Previously, we developed a constructive method for modeling surfaces of rotation with axes, which were second-order curves such as circle, ellipse, parabola and hyperbola [1]. We also described the principle of constructing a mathematical model [23] corresponding to this constructive technique [2], and expressed the method in mathematical form. In this paper, we applied the previously developed mathematical model that allows us to determine the trajectory of rotation of a point around an elliptical axis to some special cases of the location of this point and identified the features of each of them. We applied the previously accepted terminology and the system of designating points, straight and curved lines involved in the search for circular trajectories of rotation of points. We analyzed the cases of the location of the generating point on the coordinate axes. We determined in mathematical form the trajectory of the point located in these positions. This entry is represented as systems of parametrically given equations. The article also describes a step-by-step algorithm used to find the equation of a circle, which is the trajectory of rotation of a point around an elliptic axis. We applied this algorithm to various positions of the generating point relative to the elliptic axis foci. We applied the previously developed criteria for selecting near and far centers of rotation relative to one of the focuses of the ellipse. The results of these mathematical studies will be used in the future to create a computer program capable of generating digital 3D-models of surfaces formed by the rotation of arbitrary sets forming points around the curves of the axes of the second order.

A Mathematical Description of the Rotation of a Point Around an Elliptic Axis in Some Special Cases

Based on the published data, the essence of computational graphics has been laid down. Have been reported examples of new results obtained only through accurate computer constructions and measurements. The work content is a presentation of new ways to achieve the goal by solving non-traditional engineering problems. The author's method of projection with time-stamps, which, in fact, is a computer descriptive geometry, allows solve multi-parameter (not to be confused with multi-dimensional) problems with 9 variables [1–3; 13]. The author’s method of two-axis equal-sized evolvement [11; 12; 17] allows quantitatively measurements of solid angles. The addition of trigonometric functions (sinuses, sinusoids, etc.) can also be considered as a novelty [10; 11; 14]. At the junction of analytic (AG) and descriptive geometries have been calculated parameters of dodecahedron and has been given its mathematical description. In the traditional AG task, the required parameters have been calculated graphically, including a point’s speed of movement. Has been presented the author’s method for determining the instantaneous center in theoretical mechanics. For the first time, the equality of the angles of rotation for points and the link as a whole has been established, and a continuous centroid has been built. By decomposition of vectors a new way for summing up theirs vertical projections has been demonstrated. The developed method of projections with time-stamps allows simultaneously consider such parameters as spatial coordinates of moving objects (two or more) in time, their speeds and even sizes, including the variable ones. Has been shown the possibility for graphical programming while solving systems of equations, as well as for graphical solution of algebraic and stereometric problems. This publication aims to disseminate computer methods for engineering problems solving.

Computational Graphics in Solving of Non-Traditional Engineering Problems

Application of the Dupin cyclide in temple architecture

In this research is solved the problem for determining of dependencies describing the strength redundancy of a part obtained by means of topology optimization using the SIMP method under a variety of grid’s finite elements. For this purpose, in the research was performed a digital experiment, during which almost fifty variants of part’s computer models were obtained, and their mechanical properties were studied. Based on the obtained data were constructed plots for the strength efficiency of topological optimization, which reflect fractal properties of part’s strength parameters changing. \nUpon reaching the research goal were solved the problems of software selection and applying a programs combination, which allowed automate the creation of models based on the topology optimization results. The main tool for topology optimization was the Autodesk Fusion 360 product, providing a free access to cloud computing, and Autodesk ReCap Photo was used when models converting. \nOn the results of the experiment were formulated recommendations for obtaining the part’s optimized topology without critical defects of shape, using the SIMP method. With great probability, these recommendations are important when using other methods for topological optimization, such as ESO, BESO, or Level-Set. \nThe received recommendations were tested in solution the problem of increasing the structures’ strength efficiency on the example of the rocker-Bogie wheel suspension using in modern Curiosity-type Mars rovers. The topology optimization results are openwork parts that can withstand heavy loads at low weight. This was confirmed by strength analysis, which had showed an increase in specific strength up to 13.5 times, relative to the prototype used in the Curiosity-type Mars rover’s suspension.

Fractal Plots of Topology Optimization Efficiency in Solving of the Problem for Strength Dependence on the Grid

The background of description of the method for geometric objects rotation around the curved axes in the form of curves of the second order is being considered. The above-described method got the name of "quasi-rotation". The algorithm of geometric construction of the two-two-digit interrelation on the basis of the circular axis of symmetry is provided. The described interrelation is analogous to the mirror symmetry regarding a straight linear axis. The method is suitable for construction of some well-known algebraic curves. The analogies between rotation and quasi-rotation are determined and described. The formula of the same plane preimage-to-image interrelation, generated by the quasi-rotation around the curved axis of the second-order curve is provided. The presented formula allows to derive the equations that effectively describe images of the given sets. The method for geometric construction of the images of geometrical objects lying within the plane of the second-order curve at their quasi-rotation for a given angle regarding the curve is described. The obtained results describe geometrical basics for the quasi-rotation. They serve as the basis for shape-forming of algebraic surfaces of high orders, and can be utilized as the theoretical basis for the computer-aided automated geometric projection.

N-n-digit interrelations between the sets within the R 2 plane generated by quasi-rotation of R 3 space

Method of Rotation of Geometrical Objects Around the Curvilinear Axis

Previously, the method of rotating of flat geometric objects around curvilinear axes was described by us. The next step in the path of our research should be the development of methods for the automated creation of surfaces digital models obtained by the described rotation method. We have created models of surfaces, the axis and the forming curve of which are circles lying in the same plane. Several cases of mutual disposition for such circles were analyzed. Modeling was carried out using constructive techniques. Surfaces were created using the “surface by section” operation. The centers of such circular sections belong to the axis of rotation, if it is a circle. Using the special tools incorporated in the KOMPAS-3D program, we have cut the surfaces modeled in this way by planes, and obtained a number of flat sections. Taking into account the difficulties occurring during the study of such complex geometric objects by means of flat graphic constructions, as well as graphic computer modeling, we have realized the need to create a mathematical apparatus describing these objects’ shape. The required mechanism should be applicable to any pair of second-order curves interconnected as “axis — generatix”. We have considered an elementary example – the rotation of a point around a curve elliptical axis. In this paper a solution for the problem of finding a system of equations describing a set of point positions, which it will successively take when rotating around the elliptic axis, is presented. It is possible to apply a similar mathematical apparatus to axes having the form of other quadrics, for example, hyperbolas or parabolas, as well as to generatices consisting of more than one point, that is, to forming curves.

Mathematical Interpretation for a Method of Rotation of a Point Around a Second Order Curved Axis

Descriptive Geometry in Modern Education

The present paper details an approach to modeling of quasi-rotation surface segments with pre-defined conditions. The desired surfaces and their segments are formed parametrically through Maple computer algebra system. Various analytical approaches to trimming of the excessive parts of quasi-rotation surfaces are indicated. Geometric modeling is performed with use of a previously developed algorithm. An example illustrating the initial stage of design of an overpass between two separate buildings is provided. The design objective contains the basic geometric parameters of the desired architectural element. Three control models were created with the same initial parameters through basic CAD software methods. All of the constructed models are thin-walled. Mechanical properties of the scale 3D models of the target element were compared in equal conditions by means of CAD software. The prototype created through the use of a quasi-rotation surface segment was the most resistant to load. The paper also features a rendered concept of a complex of three towers connected with three overpasses. The acquired results allow us to conclude that quasi-rotation is an effective method of formation of models of architectural elements.

https://doi.org/10.1088/1742-6596/2182/1/012002

I. Beglov

Papers

Method of Rotation of Geometrical Objects Around the Curvilinear Axis

Mathematical Interpretation for a Method of Rotation of a Point Around a Second Order Curved Axis

A Mathematical Description of the Rotation of a Point Around an Elliptic Axis in Some Special Cases

Descriptive Geometry in Modern Education

Application of quasi-rotation surface segments in architectural prototyping