Mathematics and Statistics Vol. 12(5), pp. 420 - 427
DOI: 10.13189/ms.2024.120503
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On One Property of a Conditional Full Angle

Anvarjon Sharipov ^1,2,*, Fayzulla Topvoldiyev ³, Zokirkhuja Usmonkhujaev ^4
1 Department of Geometry and Topology, Faculty of Mathematics, National University of Uzbekistan, 100174, Tashkent, Uzbekistan
² Branch of National Research Nuclear University MEPhI in Tashkent, 1, Khuroson str., 100214, Tashkent, Uzbekistan
³ Department of Mathematics, Faculty of Mathematics and Informatics, Fergana State University, 150100, Fergana, Uzbekistan
⁴ Department of High Mathematics, Tashkent State Technical University Named after Islam Karimov, 100095, Tashkent, Uzbekistan

ABSTRACT

One of the main directions of modern differential geometry and topology is the so-called geometry "in large", which is a field that studies geometric objects as a whole. As one of the first results of this field, it is possible to recognize the equality of the polyhedron consisting of two closed, congruent sides with the same constituent sides, proved by O. Cauchy in 1813. The problems of restoring polyhedron and surfaces according to the given geometric characteristics are also included in the problems of geometry "in large". As a geometric characteristic, it is possible to obtain external curvature, internal curvature, and area of a spherical image, area of a cylindrical image of a polyhedron and surface, or any other value related to a geometric object. Geometric methods related to the theory of polyhedra are widely used not only for polyhedra, but also in the general theory of surfaces. Some surfaces are formed from the limit states of polyhedras. An analogy of some properties of polyhedra can be made for surfaces. Therefore, studying the properties of polygons helps to study such properties of surfaces. This article studied the properties of polyhedron isometric on sections. In particular, the concept of a conditional full angle given at the vertex, which is important for the restoration of isometric polygons in terms of their external curvature, is included. It is known that the concept of isometry on sections depends to given direction. The concept of a conditional full angle is generalized for trihedral angles that have edges perpendicular to this direction (a special case) and do not have a support plane perpendicular to this direction. The monotonicity of this conditional full angle for a trihedral angle is proved for the general case.

KEYWORDS
Defect, Conditional Full Angle, Invariant, Conditional Curvature, Monotonicity, Isometry on Sections

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Anvarjon Sharipov , Fayzulla Topvoldiyev , Zokirkhuja Usmonkhujaev , "On One Property of a Conditional Full Angle," Mathematics and Statistics, Vol. 12, No. 5, pp. 420 - 427, 2024. DOI: 10.13189/ms.2024.120503.

(b). APA Format:
Anvarjon Sharipov , Fayzulla Topvoldiyev , Zokirkhuja Usmonkhujaev (2024). On One Property of a Conditional Full Angle. Mathematics and Statistics, 12(5), 420 - 427. DOI: 10.13189/ms.2024.120503.