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

TLDR: In this paper, the authors 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.

Abstract: 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.