The Mathematics of Perspective: An Introduction to the Cross Ratio

David A. Thomas

Department of Mathematics

University of Idaho

Summary

Every time you look at a building from a different location, you see it from a different perspective. In addition to revealing different details of the building, each perspective provides important visual clues about the relative distance to objects in our field of view. These clues arise in the form of apparent distortions. For instance, features that are known to be of equal length, area, or angle appear to be different when seen in perspective. Far from misleading, these distortions are essential to our depth perception and understanding of our surroundings.

This paper focuses on an aspect of perspective that we cannot experience directly, a division of space that is independent of our perspective or point of view. Mathematicians call this division the cross ratio. As presented in this paper, the cross ratio

² Is a ratio of ratios

² May be computed using collinear segments

² May be computed using concurrent angles

² Is independent of the position of the observer

² Is invariant under projective transformations

² Is the basis for the paper strip technique used by cartographers

² Is an important mathematical basis for 3-D computer graphics

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