On the basis of model numerical realizations, we propose and discuss the ideas and procedures of projectively-invariant
description of ovals, which use our previously developed theory for nding eight projectively-stable points on a gure
contour, given that one invariant point has been found earlier (inside the oval or on its contour, regardless of the
presence or absence of features of its implicit symmetry). This makes it possible to use this octet for standard
description of the curve as a compact descriptor of shape in technical schemes of planar gure classi cation on the
basis of the criterion of their projective equivalence. Using the example of a radially symmetric oval, new features of
the triad of external invariant points have been described (these points are called elliptical; their positions are
determined by the marked internal point of the gure). This triad provides calculating the invariant octet of the oval
contour. For an non-symmetric oval with a selected point on its contour, we propose a procedure for fast (o(n)
asymptotics) detection of the stable internal point, the coordinates of which determine the descriptor octet sought for.
We also consider and discuss some numerical schemes capable of forming such an octet on a curve showing either axial or
rotational symmetry. The overall result of the work is the description of the concept of using the mentioned theoretical
propositions in the problem of recognition of ovals.

*Key words:*
oval, invariant, projective transformation, wurf mapping, track of elliptical points, cyclic descriptor graph, theorems
on the number of invariant points of an oval

*Cite:*
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).

## References:

- Akimova G.P., Bogdanov D.S., Kuratov P.A.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. 2014. V. 64(1). P. 75–83 [in Russian]
- Gill P., Murray W., Wright M. Practical optimization. New York. Academic Press, 1981. 401 p. [in English]
- Deputatov V. N. On the nature of planar wurfs // Matematichesk sbornik. 1926. V. 33 (1). P. 109– 118. [in Russian]
- Kartan Je. The method of a moving ranging mark, the theory of continuous groups and generalized spaces. Moscow, Leningrad. Gosudarstvennoe tehniko-teoreticheskoe izdatel’stvo, 1933. 72 p. [in Russian]
- Nikolayev P.P. Recognition of projectively transformed planar gures. II. An oval in a composition with a dual element of a plane // Sensornye sistemy. 2011a. V. 25(3). P. 245–266 [in Russian]
- Nikolayev P. P. Recognition of projectively transformed planar figures. III. Processing of axisymmetric ovals by means of polar line analysis methods // Sensornye sistemy. 2011b. V. 25(4). P. 275– 296 [in Russian]
- Nikolayev P.P. A method for projectively invariant description of ovals having axial or central symmetry // Informacionnye tekhnologii i vychislitel’nye sistemy. 2014. I. 2. P. 46–59 [in Russian])
- Nikolayev P. P. Recognition of projectively transformed planar gures. VIII. On computation of the ensemble of correspondence for “rotationally symmetric” ovals // Sensornye sistemy. 2015а. V. 29(1). P. 28–55 [in Russian]
- Nikolayev P. P. Recognition of projectively transformed planar gures. IX. Methods for description of ovals with a xed point on the contour // Sensornye sistemy. 2015b. V. 29(3). P. 213– 244 [in Russian]
- Nikolaev P.P. A Projective Invariant Description of Ovals with Three Possible Symmetry Genera // Vestnik RFFI. 2016. V. 4 (92). P. 38–54 [in Russian]
- Ovsienko I.U., Tabachnikov S.L. Projective Differential Geometry Old and New From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Moscow. MCCME, 2008. 280 p. [in Russian]
- Savchik A. V., Nikolaev P. P. The theorem on the intersection of the T- and H-polars // Informacionnye processy. 2016. V. 16 (4). P. 430–443 [in Russian]
- Alt H., Godau M. Computing the Frechet distance between two polygonal curves // International Journal of Computational Geometry and Applications. 1995. V. 5 (1–2). P. 75–91.
- Boutin M. Polygon recognition and symmetry detection // Found. Comput. Math. 2003. V. 3. 227–271.
- Faugeras O. Cartan’s moving frame method and its application to the geometry and evolution of curves in the euclidean, a ne and projective planes // Applicat. Invar. Comput. Vision / Springer Verlag, Lecture Notes in Computer Science. 1994. V. 825. P. 11–46.
- Fels M., Olver P.J.Moving coframes. I. A practical algorithm // Acta Appl. Math. 1998. V. 51. P. 161–213.
- Hann C. E., Hickman M. S. Projective curvature and integral invariants // Acta Appl. Math. 2002. V. 74. No 2. P. 177–193.
- Musso E., Nicolodi L. Invariant signature of closed planar curves // J. Math. Imaging and Vision. 2009. V. 35. No 1. P. 68–85.
- Olver P. J. Equivalence, Invariants and Symmetry // Cambridge. Cambridge Univ. Press. 1995. 525 p.
- Olver P. J. Geometric foundations of numerical algorithms and symmetry // Appl. Alg. Engin. Comp. Commun. 2001. V. 11. P. 417–436.