# The Cross Ratio

## November-December 1998

Geometry is the study of what properties of a figure stay the same as the figure undergoes some transformation. For example, in Euclidean geometry, the allowable transformations are rotations and translations. Properties that stay constant include distances and angles. For projective geometry, such as we use with homogeneous coordinates, the transformations include perspective projections. In this case, one thing that emphatically does not stay the same is geometric length. A property that does, however, is something called the cross ratio. Take a look at Figure 12.1 where lines p, q, r, and s all intersect at the same point; the cross ratio is the ratio of the ratios of the following distances: ...

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