Classical Geometry: Euclidean, Transformational, Inversive, and Projective

CHAPTER 15

CROSS RATIOS

15.1 Cross Ratios

Directed Distances

Recall that we denote directed distances, or signed distances, with a bar over the distance, as in , and that

Also recall the following facts:

1. Given points A, B, and C on a line, if = , then B = C.

2. Given points A, B, C, and X on a line, if / = , then B = X.

Directed Distances and Harmonic Conjugates

Recall that given points A, B, C, and D, then B and D are harmonic conjugates with respect to A and C if and only if

as in the figure below.

Here, we are using unsigned distances; for signed distances, B and D are harmonic conjugates ...