A note on Three-dimensional pursuit-evasion game with bounded curvature

The system of kinematic equations which describes Isaacs' Game of Two Cars has been extended to the case of a three-dimensional pursuit-evasion game between players with bounded curvature. It is shown that in the three-dimensional version of the Homicidal Chauffeur game, both players, when playing optimally, strive to steer the system into a common plane and will remain on this plane until game termination. The situation is somewhat different for the three-dimensional version of the Game of Two Cars, where it is not always advantageous to steer an initially three-dimensional dynamic system into a plane. Nevertheless, if at any instant the velocity vectors and the line of sight between players are coplanar, both players strive to remain on this plane.