On the minimum separation between any pair of flight paths

Abstract

The paper presents methods to determine the time, positions, and distance of closest approach for two vehicles following arbitrary trajectories in two or three dimensions. The distance of closest approach of two vehicles following arbitrary curved trajectories is shown to be determined by two conditions: (i) the relative velocity must be orthogonal to the relative position for the distance to be a non-zero extremum; (ii) the radial acceleration including centripetal terms must have a direction that increases the separation for the extremum to be a minimum. This theorem on the distance of closest approach simplifies in the case of rectilinear trajectories and uniform motion. To illustrate the general theory three examples are given: (i) the two-dimensional motion of surface vehicles changing the velocity of one of them so as to enforce a given minimum separation distance; (ii) the three-dimensional motion of two aircraft, one flying horizontally and the other climbing, changing the vertical velocity of the latter to ensure a minimum separation distance set "a priori"; (iii) the case of an aircraft flying with constant velocity on a straight line so that its closest approach to another aircraft flying in a circular holding pattern in the same plane occurs at a given time chosen "a priori".