Two methods of characterizing the visibility of a moving point

Two methods are presented of determining the visibility (observability) of an object moving in space with an obstacle that hinders the motion and the perception of the object by an observer. The first method is based on taking into account the distance from the object to all possible observers. The second method uses not only the distance but also the size of the circular cone with the vertex at the observation point that contains a spherical neighborhood of the object. The directional differentiability of the functions characterizing the visibility of the object is established. The calculation of the derivatives is reduced to an extremal problem, for which “refinement” theorems are given.