Functional-Voxel Modelling of Bezie Curves

The article describes the model of group pursuit of a single target by the chase method. All objects participating in the pursuit model move with a constant modulo speed. One of the participants in the process moves along a certain trajectory and releases objects at specified intervals, the task of which is to achieve the goal by the chase method. All objects have restrictions on the curvature of the motion path. A single target, in turn, is tasked with achieving the target that releases objects using the parallel approach method. For each pursuing object, a detection area is formed in the form of two beams. The object's velocity vector is directed along the bisector of the angle formed by such rays. If the target enters the detection area, then the object starts pursuit and the velocity vector is directed to the target. If the target leaves the detection area, then the object makes a uniform and rectilinear movement. The task is to implement a dynamic model of multiple group pursuit, where each object has its own tasks, implemented by the chase method. As an example, where the model developed in the article could be in demand, the following example can be given. Consider the movement of a low-maneuverable object that is overtaking a faster target. As a means of protection, instead of releasing passive heat traps, it is proposed to drop a variety of autonomously controlled weapons. An analysis of existing studies has shown that such means of protecting aircraft do not exist. The results of the research can be in demand in the design of unmanned aerial vehicles with elements of autonomous control and artificial intelligence.

Geometric Model of Group Pursuit of a Single Target by the Chase Method

Dubanov

2022

Review of the conclusion of a land surveyor expert as evidence in a lawsuit

Beglov

2023

. This article presents the results of a study of the geometric properties of the Nicomed conchoid and the oblique conchoid. In this paper, the oblique conchoid is modeled in a new way, namely by quasi-symmetry with respect to the elliptic axis. The method used is a fourth-order transformation of the plane relative to the second-order curve. That is, a straight line with quasi-symmetry is mapped into a fourth-order curve. The image of a straight line in this case consists of two branches that tend to two asymptotes. Quasi–symmetry makes it possible to obtain an oblique conchoid, as a special case under certain conditions, and in the general case, many other conchoidal curves. The use of this method made it possible to discover new geometric properties of conchoidal curves, in particular, to find a previously undescribed constructive correspondence between points belonging to different branches of the oblique conchoid. The paper formulates and proves three statements, namely: 1) The image of a straight line with its quasi-symmetry with respect to a circle is a Nicomedes conchoid, 2) the image of a circle with its quasi-symmetry with respect to a circle is a curve of the sixth order, 3) the image of a straight parallel major semiaxis of an ellipse with its quasi-symmetry with respect to a given ellipse is two symmetrical oblique conchoids with respect to the minor semiaxis of an ellipse. Also, the equations of the curves under consideration and their asymptotes in the general case are derived. The results of the research carried out in this paper expand the possibilities of using conchoidal curves in solving problems of engineering geometry. For example, when modeling various physical phenomena and processes, as well as in engineering and architectural design.

Integration of Three-Dimensional Reflections from Curved Mirrors Using Computer Algebra Tools

Suncov,

Zhikharev,

Efremov

2024

This article is a continuation of the study of the process of reflection of various objects from curved mirrors. So, earlier in the works [18; 20], a geometric method of constructing the results of reflections was described, which was implemented mathematically in the article [38] using the principles of analytical geometry [6; 11–14; 30]. The obtained analytical equations of the reflection results were visualized in the Wolfram Mathematica [24] program with the ability to dynamically change the parameters of the mirror and the reflected object. However, in the listed works, only cases of reflection on the plane were considered. In this study, attention is paid to a more complex case — reflection in three-dimensional space. The article considered the reflection of a point from surfaces of the second order: a cylinder, a cone, a single-cavity and double-cavity hyperboloids, a sphere, elliptical and hyperbolic paraboloids, and from a torus — a surface of the fourth order. As before, the reflection result obtained in each of the cases is accompanied by a program code for Wolfram Mathematica, which allows the reader to independently simulate the reflection process with different initial parameters. In addition, the relationships between the results obtained were analyzed — both the relationships between the results of various three-dimensional reflections, and the relationship of the results of three-dimensional reflections with the results of similar plane reflections. In particular, on the basis of this, a hypothesis was formulated about the relationship between the curvature of the Gaussian mirror and the dimension of the object obtained as a result of reflection. Based on the results of the work, conclusions were drawn and prospects for further research were outlined. One of them is to obtain an analytical mechanism for describing complex geometric surfaces using a set of simpler objects. This feature will increase the efficiency of specialists when working with reflections from complex surfaces in areas such as aircraft construction (for creating aerodynamic surfaces and air ducts), medicine [40], shipbuilding [7; 31; 42], etc.