Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane

Vladimir Vyshnyepolskiy; E. Zavarihina; K. T. Egiazaryan

doi:10.12737/2308-4898-2022-9-4-22-34

UDC 51

LOCI EQUIDISTANT FROM TWO GIVEN GEOMETRIC FIGURES. PART 5: LOCI EQUIDISTANT FROM A SPHERE AND A PLANE Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane

АА Аа

Published in Geometry & Graphics · Volume 9, Issue 4 · Pages 22–34 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/2308-4898-2022-9-4-22-34

Received: 07.04.2022 Accepted: 14.04.2022 Published: 14.04.2022 Language of publication: ru-RU

Authors

Vladimir Vyshnyepolskiy

E. Zavarihina

K. T. Egiazaryan

Vladimir Vyshnyepolskiy

Academic degree: candidate of technical sciences

Science rang: docent

MIREA — Russian Technological University Head · Department of Engineering Graphics · Russian Federation · Moscow

E. Zavarihina

Moscow Aviation Institute (National Research University) Senior Lecturer · Russian Federation · Москва

K. T. Egiazaryan

MIREA — Russian Technological University assistant · Russian Federation · Moscow

ABSTRACT

In this paper have been investigated the loci equidistant from sphere and plane, and properties of obtained surfaces have been studied. Four options for possible mutual arrangement of plane and sphere have been considered: the plane passes through the center of the sphere; the plane intersects the sphere; the plane is tangent to the sphere; the plane passes outside the sphere. In all options of the mutual arrangement of the sphere and the plane, the loci are two surfaces - two coaxial confocal paraboloids of revolution. The general properties of the obtained paraboloids of revolution have been studied: foci and vertices positions, axes of rotation, the distance from the sphere center to the vertices of the paraboloids, the distance between the vertices of the paraboloids, and the position of the directorial planes have been defined. Have been derived equations for the surfaces of the loci equidistant from the sphere and the plane: various paraboloids of revolution. The loci in each of the four options for the possible mutual arrangement of the plane and the sphere are as follows. 1. The original plane passes through the sphere center – two coaxial confocal multidirectional paraboloids of revolution symmetric relative to the original plane. 2. The initial plane intersects the sphere – two coaxial confocal multidirectional but not symmetrical paraboloids of revolution, since the circle of intersection of the plane and the sphere does not coincide with the diameter of the sphere great circle. 3. The plane is tangent to the sphere – a paraboloid of revolution and a straight line (more precisely, a second order zero-quadric – a cylindrical surface with zero radius) passing through the tangency point of the plane and the sphere and the sphere center. 4. The plane passes outside the sphere – the equidistant loci will be two coaxial confocal unidirectional paraboloids of revolution.

KEYWORDS

geometry; descriptive geometry, loci, locus, analytical geometry, line, sphere, paraboloid of revolution

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JATS XML

Zamestitel' glavnogo redaktora, chlen redsoveta

Vladimir Vyshnyepolskiy

Glavnyy redaktor

A. A. Ishchenko

Vypuskayuschiy redaktor

Dar'ya Sklyankina

501@infra-m.ru

Deputy editor-in-chief, member of the editorial Board

N. Sal'kov

nikolaysalkov@mail.ru

GOST

Vyshnyepolskiy V. I., Zavarihina E. V., Egiazaryan K. T. Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane // Geometry & Graphics. 2022. no. 4. pp. 22-34. DOI: https://doi.org/10.12737/2308-4898-2022-9-4-22-34. (Date of access 11.11.2026).

MLA

Vladimir Vyshnyepolskiy, Zavarihina E. V., Egiazaryan K. T.. "Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane." Geometry & Graphics 9.4 (2022): 22-34. DOI: https://doi.org/10.12737/2308-4898-2022-9-4-22-34.

APA

Vyshnyepolskiy V. I., Zavarihina E. V., Egiazaryan K. T. (2022). Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane. Geometry & Graphics, 9(4), 22-34. https://doi.org/10.12737/2308-4898-2022-9-4-22-34

Chicago

Vladimir Vyshnyepolskiy, Zavarihina E. V., Egiazaryan K. T.. "Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane." Geometry & Graphics 9, no. 4 (2022): 22-34. https://doi.org/10.12737/2308-4898-2022-9-4-22-34.

WIKI

{{cite journal | author = Vyshnyepolskiy V. I., Zavarihina E. V., Egiazaryan K. T. | title = Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane | journal = Geometry & Graphics | year = 2022 | volume = 9 | issue = 4 | pages = 22-34 | doi = 10.12737/2308-4898-2022-9-4-22-34 }}

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LOCI EQUIDISTANT FROM TWO GIVEN GEOMETRIC FIGURES. PART 5: LOCI EQUIDISTANT FROM A SPHERE AND A PLANE Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane

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