Visualization of geodesics on a weekly deformed sphere
"Математика. Компьютер. Образование". Cб. трудов XII международной конференции. Под общей редакцией Г.Ю. Ризниченко Ижевск: Научно-издательский центр "Регулярная и хаотическая динамика", 2005. Vol. 2, 466pp. Pp. 606-615.
We aim at visualizing geodetic lines on a surface, which is close to the standard sphere. We offer a means of the asymptotic representation of geodesics based on the fact that a short segment of the geodesic can be well approximated by an appropriate “great circle” of the sphere. We apply the method to a surface generated by a specific deformation of the sphere. We obtain a graphic description of geodesics with the help of an auxiliary Hamiltonian system. The main point about the auxiliary system is that it is much simpler than the initial one for geodesics, and even admits of the exact solution. The topology of solutions to the system is described by the separatrix graph; the vertices corresponding to geodesics close to great circles and the edges (separatrixes) to geodetic coils joining them. The variation of parameters determining the deformation of the sphere generates several topological types of the graphs.