projectively invariant W-mapping, cevian

DOI: 10.31857/S0235009221010054

Cite: Nikolaev P. P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. xv. metody poiska osei i tsentrov ovalov s simmetriyami, ispolzuyushchie set dualnykh par libo triady chevian [Recognition of projectively transformed planar figures. xv. methods for searching for axes and centers of ovals with symmetries, using a set of dual pairs of cevian triads]. Sensornye sistemy [Sensory systems]. 2021. V. 35(1). P. 55–78 (in Russian). doi: 10.31857/S0235009221010054

References:

  • Akimova G.P., Bogdanov D.S., Kuratov P.A. Zadacha proektivno invariantnogo opisanija ovalov s nejavno vyrazhennoj central’noj i osevoj simmetriej i princip dvojstvennosti Pljukkera [Task projectively the invariant description of ovals with implicitly expressed central and axial symmetry and the principle of a duality of Plucker]. Trudy ISA RAN [Proceedings of the ISA RAS]. 2014. V. 64 (1). P. 75–83 (in Russian)
  • Balitsky A.M., Savchik A.V., Gafarov R.F., Konovalenko I.A. O proektivno invariantnyh tochkah ovala s vydelennoj vneshnej prjamoj. [On projective invariant points of oval coupled with external line]. Problemy peredachi informacii [Problems of Information Transmission]. 2017. V. 53(3). P. 84–89 (in Russian)
  • Glagolev N.A. Proektivnaja geometrija [Projective geometry]. Moscow, Vysshaja shkola [High school]. 1963. 344 p. (in Russian)
  • Deputatov V.N. K voprosu o prirode ploskostnyh vurfov [On the nature of the plane wurfs]. Matematicheskij sbornik [Mathematical collection]. 1926. V. 33 (1). P. 109–118. (in Russian)
  • Kartan Je. Metod podvizhnogo repera, teoriya nepreryvnykh grupp i obobshchennye prostranstva. Sb. Sovremennaya matematika. Kniga 2-ya [The method of a moving ranging mark, the theory of continuous groups and generalized spaces]. Moscow, Leningrad, Gosudarstvennoe tehniko-teoreticheskoe izdatel’stvo [State technical and theoretical publishing]. 1933. 72 p. (in Russian)
  • Modenov P.S. Analiticheskaja geometrija [Analytic geometry]. Moscow, Izdatel’stvo moskovskogo universiteta [Moscow University Press]. 1969. 699 p. (in Russian)
  • Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. II. Oval v kompozitsii s dual’nym ehlementom ploskosti. [Recognition of projectively transformed planar figures. II. An oval in a composition with a dual element of a plane]. Sensornye sistemy [Sensory systems]. 2011. V. 25 (3). P. 245–266. (in Russian)
  • Nikolaev P.P. Metod proektivno invariantnogo opisaniya ovalov s osevoi libo tsentral’noi simmetriei [A method for projectively-invariant description of ovals having axial or central symmetry]. Informatsionnye tekhnologii i vychislitel’nye sistemy. 2014. № 2. P. 46–59. (in Russian)
  • Nikolaev P.P. O zadache proektivno invariantnogo opisaniya ovalov s simmetriyami trekh rodov [A projective invariant description of ovals with three possible symmetry genera]. Vestnik RFFI [RFBR Information Bulletin]. 2016. V. 92 (4). P. 38–54. https://doi.org/10.22204/2410-4639-2016-092-04-38-54 (in Russian)
  • Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. X. metody poiska okteta invariantnykh tochek kontura ovala – itog vklyucheniya razvitoi teorii v skhemy ego opisaniya [Recognition of projectively transformed planar figures. X. Methods for finding an octet of invariant points of an oval contour – the result of introducing a developed theory into the schemes of oval description]. Sensornye sistemy [Sensory systems]. 2017. V. 31 (3). P. 202–226 (in Russian)
  • Nikolaev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. XII. O novykh metodakh proektivno invariantnogo opisaniya ovalov v kompozitsii s lineinym elementom ploskosti [Recognition of projectively transformed planar figures. XII. On new methods for projectively-invariant description of ovals in composition with a linear element of a plane]. Sensornye sistemy [Sensory systems]. 2019. V. 33 (1). P. 15–29. https://doi.org/10.1134/S0235009219010104 (in Russian)
  • Nikolaev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. XIV. novye metody proektivno invariantnogo opisaniya ovalov s privlecheniem h-polyary i dualnykh tochek [Recognition of projectively transformed planar figures. XIV. New methods for projectively-invariant description of ovals, using an h-polar curve and dual points]. Sensornye sistemy [Sensory systems]. 2020. V. 34(3). P. 226–253. https://doi.org/10.31857/S0235009220030063 (in Russian)
  • Ovsienko I.Ju., Tabachnikov S.L. Proektivnaya differentsial’naya geometriya. Staroe i novoe: ot proizvodnoi Shvartsa do kogomologii grupp diffeomorfizmov [Projective differential geometry. Old and new from the schwarzian derivative to the cohomology of diffeomorphism groups]. Moscow, MCNMO. 2008. 280 p. (in Russian)
  • Savchik A.V., Nikolaev P.P. Metod proektivnogo sopostavleniya dlya ovalov s dvumya osobymi tochkami [Projective correspondence method for an oval with two fixed points]. Informatsionnye tekhnologii i vychislitel’nye sistemy. 2018. № 1. P. 40–47. (in Russian)
  • Brugalle E. Symmetric plane curves of degree 7: Pseudoholomorphic and algebraic classifications. Journal fur Die Reine und Angewandte Mathematic (Crelles Journal). 2007. V. 612. P. 1–38. https://doi.org/10.1515/CRELLE.2007.086
  • Itenberg I.V., Itenberg V.S. Symmetric sextics in the real projective plane and auxiliary conics. Journal of Math. Sciences. 2004. V. 119 (1). P. 78–85. https://doi.org/10.1023/B:JOTH.0000008743.36321.72
  • Faugeras O. Cartan’s moving frame method and its application to the geometry and evolution of curves in the euclidean, affine and projective planes. Joint EuropeanUS Workshop on Applications of Invariance in Computer Vision. Berlin, Heidelberg. Springer, 1993. P. 9–46.
  • Hauer M., Jüttler B. Projective and affine symmetries and equivalences of rational curves in arbitrary dimension. Journal of Symbolic Computation. 2018. V. 87. P. 68–86. https://doi.org/10.1016/j.jsc.2017.05.009
  • Hann C.E., Hickman M.S. Projective curvature and integral invariants. Acta Appl. Math. 2002. V. 74 (2). P. 177–193. https://doi.org/10.1023/A:1020617228313
  • Hoff D., Olver P.J. Extensions of invariant signatures for object recognition. J. Math. Imaging Vision. 2013. V. 45. P. 176–185. https://doi.org/10.1007/s10851-012-0358-7
  • Lebmeir P., Jurgen R.-G. Rotations, translations and symmetry detection for complexified curves. J. Computer Aided Geometric Design. 2008. V. 25. P. 707–719. https://doi.org/10.1016/j.cagd.2008.09.004
  • Musso E., Nicolodi L. Invariant signature of closed planar curves. J. Math. Imaging and Vision. 2009. V. 35 (1). P. 68–85. https://doi.org/10.1007/s10851-009-0155-0
  • Olver P.J. Geometric foundations of numerical algorithms and symmetry. Appl. Alg. Engin. Comp. Commun. 2001. V. 11. P. 417–436. https://doi.org/10.1007/s002000000053
  • Sanchez-Reyes J. Detecting symmetries in polynomial Bezier curves. Journal of Computational and Applied Mathematics. 2015. V. 288. P. 274–283. https://doi.org/10.1016/j.cam.2015.04.025