Four-Dimensional Space’s Hypersurface Mapping Singularity

The problem for construction of straight lines, which are tangent to conics, is among the dual problems for constructing the common elements of two conics. For example, the problem for construction of a chordal straight line (a common chord for two conics) ~ the problem for construction of an intersection point for two conics’ common tangents. In this paper a new property of polar lines has been presented, constructive connection between polar lines and chordal straight lines has been indicated, and a new way for construction of two conics’ common chords has been given, taking into account the computer graphics possibilities. The construction of imaginary tangent lines to conic, traced from conic’s interior point, as well as the construction of common imaginary tangent lines to two conics, of which one lies inside another partially or thoroughly is considered. As you know, dual problems with two conics can be solved by converting them into two circles, followed by a reverse transition from the circles to the original conics. This method of solution provided some clarity in understanding the solution result. The procedure for transition from two conics to two circles then became itself the subject of research. As and when the methods for solving geometric problems is improved, the problems themselves are become more complex. When assuming the participation of imaginary images in complex geometry, it is necessary to abstract more and more. In this case, the perception of the obtained result’s geometric picture is exposed to difficulties. In this regard, the solution methods’ correctness and imaginary images’ visualization are becoming relevant. The paper’s main results have been illustrated by the example of the same pair of conics: a parabola and a circle. Other pairs of affine different conics (ellipse and hyperbola) have been considered in the paper as well in order to demonstrate the general properties of conics, appearing in investigated operations. Has been used a model of complex figures, incorporating two superimposed planes: the Euclidean plane for real figures, and the pseudo-Euclidean plane for imaginary algebraic figures and their imaginary complements.

Dual Problems with Conics

Girsh

2020

New Problems of Descriptive Geometry

Girsh

2020

Complex geometry consists of Euclidean E-geometry (circle geometry) and pseudo-Euclidean M-geometry (hyperbola geometry). Each of them individually determines an open system in which a correctly posed problem may give no solution. Analytical geometry is an example of a closed system, in which the previously mentioned problem always gives a solution as a complex number, whose one of the parts may turn out to be zero. Development of imaginary solutions and imaginary figures is a new task for descriptive geometry. Degenerated conics and quadrics set up a new class of figures and a new class of descriptive geometry’s problems. For example, a null circle, null sphere, null cylinder, and a cone as a hyperboloid degenerated to an asymptote. The last ones necessarily lead to imaginary solutions in geometric operations. In this paper it has been shown that theorems formulated in one geometry are also valid in conjugate geometry as well, while the same figures of conjugated geometries visually look different. So imaginary points exist only by pairs, the imaginary circle is not round one, the centers of dissimilar circles’ similarity do not belong to the centerline and other examples. For solution, a number of problems on geometric relations, and operations with degenerated conics and quadrics, as well as several problems from 4D-geometry are proposed. Solutions for above mentioned problems are given in section 9. In this paper some examples of new problems for descriptive geometry have been considered. It has been shown that the new problems require access to a complex space. New figures consist of two parts, a real figure and a figure of its imaginary complement.

Geometric Modeling of Objects’ Thermal Characteristics by the Functional-Voxel Method

Plaksin¹,

Pushkarev

2020

In this paper the influence of objects’ thermal processes on their correspondence to a given geometry has been considered, and an alternative apparatus for geometric modeling of bodies’ temperature stress and thermal expansion after effect of a heat source, based on a functional-voxel approach, has been proposed as well. A discrete geometric model of temperature stress at a point of thermal loading in an isotropic heat-conducting body for a functional-voxel representation has been developed, allowing simulate a single action of a heat source to obtain local geometric characteristics of thermal stress in the body. This approach, unlike traditional approaches based on the FEM, allows apply the temperature load at the object’s point taken by itself. A discrete geometric model for expansion at the point of thermal loading in an isotropic heat-conducting body for a functional-voxel representation has been developed, which allows simulate the change of an object’s local geometric characteristics during the process of material expansion from a single effect of a heat source to obtain a value upon the body volume changing. This approach, unlike traditional approaches based on the FEM, allows simulate a change in the body’s surface geometry from thermal expansion at a point taken by itself without errors arising from calculations using a mesh. Have been proposed algorithms for functional-voxel modeling of temperature stress and expansion under distributed thermal loading. These algorithms allow construct a loading region of complex configuration based on the spatial distribution and scaling of the temperature stress’s geometric model for a single point of thermal loading, uniformly form a contour (surface) after material expansion, and obtain information about the change in products’ length (volume) based on information about each point of functional space. Has been presented an example of using the proposed approach for solving a processing tool’s correction problem based on the temperature in the cutting zone and material thermal reaction. The geometric model can be used to the automated design of a processing tool path for parts cutting on CNC machines.