Construction of geodesics circle for surfaces of revolution of constant Gaussian curvature

The investigation of geodesic lines is connected with the need to solve a system of nonlinear differential equations. For surfaces of revolution this system reduces to a single differential equation of the second order. The work is devoted to the construction of geodesic lines for surfaces of revolution of constant Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minging top, the Minding coil, the pseudosphere (Beltrami surface). There are also three types of surfaces of constant positive Gaussian curvature. The studied surfaces and their geodesics are described by means of elliptic integrals. Using a mathematical package, the surfaces of rotation of constant Gaussian curvature and their geodesics are constructed.