For scenes where the input object is a composition of an arbitrary oval with point P fixed in its field (called a test
pole), and this P determines the position (outside the oval) of a projectively invariant curve, the so-called T-polar
curve, methods of engaging the triad of wurf functions, w1(n), w2(n), and w3(n) (n being the number of vertices used to
approximate the oval), are described. These functions allow detecting on the T-polar curve a set of its projectively-
invariant points, previously called elliptic points (EP) or dual points (DP), which can be used to compute descriptors
of the “Oval + P” composition, while obtaining solely a triad of new wurf functions is enough to quickly calculate
several independent forms of wurf mapping of such a composition. The developed fast processing algorithms have been
successfully tested on a representative series of numerical models. The proposed approaches of discrete analysis of the
curve use the previously obtained theoretical statements. For an oval with a specified internal point, intP, these
statements guarantee the presence of the EP triad, while for an oval combined with extL (externally positioned line),
they guarantee the presence of two pairs of DP. On the basis of the new features obtained for the elements of implicit
symmetry (IS) of the oval (the axis or the center), fast iterative schemes of IS detection are described, which
implement the projectively-invariant description of the “oval + IS” curve. Thus, such an “old tool” as the T-polar curve
has introduced new possibilities of invariant recognition of figures of the oval family.

*Key words:*
test pole, elliptic and dual points of a T-polar curve, wurf function, projectively-invariant mapping, axial or radial
symmetry of an oval

DOI: 10.1134/S0235009219030077

*Cite:*
Nikolaev P. P.
Raspoznavanie proektivno preobrazovannykh ploskikh figur. xiii. novye metody proektivno invariantnogo opisaniya ovalov s ispolzovaniem t-polyary
[Recognition of projectively transformed planar figures. xiii. new methods for projectively-invariant description of ovals, using a t-polar curve].
*Sensornye sistemy* [Sensory systems].
2019.
V. 33(3).
P. 238-266 (in Russian). doi: 10.1134/S0235009219030077

## 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).
- Gill P., Murray W., Wright M. Practical optimization. NewYork, AcademicPress. 1981. 401 p.
- Glagolev N.A. Proektivnaja geometrija [Projective geometry]. Moscow, Vysshaja shkola [High school]. 1963. 344p. (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. 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. 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. Recognition of projectively transformed planar figures. III. Processing of axisymmetric ovals by means of polar line analysis methods. Sensornye sistemy [Sensory systems]. 2011. V. 25 (4). P. 275–296 (in Russian).
- Nikolayev P.P. Recognition of projectively transformed planar figures. IV. Methods for forming a projective-invariant description of axisymmetric ovals. Sensornye sistemy [Sensory systems]. 2012. V. 26 (4). P. 280–303 (in Russian).
- Nikolaev P.P. Recognition of projectively transformed planar figures. VI. Invariant representation and methods for detecting the center image of an oval having implicit central symmetry. Sensornye sistemy [Sensory systems]. 2014. V. 28 (1). P. 45–74 (in Russian).
- Nikolaev P.P. Recognition of projectively transformed planar figures. IX. Methods for description of ovals with a fixed point on the contour. Sensornye sistemy [Sensory systems]. 2015. V. 29 (3). P. 213–244 (in Russian)
- Nikolaev P.P. 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).
- Ovsienko I.U., Tabachnikov S.L. Proektivnaja differencial’naja geometrija. Staroe i novoe: ot proizvodnoj Shvarca do kogomologij grupp diffeomorfizmov [Projective differential geometry old and new from the schwarzian derivative to the cohomology of diffeomorphism groups]. Moscow, MCNMO [MCCME]. 2008. 280 p. (in Russian).
- Savchik A.V., Nikolayev P.P. Teorema o peresechenii T- i H-polar [The theorem on the intersection of the T- and H-polars]. Information processes. 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.
- Brugalle E. Symmetric plane curves of degree 7: Pseudoholomorphic and Algebraic. Journal fur Die Reine und Angewandte Mathematic (Crelles Journal). 2007. V.612. P. 1–38.
- 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.
- 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.
- Hann C.E., Hickman M.S. Projective curvature and integral invariants. Acta Appl. Math. 2002. V. 74 (2). P.177–193.
- Hoff D., Olver P.J. Extensions of invariant signatures for object recognition. J. Math. Imaging Vision. 2013. V.45. P. 176–185.
- Musso E., Nicolodi L. Invariant signature of closed planar curves. J. Math. Imaging and Vision. 2009.V. 35 (1). P.68–85.
- Olver P.J. Geometric foundations of numerical algorithms and symmetry. Appl. Alg. Engin. Comp. Commun. 2001. V. 11. P. 417–436.
- Olver P.J. Recursive moving frames. Results Math. 2011. V.60. P. 423–452.