Restoration From The External Curvature Of A Surface With A Single Vertex
DOI:
https://doi.org/10.56143/ujmcs.v1i1.5Keywords:
surface, convex surface; point; ribbed; vertex; external curvature; tangent cone; polyhedra; support plane; tangent plane.Abstract
It is known that the points of a convex surface are divided into three classes: regular points, edge points, and vertices. There exists a class of surfaces that have a vertex at a single point, with all other points being regular. Such surfaces are called single-vertex surfaces, and the article provides some properties of the external curvature of such surfaces. Furthermore, the existence and uniqueness of a single-vertex surface have been proven, where the boundary consists of some closed spatial curve, and the external curvature is equal to a function defined on all Borel sets, given in advance. In this case, certain necessary conditions are imposed on the external curvature function.
References
[1] Aleksandrov A.D. : Intrinsic geometry of convex surfaces.. Taylor and Francis Group. pp. 235-269 (2006).
[2] Alexandrov A.D.: Convex Polyhedra.Selected Works. Vol-2 Novosibirsk. Science. pp. 171-213 (2007).
[3] Artykbaev A., Sobirov J. A development of a polyhedron in the Galilean space A development of a polyhedron in the Galilean space, Bulletin of National University of Uzbekistan Mathematics and Natural Sciences, 2021, DOI:10.56017/2181-1318.1148
[4] Sharipov A., Topvoldiyev F. On one invariant of polyhedra isometric on sections, AIP Conference Proceedings, 2023, DOI:10.1063/5.0144737
[5] Artikbayev A., Tillayev D.R. Property of external curvature of convex surfaces with a finite number of vertices. Modern problems of Mathematics and Mechanics. 26-28 April, 2023 Baku, Azerbaijan.
[6] Artikbayev A., Ibodullayeva N. Generalized Extrinsic Curvature of a Surface, AIP Conf. Proc. 3004, 030007 (2024).
[7] Bakelman I. Ya., Werner A. L., Kantor B. E. Introduction to Global Differential Geometry.— Moscow: Nauka, 1973.(in russian)
[8] Pogorelov A. External Geometry of Convex Surfaces. Moscow, Nauka, 1969, pp. 496–569.(in russian)
[9] Artikbaev A., Tillaev D. Properties of External Curvature of Surfaces with Vertices // Reports of the Academy of Sciences of the Republic of Uzbekistan. 2023. Vol. 4, No. 1, pp. 3–6.(in russian)
