About: Point-pair separation

In a cyclic order, such as the real projective line, two pairs of points separate each other when they occur alternately in the order. Thus the ordering a b c d of four points has (a,c) and (b,d) as separating pairs. This point-pair separation is an invariant of projectivities of the line. The concept was described by G. B. Halsted at the outset of his Synthetic Projective Geometry: Given any pair of points on a projective line, they separate a third point from its harmonic conjugate. A pair of lines in a pencil separates another pair when a transversal crosses the pairs in separated points.