Recognition of projectively transformed planar figures. XVII. Using plucker’s reciprocity theorem to describe ovals with an external fixed point-Reference-Cited by-同舟云学术

Recognition of projectively transformed planar figures. XVII. Using plucker’s reciprocity theorem to describe ovals with an external fixed point

Published:2024-08-01 Issue:2 Volume:38 Page:62-93
ISSN:0235-0092
Container-title:Сенсорные системы
language:
Short-container-title:Sensornye sistemy

Author:

Nikolaev P. P.¹²

Affiliation:

1. A. A. Kharkevich Institute of Information Transmission Problems of the Russian Academy of Sciences

2. Smart Engines Service LLC

Abstract

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.

Publisher

The Russian Academy of Sciences

Reference27 articles.

1. 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).

2. 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

3. Glagolev N.A. Proektivnaya geometriya [Projective geometry]. Moscow, Vysshaya shkola [High school]. 1963. 344 p. (in Russian).

4. 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).

5. 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).