The Mathematics of Collision Avoidance in Two Dimensions

Abstract

The geometry of collision at sea has been dealt with in a series of papers published in the Journal, notably by Sadler (10, 306), Calvert (13, 127), Garcia-Frias (13, 316) and Morell (14, 163); and a further contribution from Calvert (to which reference is made in this paper) will be published in the next number.The object of the present paper is to examine whether anti-collision manœuvres, here considered for the case of two craft moving in a plane, can be formulated on a rigorous logical basis. If they can, then clearly a proper appreciation of the geometry of collision is a prerequisite to the formulation of any rules and regulations. In this important paper the author gives a precise and complete answer to the basic problem, and he proves mathematically that it is the only answer. As Dr. Hollingdale freely acknowledges, however, this can only be a contribution to the study of the collision problem, which involves innumerable operational factors in addition to the geometry of the situation.The convention adopted in the present paper is that each craft shall manœuvre so that if the other craft stands on, the sight line always rotates in the anti-clockwise direction. The analysis shows that a simple set of manœuvres can in fact be developed on this basis and that such manœuvres are the only ones that, geometrically, meet all the specified requirements. In all cases the combined manœuvre converts a collision situation into a ‘miss’ of at least a specified magnitude.