graduate student
Saint-Petersburg, St. Petersburg, Russian Federation
The change in the position of a point P1, involutional related to a fixed point P, is investigated in response to a change in one of the conics defining a projective involution on the plane. The involution under consideration is defined by a pair of conics K1, K2, whereby conic K1 causes a continuous transformation due to consideration of the position of one of its support points E, while conic K2 remains unchanged. The purpose of the work is a detailed analysis of the geometric trajectory of point P1 under various modes of motion of point E, as well as the manifestation of patterns characterizing the stability and typology of the resulting changes. The research methodology is based on a combination of classical constructive methods of projective theory, analytical descriptions of the corresponding transformations, and numerical experiments with visualization of results in the Python/Matplotlib environment. Approaches to digital visualization, computational modeling, and interactive support for graphical research, which are widely presented in [7; 8; 14; 20], are applied. In this study, three scenarios for the motion of point E are proposed: linear, curvilinear, and parametrically controlled. This allows us to track the influence of different types of conic deformation on the motion of the conjugate point. It is established that the trajectory P1 is continuous and piecewise smooth, and that small displacements of points E result in quasilinear behavior close to an affine dependence. The key situations that arise during the emergence of conic K1 are identified, leading to abrupt changes in the direction of motion of point P1, which leads to the emergence of a trajectory and a change in its topological type. The observed functional relationship between the positions of points E and P1 changes the algebraic transformations of the degree of reduction, which is consistent with the hypothesis of its projective nature and complements the subsequent analytical description. The obtained results have practical implications for optimizing geometric calculations in CAD systems, increasing the stability of computer vision algorithms, correcting projective manipulation algorithms, and developing fast geometric modeling methods for engineering graphics and computational geometry.
projective geometry, connective point, conic, deformation of a conic, algebraic value, trajectory of points, principal points
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