JP Journal of Geometry and Topology

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RESTORATION SURFACES WITH VERTICES BASED ON GIVEN EXTERNAL CURVATURE

Authors

  • Abdullaaziz Artikbayev
  • Donyorbek Tillayev

Keywords:

surface, convex surface, point, ribbed, vertex, external curvature, tangent cone, polyhedra, support plane, tangent plane

DOI:

https://doi.org/10.17654/0972415X25007

Abstract

A surface with vertices is defined as a convex surface whose regularity is broken at certain points. The problem of restoration of a surface based on a given external curvature was posed and solved by A. D. Alexandrov in the class of convex polyhedra. This problem has also been solved for regular surfaces. In this article, the problem is considered for surfaces exhibiting a violation of regularity at specific points within their domain of definition. The problem is solved when the external curvature is given over a convex domain in the plane and within the framework of the solution to the Monge-Ampère equation for the surface under consideration. In this case, the vertices are projected onto the given points in the plane.

Received: May 28, 2025
Revised: July 11, 2025
Accepted: July 21, 2025

References

[1] A. D. Alexandrov, Intrinsic Geometry of Convex Surfaces, Taylor and Francis Group, 2006.

[2] M. Gopi, S. Krishnan and C. T. Silva, Surface reconstruction based on lower dimensional localized Delaunay triangulation, Computer Graphics Forum 19 (2000), 467-478.

[3] D. Attali, r-regular shape reconstruction from unorganized points, Proceedings of the ACM Symposium on Computational Geometry, 1997, pp. 248-253.

[4] B. M. Sultanov, A. Kurudirek and Sh. Sh. Ismoilov, Development and isometry of surfaces Galilean space G3, Mathematics and Statistics 11 (2023), 965-972. doi:10.13189/ms.2023.110612.

[5] A. Artikbayev and N. Ibodullayeva, Generalized External Curvature of a Surface, AIP Conference Proceedings, Uzbekistan, 2024.

[6] F. Topvoldiyev and A. Sharipov, On Defects of Polyhedra Isometric on Sections at Vertices, AIP Conference Proceedings, Uzbekistan, 2024.

[7] N. Amenta and M. Bern, Surface reconstruction by Voronoi filtering, Discrete and Computational Geometry 22 (1999), 481-504.

[8] N. Amenta, M. Bern and M. Kamvysselis, A new Voronoi-based surface reconstruction algorithm, Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques (1998), 415-421.

[9] N. Amenta, S. Choi, T. K. Dey and N. Leekha, A simple algorithm for homeomorphic surface reconstruction, Proceedings of the Sixteenth Annual Symposium on Computational Geometry (2000), 213-222.

[10] A. B. Pogorelov, External Geometry of Convex Surfaces, Science, Moscow, 1991.

[11] John A. Thorpe, Elementary Topics in Differential Geometry, Springer, Berlin, 2010.

[12] Victor A. Toponogov, Differential Geometry of Curves and Surfaces, Birkhäuser, Boston, 2006.

[13] Dirk J. Struik, Lectures on Classical Differential Geometry, Second Edition, Dover Publications, 1988.

[14] A. Artikbayev and D. Tillayev, Properties of external curvature of surfaces with vertices, Reports of the Academy of Sciences of the Republic of Uzbekistan 4 (2023), 9-12 (in Russian).

[15] D. R. Tillayev, Properties of the external curvature of a surface with vertices, Scientific Bulletin: Physical and Mathematical Research 6 (2024), 110-113.

[16] D. R. Tillayev, Reconstruction from the external curvature of a surface with a single vertex, Uzbekistan Journal of Mathematics and Computer Science 1 (2025), 50-56.

[17] A. D. Alexandrov, Convex Polyhedra, Springer Monographs in Mathematics, 2005.

[18] A. Sharipov and F. Topvoldiyev, On invariants of surfaces with isometric on sections, Mathematics and Statistics 10 (2022), 523-528.

[19] Sh. Ismoilov and B. Sultonov, Invariant geometric characteristics under the dual mapping of an isotropic space, Asia Pacific Journal of Mathematics 10 (2023), 1-12.

[20] A. Artikbayev and Sh. Ismoilov, Surface recovering by a give total and mean curvature in isotropic space , Palestine Journal of Mathematics 11 (2022), 351-361.

[21] A. Artikbayev and S. Saitova, Interpretation of geometry on manifolds as a geometry in a space with projective metric, Journal of Mathematical Science 265 (2022), 1-10.

[22] A. Artikbayev and B. Mamadaliyev, Geometry of two-dimensional surfaces in the five-dimensional pseudo-Euclidean space of index two, AIP Conference Proceedings, 2023.

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Published

2025-08-29

Issue

Vol. 31 No. 2 (2025)

Section

Articles

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How to Cite

RESTORATION SURFACES WITH VERTICES BASED ON GIVEN EXTERNAL CURVATURE. (2025). JP Journal of Geometry and Topology, 31(2), 63-78. https://doi.org/10.17654/0972415X25007

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