APPROXIMATE CONSTRUCTION OF GEODESIC LINES ON ROTATION SURFACES

Abstract

At present, geodesic lines attract the attention of many scientists. In theoretical studies, the geodesic line plays an important role as a line of the internal geometry of the surface. In practice, the properties of geodesic lines are used to determine the shortest distances or reinforce covering and so on. When two surfaces are conjugated, their contact line is also a geodesic. This property can be used to design conjugate cyclic helical surfaces that occur, for example, in Novikov gears. The search for geodetic lines on non-expandable surfaces is quite complicated. In analytical terms, this problem is reduced to the making and solving of differential equations, the explicit solution of which can only be found in rare cases. The authors propose a graph-analytical method for finding geodesics on a surface of rotation of a general form using sweeps. The accuracy of this method directly depends on the accuracy of constructing conditional (approximate) sweeps of non-expandable surfaces of rotation. In this case, it is advisable to use the method of constructing scans using the integral calculus. The proposed graph-analytical method for finding geodesics can be used for other surfaces. Approximate geodesic lines on surface using sweep can be constructed in two ways: in the first, the statement of the Clareau theorem is taken into account, and in the second, segments of the conditional approximate sweep of the surface are connected so that a continuous straight line can be drawn. For a detailed description of the two methods, two surfaces are considered - a paraboloid of rotation and a catenoid. It is possible to determine the general appearance of geodesic lines on surfaces without constructing their sweeps, and use only the principle of constructing geodesic lines on it. The approximate geodesic line is a broken line, the length of which is easy to calculate, knowing the necessary geometric dimensions of the surface. The proposed graph-analytical method for determining geodetic lines is quite simple. It is easy to apply to general surfaces of rotation or “corrugated” surfaces.