Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis

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

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

Vyshnyepolskiy¹,

Zavarihina

Peh³

2021

The article deals with the geometric locations of points equidistant from two spheres. In all variants of the mutual position of the spheres, the geometric places of the points are two surfaces. When the centers of the spheres coincide with the locus of points equidistant from the spheres, there will be spheres equal to the half-sum and half-difference of the diameters of the original spheres. In three variants of the relative position of the initial spheres, one of the two surfaces of the geometric places of the points is a two-sheet hyperboloid of revolution. It is obtained when: 1) the spheres intersect, 2) the spheres touch, 3) the outer surfaces of the spheres are removed from each other. In the case of equal spheres, a two-sheeted hyperboloid of revolution degenerates into a two-sheeted plane, more precisely, it is a second-order degenerate surface with a second infinitely distant branch. The spheres intersect - the second locus of the points will be the ellipsoid of revolution. Spheres touch - the second locus of points - an ellipsoid of revolution, degenerated into a straight line, more precisely into a zero-quadric of the second order - a cylindrical surface with zero radius. The outer surfaces of the spheres are distant from each other - the second locus of points will be a two-sheet hyperboloid of revolution. The small sphere is located inside the large one - two coaxial confocal ellipsoids of revolution. In all variants of the mutual position of spheres of the same diameters, the common geometrical place of equidistant points is a plane (degenerate surface of the second order) passing through the middle of the segment perpendicular to it, connecting the centers of the original spheres. The second locus of points equidistant from two spheres of the same diameter can be either an ellipsoid of revolution (if the original spheres intersect), or a straight (cylindrical surface with zero radius) connecting the centers of the original spheres when the original spheres touch each other, or a two-sheet hyperboloid of revolution (if continue to increase the distance between the centers of the original spheres).

Functional-Voxel Modelling of Bezie Curves

Sycheva

2022

The problem of this research is the impossibility of applying parametric functions in theoretical-multiple modeling, that significantly narrows the range of problems solved by analytical models and methods of computational geometry. To expand the possibilities of R-functional modeling application in the field of computer-aided design systems, it is proposed to solve the problem of finding an appropriate representation of parametric curves using functional-voxel computer models. The method of functional-voxel modeling is considered as a computer graphic representation of analytical functions’ areas on the computer. The basic principles and examples of combining R-functional and functional-voxel methods with obtaining R-voxel modeling have been presented. In this case, R-functional operations have been implemented on functional-voxel models by means of functional-voxel arithmetic. Based on the described approach to modeling of theoretical-multiple operations for the function area represented by graphical M-images, two approaches to construction a functional-voxel model of the Bezier curve have been proposed. The first one is based on the sequential construction of the curve’s interior by intersection a positive area of half-planes, which enumeration is performed by De Castiljo algorithm. This approach is limited by the convexity of the curve’s reference polygon. This problem’s solution has been considered. The second approach is based on the application of a two-dimensional function for local zeroing (FLOZ), i.e., a nil segment on the positive area of function values. By consecutive unification of such segments it is proposed to construct the required parametrically given curve. Some features related to operation and realization of the proposed approaches have been described and illustrated in detail. The advantages and disadvantages of described approaches have been highlighted. Assumptions about applicability of proposed algorithms for Bezier curve functional-voxel modeling in solving of various geometric modeling problems have been made.