The Mathematics of Perspective:  An Introduction to the Cross Ratio

David A. Thomas

Department of Mathematics

University of Idaho

Introduction

Even though the world is a finite physical space, there appear to be an infinite number of perspectives on the world’s people, places, structures, and processes.  Artists, architects, and designers specialize in creating and communicating perspectives that reflect their individual interests and values.  In many cases, creating these perspectives involves representing 3-dimensional objects on 2-dimensional surfaces such as canvas, illustration board, computer monitors, or movie screens.  Since the Renaissance, artists, architects, scientists, and mathematicians have been developing concepts and procedures to guide this process.  This paper focuses on a mathematical concept underlying many of these efforts, the cross ratio.

Table of Contents

Part I

 

History

Part II

 

Dividing Space: Collinear Points

Part III

 

Dividing Space: Non-Collinear Points

Part IV

 

Projective Transformations

Part V

 

Martian Canyon System:  Two Views

Part VI

Martian Canyon System:  In the Cross Hairs

 

Part VII

Summary

 

Bibliography