Any two planes, except those that are parallel to each other, will mutually intersect along a straight line. Figure 13a shows two planes (plane 1 and plane 2) and their line of intersection, L. When these planes are each shown as passing through the centre of a sphere (Fig. 13b), their line of intersection L is seen to correspond to the line coming from the centre of the sphere to the intersection point of the great circles for the two planes.

The line of intersection of any two planes is therefore found stereographically by plotting the two planes as great circles. The point where the great circles cross each other is the stereographic projection of the line of intersection. The plunge and plunge direction of the line of intersection are then obtained by following, in reverse, the procedure given on p. 22.

For example, if two limbs of a chevron-style fold are measured (Fig. 13c), the orientation of the hinge line, h, is readily calculated from the intersection of the two great circles representing the limbs (Fig. 13d). In Figure 13d plotting the great circles for the limbs of a fold with orientations 101/50N and 065/60S yields a line of intersection (the fold hinge line) which plunges 23° in direction 080°.

Potential applications of this construction are numerous. Figure 13e and 13f show how measurements of bedding and cleavage planes at an outcrop allow the determination of the plunge and plunge direction of L, the bedding–cleavage intersection lineation.