On the method of packing geodesic circles into a spherical segment using a plane projection

The article discusses the problem of finding the optimal location for a given number of equal geodesic circles without overlapping on a spherical segment. The optimality criterion is to maximize the radius of the circles. This formulation is known as the Tammes problem, and it is a variant of the classical densest packing problem of equal objects in a specified container. We propose a new two-stage numerical method to solve the issue under consideration. The first stage is the construction of an initial approximation using the best-of-known optimal packings of equal circles into a circle of a larger radius, which centers are projected onto a spherical segment in a specific manner. A theorem on the properties of projection is proved, establishing a connection between the radii of elements of plain and spherical packings. The second stage is the improving procedure based on the billiard modeling. The article presents illustrative numerical calculations for spherical segments of different angular sizes. The comparison of results obtained with known ones shows that the proposed method allows finding optimal or close-to-optimal packings in significantly less time than when using the traditional multistart method. Besides, an applied problem of designing a spherical focal surface, which arises in engineering optics, is solved.