Imaginary Straight Lines in Cartesian Coordinate System

In this article, the study of the geometry of the flat shapes reflection from curved lines located in the plane of these shapes continues. The paper is devoted to a more detailed description of reflection from the analytical geometry point of view. In addition, the range of proposed tasks has been significantly expanded. An algorithm for reflecting zero-dimensional and one-dimensional objects from plane curves is compiled, and corresponding illustrations are given. For the first time, the authors have obtained equations that allow us to construct reflections of a point from second-order curves: a circle, an ellipse, a parabola and a hyperbola, as well as from high-order curves: Bernoulli lemniscates and cardioids [17], [24], [13], [25], [23], [22]. In addition, equations for the reflection results of one-dimensional objects: a segment and a circle, from the same plane curves were obtained. Similar studies are being conducted in the works [2], [1], [32], [28], [3], [4]. All equations are accompanied by blueprints of special cases of reflections obtained using the Wolfram Mathematica mathematical package [18], [19]. In addition, the application contains the source codes, which gives the reader to configure the reflection parameters themselves on condition having access this program, as well as visually assess the change in the reflection pattern when changing these parameters for various types of flat mirrors. This article demonstrates the possibilities that the obtained equations provide, and the prospects for further work, which consist in obtaining new equations of objects reflected from other flat mirrors.

Investigation of Reflection from Curved Mirrors on a Plane in the Wolfram Mathematica

Suncov

Zhikharev

2021

Building a Sphere from Imaginary Points

Girsh

2022

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].

New Problems of Descriptive Geometry. Continuation

Girsh

2022

Complex geometry is a synthesis of Euclidean E-geometry (circle geometry) and pseudo-Euclidean M-geometry (hyperbola geometry). Each of them individually defines a non-closed system in which a correctly posed problem may not give a solution. Analytical geometry represents a closed system. In it, a correctly posed problem always gives solutions in the form of complex numbers, for each of which, one of the parts may be equal to zero. Finding imaginary solutions and imaginary figures formed by a set of such solutions is a new problem in descriptive geometry. Degenerated conics and quadrics, or curves and surfaces of higher orders, constitute a new class of figures and a new class of problems in descriptive geometry. For example, null-circle, null-sphere, null-cylinder, null-torus. In this paper the problem for studying the shape of second (conics, quadrics), third (conoid), and fourth (torus) order figures is posed. The latest suggest a meeting with new geometric properties of figures. Geometric operations are still immersed in the complex space E + M or real - imaginary. The examples under consideration continue a series of degenerated figures in which the null-circle splits into isotropic lines. Isotropic lines are taken as generators of surfaces. They form a cone of revolution and a hyperbolic paraboloid (an oblique plane).