Recognition of projectively transformed planar figures. XVII. Using plucker’s reciprocity theorem to describe ovals with an external fixed point
- Авторлар: Nikolaev P.1,2
-
Мекемелер:
- A. A. Kharkevich Institute of Information Transmission Problems of the Russian Academy of Sciences
- Smart Engines Service LLC
- Шығарылым: Том 38, № 2 (2024)
- Беттер: 62-93
- Бөлім: ТЕХНИЧЕСКОЕ ЗРЕНИЕ
- URL: https://journals.rcsi.science/0235-0092/article/view/260786
- DOI: https://doi.org/10.31857/S0235009224020059
- EDN: https://elibrary.ru/DDKZPO
- ID: 260786
Дәйексөз келтіру
Аннотация
An approach to a projectively invariant description of a family of ovals (o) in scenes where the figure o is given in a composition with an external point, P, fixed in its plane is considered, and in cases where o has hidden symmetries (central or axial), the position of P is not specified in the form of an additional condition defining the scene, but can be calculated through the symmetry parameters. The invariant description, as a general universal method for numerical processing of compositions like “o + ext-P”, is proposed to be implemented in the form of Wurf mappings.The method uses the apparatus of dual pairs (DP) and wurf functions,previously developed and described by us, which are a product of decomposition of statements of the reciprocity theorem proposed by J. Plьcker to describe the properties of quadratic curves (conics).Illustrated examples of special cases of the “o + ext-P”composition are considered and discussed, actually completing the topic of studying the scenes like “an oval and a linear element of the plane”, which are classified according to the types of symmetry of o.
Толық мәтін
Авторлар туралы
P. Nikolaev
A. A. Kharkevich Institute of Information Transmission Problems of the Russian Academy of Sciences; Smart Engines Service LLC
Хат алмасуға жауапты Автор.
Email: nikol@iitp.ru
Ресей, Moscow; Moscow
Әдебиет тізімі
- 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 pryamoj. [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). https://doi.org/10.1134/S0032946017030097
- Glagolev N.A. Proektivnaya geometriya [Projective geometry]. Moscow, Vysshaya 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. Analiticheskaya geometriya [Analytic geometry]. Moscow, Izdatel’stvo moskovskogo universiteta [Moscow University Press]. 1969. 699 p. (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. No. 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. doi: 10.22204/2410-4639-2016-092-04-38-54 (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).
- Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. VIII. O vychislenii ansamblya rotacionnoj korrespondencii ovalov s simmetriej vrashheniya [Recognition of projectively transformed planar figures. VIII. On computation of the ensemble of correspondence for “rotationally symmetric” ovals]. Sensornye sistemy [Sensory systems]. 2015. V. 29(1). P. 28–55. (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. (in Russian). https://doi.org/10.1134/S0235009219030077
- Nikolaev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. XV. Metody poiska osej i centrov ovalov s simmetriyami, ispol’zujushhie set dual’nyh 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). https://doi.org/10.31857/S0235009221010054
- Nikolayev P.P. Raspoznavanie proektivno preobrazovannykh ploskikh figur. XVI. Oktet proektivno stabil’nykh vershin ovala i novye metody etalonnogo ego opisaniya, ispol’zuyushchie oktet. [Recognition of projectively transformed planar figures. XIV. The octet of projectively stable vertices of the oval and new methods for its reference description using the octet]. Sensornye sistemy [Sensory sistems]. 2022. V. 36(1). P. 61–89. (in Russian). https://doi.org/10.31857/S023500922201005X
- Savelov A.A. Ploskie krivye. Sistematika, svojstva, primeneniya [Flat curves. Systematics, properties, applications]. M. Gos. izd-vo fiziko-matematicheskoj literatury [Moscow. State publishing house of physical and mathematical literature], 1960. 293 p. (in Russian).
- Savchik A.V., Nikolaev P.P. Teorema o peresechenii T- i H- polyar [The Theorem of T- and H-Polars Intersections Count]. Informacionnye process [Information processes]. 2016. V. 16(4). P. 430–443 (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
- Carlsson S. Projectively invariant decomposition and recognition of planar shapes. International Journal of Computer Vision. 1996. V. 17(2). P. 193–209. https://doi.org/10.1007/BF00058751
- 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 European-US Workshop on Applications of Invariance in Computer Vision. Berlin, Heidelberg. Springer, 1993. P. 9–46. https://doi.org/10.1007/3-540-58240-1_2
- Gardner M. Piet Hein’s Superellipse, Mathematical Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American. New York. Vintage Press, 1977. 240–254 p.
- Hann C.E., Hickman M.S. Projective curvature and integral invariants. Acta Applicandae Mathematica. 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. Journal of mathematical imaging and vision. 2013. V. 45. P. 176–185. https://doi.org/10.1007/s10851-012-0358-7
- Itenberg I.V., Itenberg V.S. Symmetric sextics in the real projective plane and auxiliary conics. Journal of Mathematical Sciences. 2004. V. 119(1). P. 78–85. https://doi.org/10.1023/B:JOTH.0000008743.36321.72
- Lebmeir P., Jurgen R.-G. Rotations, translations and symmetry detection for complexified curves. 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. Journal of mathematical 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. Applicable Algebra in Engineering, Communication and Computing. 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