Demonstration of Common Elements of Involution on a Simple Example

Umbetov, N.

doi:10.12737/2308-4898-2022-10-2-27-34

“…In our time, there is an understanding that the situation must be urgently corrected. Work on the subject of constructive geometric modeling continues [1,2,5,6,[36][37][38][39][40][41][42][43][44][45][46][47]. Relentless attempts are being made to revive geometric knowledge in higher education.…”

Section: Introductionmentioning

confidence: 99%

Problems of Graphic Design of the Results of a Geometric Experiment

Boyashova,

Musaeva,

Voloshinov

2023

Proceedings of the 33rd International Conference on Computer Graphics and Vision

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The article is devoted to the problems of graphical representation of drawings for purposes of interactive automated constructive geometry modeling. The authors demonstrate the necessity for studying principles of models creation to develop special informational technologies for visualizing, interpretation and processing of geometrical data. The article presents a retrospective review of methods for displaying geometric information, adopted among specialists in constructive geometric modeling. It is shown the importance to support scientific research in a field of geometry modelling, especially for multidimensional space for science, education and applied tasks implementation. The article formulates the goals and objectives of the further development of the system of constructive geometric modeling Simplex.

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Building a Sphere from Imaginary Points

Girsh

2022

Geometry &Amp; Graphics

2

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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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Building a Sphere from Imaginary Points

Girsh

¹

2023

Geometry &Amp; Graphics

0

View full text Add to dashboard Cite

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].

show abstract

Demonstration of Common Elements of Involution on a Simple Example

Cited by 4 publications

References 9 publications

Problems of Graphic Design of the Results of a Geometric Experiment

Problems of Graphic Design of the Results of a Geometric Experiment

Building a Sphere from Imaginary Points

Building a Sphere from Imaginary Points

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