Some properties of the curve of constant bearing

If Z is a fixed point on the surface of the earth (assumed spherical) and P is the North Pole, then the locus of a point X, which moves in such a way that the angle a between the great circle arcs PX, ZX is constant, is called a curve of constant bearing. a is measured clockwise from XP; it is then the great circle bearing of Z from X as defined in navigation, and lies within the range 0° to 360°. Curves of constant bearing are of some importance in navigation because, if a ship or aircraft at X takes a bearing of a radio station at Z, the position line so obtained is an arc of such a curve. Nevertheless, few properties of the curves seem to be recorded; the reason is probably that practical navigators are interested not in the actual curves in their entirety but in the projections on a Mercator chart of comparatively short lengths of them. In this note some simple properties of the curves are obtained; the derivation of the results is very straightforward and, needless to say, no originality is claimed. We begin by writing down the equation of a curve of constant bearing. Let the latitudes of X and Z be , 0 respectively, and let the meridian of Z be that of zero longitude; the longitude A of X is considered as positive or negative according as it is Easterly or Westerly. Since, by convention, the angles of a spherical triangle cannot exceed 180°, two cases a < 180° (fig (i)) and a > 180° (fig. (ii)) must be considered. Then, in both cases, by the Four Part Formula of Spherical Trigonometry, the equation of the curve of constant bearing is