On symmetry of projective curves

A. V. Seliverstov

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2016), pp. 59-66

Résumé

The known methods to find symmetry for convex ovals have been extended for a new class of plane projective curves, including real cubic curves. The proposed method uses only the description of tangent lines, but does not require the calculation of the curvature. This approach increases the accuracy of the calculations and allows you to use graphical methods. The recognition of symmetrically arranged points allows to coordinate pictures taken from different angles, as we study the symmetry of projective curves. The results can be used to illustrate the methods of descriptive geometry.

Détail

Keywords: symmetry, projective curve, descriptive geometry, graphics, pattern recognition.

Bibliographie

[1] Kotov A. P., Fursov V. A., Goshin Ye. V., “Technology for fast 3d-scene reconstruction from stereo images”, Computer Optics, 39:4 (2015), 600–605 (in Russian) | DOI

[2] Goshin Ye. V., Fursov V. A., “3D scene reconstruction from stereo images with unknown extrinsic parameters”, Computer Optics, 39:5 (2015), 770–775 (in Russian) | DOI

[3] Mishkin D., Matas J., Perdoch M., “MODS: Fast and robust method for two-view matching”, Computer Vision and Image Understanding, 141 (2015), 81–93 | DOI

[4] Chafonova V. G., Gazeeva I. V., Tihomirova G. V., “Automatic control and digital correction of scale and rotation mismatch in stereo pairs”, Computer Optics, 40:1 (2016), 112–120 (in Russian) | DOI

[5] Sal'kov N. A., “Art and descriptive geometry”, Geometry and Graphics, 1:3-4 (2013), 3–7 (in Russian) | DOI

[6] Miloserdov E. P., Glebov M. A., “Calculation of construction parameters and algorithm design of analemmatic sundial”, Geometry and Graphics, 2:3 (2014), 14–16 (in Russian) | DOI

[7] Sela M., Aflalo Y., Kimmel R., “Computational caricaturization of surfaces”, Computer Vision and Image Understanding, 141 (2015), 1–17 | DOI

[8] Nikolayev P. P., “Recognition of projectively transformed planar figures. VIII. On computation of the ensemble of correspondence for rotationally symmetric ovals”, Sensor Systems, 29:1 (2015), 28–55 (in Russian)

[9] Prasolov V. V., Solovyev Yu. P., Elliptic Functions and Elliptic Integrals, American Mathematical Society, Providence, 1997 | MR | Zbl

[10] Hirsch A. G., “Ostensibilities in geometry”, Geometry and Graphics, 2:2 (2014), 3–8 (in Russian) | DOI

[11] Hirsch A. G., “Foci of algebraic curves”, Geometry and Graphics, 3:3 (2015), 4–17 (in Russian) | DOI

[12] Ivanov G. S., Dmitrieva I. M., “About the tasks of descriptive geometry with imaginary solutions”, Geometry and Graphics, 3:2 (2015), 3–8 (in Russian) | DOI | MR

[13] Smogorzhevskii A. S., Stolova E. S., Handbook of the Theory of Curves of the Third Order, Fizmatlit Publ., Moscow, 1961, 264 pp. (in Russian)

[14] Seliverstov A. V., “Cubic forms without monomials in two variables”, Herald of Udmurtia State University. Mathematics. Mechanics. Computer Science, 25:1 (2015), 71–77 (in Russian) | Zbl

[15] Prasolov V. V., Tikhomirov V. M., Geometry, American Mathematical Society, 2001 | MR

[16] Belim S. V., Kutlunin P. E., “Boundary extraction in images using a clustering algorithm”, Computer Optics, 39:1 (2015), 119–124 (in Russian) | DOI

[17] Semenov A. B., “On method of symbols form deformation based on planar Bezier patch”, Vestnik TvGU. Seriya: Prikladnaya Matematika [Herald of Tver State University. Series: Applied Mathematics], 2014, no. 4, 105–113 (in Russian)