Geometric Locations of Points Equally Distance from Two Given Geometric Figures. Part 4: Geometric Locations of Points Equally Remote from Two Spheres

The article is devoted to the annual All-Russian scientific and methodological conference "Problems of Engineering Geometry" and the annual All-Russian scientific and methodological seminar "Geometry and Graphics" in 2021. Statistical information about the conference and seminar is provided: the number of participants, universities, the number of cities and countries in which universities are located -participants. Using the expression proposed earlier, the activity of participation of the departments of graphic disciplines in the conference "Problems of Engineering Geometry" and the seminar "Geometry and Graphics", held in 2021, was determined. The comparison of the number of participants and reports of the conference and seminar in 2021 with the number of participants and reports is given and analyzed International Internet conferences "Quality of graphic training" at the Perm National Research Polytechnic University. The results of the All-Russian Seminars "Geometry and Graphics" and the All-Russian Conferences "Problems of Engineering Geometry" of the last two years are compared with each other. In order to compare conferences and seminars quantitatively, not qualitatively, a relationship has been proposed. The content of the reports of the participants of the conference and the seminar is briefly considered. Conclusions are drawn: 1) in 2021, in terms of the success of the seminar "Geometry and Graphics" and the conference "Problems of Engineering Geometry", we managed to move forward - the success rate increased; 2) judging by the number of reports, scientific work on the profile of the department is carried out in a small number of departments. This is due to shortcomings in the staffing of departments of graphic disciplines by teachers. One of them is a lack of understanding that the winners or participants of All-Russian and regional Olympiads who have undergone appropriate training should work as teachers.

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All-Russian Scientific and Methodological Conference «Problems of Engineering Geometry» and the Seminar «Geometry and Graphics» 2021

Vyshnyepolskiy

Kadykova

Vereschagina

2022

Geometry &Amp; Graphics

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Building a Sphere from Imaginary Points

Girsh

2022

Geometry &Amp; Graphics

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Euclidean spaces of various dimensions do not contain imaginary images and objects by definition, but are inextricably linked with them through special cases, and this leads to the need to expand the field in geometry into the region of imaginary values [1, 19, 26]. Such an extension, i.e. adding to the field of real coordinates spaces of different dimensions, the field of imaginary coordinates leads to different variants of spaces of different dimensions, depending on the chosen axiomatics. Earlier in a number of articles, examples of solving some actual problems of geometry using imaginary geometric images and objects were shown [4, 5, 6, 13, 21, 22, 29]. The article provides constructions for constructing a sphere from four predetermined points, of which one pair or both pairs of points can be imaginary complex conjugate. The construction is carried out on combined diagrams by the methods of descriptive geometry by analogy with the well-known problem of constructing a sphere from four real points. The construction of a sphere is based on seven auxiliary constructions for constructing a circle from points that can be imaginary conjugates. Both 3D problems of constructing spheres for given points and methods of 2D construction problems for determining the required imaginary points are considered. A method for calculating the parameters of the obtained sphere is described. The application of the method to other problems of descriptive geometry, for example, to the problems of finding geometric places of points, is considered. equidistant from two given surfaces. Recently, this issue has been intensively studied, for example, in the works [5, 6].

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

Vyshnyepolskiy¹,

Zavarihina

Egiazaryan³

2022

Geometry &Amp; Graphics

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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.

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Geometric Locations of Points Equally Distance from Two Given Geometric Figures. Part 4: Geometric Locations of Points Equally Remote from Two Spheres

Cited by 8 publications

References 14 publications

All-Russian Scientific and Methodological Conference «Problems of Engineering Geometry» and the Seminar «Geometry and Graphics» 2021

All-Russian Scientific and Methodological Conference «Problems of Engineering Geometry» and the Seminar «Geometry and Graphics» 2021

Building a Sphere from Imaginary Points

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

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