CONSTRUCTION OF GEODESIC LINES ON SURFACES OF ROTATION OBTAINED BY DISPLACEMENT OF THE MERIDIAN

Abstract

Geodesic surface lines are analogous to straight lines on a plane. In addition to connecting two points of the surface by the shortest distance, they are the winding trajectories of the reinforcing threads in the strengthening of high-pressure cylinders. Just as a bundle of straight lines can be drawn from a given point on a plane in different directions, so there are geodesic lines on a surface that pass through a given point in different directions. Finding geodesic surface lines in the general case comes down to solving second-order differential equations. The purpose of the study is to investigate geodesic lines on the surface formed by the rotation of a given plane curve around a vertical axis and their transformation when this curve is shifted away from or towards the axis. For surfaces of revolution, the second-order differential equation can be reduced to the first order and even reduced to an integral based on the well-known Clerot formula. However, in this case, geodesic lines in all directions can be constructed only for a limited number of surfaces of rotation, and only limited fragments of geodesic lines can be constructed on the remaining surfaces. The article considers the construction of geodesic lines using the numerical solution of a second-order differential equation. The obtained results were visualized.