On smoothness conditions and selection of the edge of a scattering surface in one class of 3D time-optimal problems

A class of time-optimal problems in three-dimensional space with a spherical velocity vectogram is considered. The target set is a parametrically defined smooth curve. Numerical and analytical approaches to constructing the bisector of the target set — the scattering surface — in the time-optimal problem are proposed. The algorithms are based on formulas for the edge points of the scattering surface, written in terms of curve invariants. It is shown that these points form the edge of the bisector and lie at the centers of the touching spheres to the curve. A theorem on sufficient conditions for the smoothness of a scattering surface is proven. The equations of the tangent plane to the bisector are found for those points from which exactly two optimal trajectories emerge. An example of solving a time-optimal problem in the form of a set of level surfaces of the optimal result function is given, highlighting the surface of their non-smoothness.