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

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

doi: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

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LOCI EQUIDISTANT FROM TWO GIVEN GEOMETRIC FIGURES. PART 5: LOCI EQUIDISTANT FROM A SPHERE AND A PLANE

Journal: GEOMETRY & GRAPHICS Volume 9 № 4 , 2022

Rubrics: SCIENTIFIC PROBLEMS OF GEOMETRY

UDK 51 Математика

Vyshnyepolskiy Vladimir ¹

Zavarihina E. ²

Egiazaryan K. T. ³

Author and publication information

Authors:

1. MIREA — Russian Technological University (Department of Engineering Graphics, Head)

Moskva, Moscow, Russian Federation

2. Moscow Aviation Institute (National Research University) (Senior Lecturer)

Moskva, Moscow, Russian Federation

3. MIREA — Russian Technological University (assistant)
graduate student

Russian Federation

Type:

Article

DOI:

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

Pages:

from 22 to 34

Status:

Published

Received:

08.04.2022

Accepted:

14.04.2022

Published:

14.04.2022

Subject area:

UDK 51 Математика

Language:

Russian

Keywords:

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

Abstract and keywords

Abstract (English):
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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