Perspective and Projective Geometry

Book Preface

Looking at the World through the Window of Mathematics
Perspective and Projective Geometry is a course that will change the way that you look at the world, and we mean that literally.
In this course, you will take photographs, you will analyze perspective pictures and draw pictures of your own, and you will explore the geometry that explains how we fit a three-dimensional world onto a two-dimensional canvas. Along the way, you will get invaluable practice with making logical arguments (that is, writing mathematical proofs of statements that are true and refuting statements that are false). By combining art and geometry, we are following in a tradition that builds on centuries of exploration. Indeed, the very first treatise on perspective art—Leon Battista Alberti’s Della Pittura (On Painting)—includes this exhortation from the author:
Itwouldpleasemeifthepainterwereaslearnedaspossibleinalltheliberalarts,but first of all I desire that he know geometry. …Our instruction in which all the perfect absolute art of painting is explained will be easily understood by the geometrician, but one who is ignorant in geometry will not understand these or any other rules of painting. Therefore, I assert that it is necessary for the painter to learn geometry. [3, p. 90]
Knowing how to look at the world is more than just a fancy, aesthetic luxury. The geometry that helped Renaissance artists create breathtaking, realistic images five or six centuries ago also helped those same societies construct maps that allowed them to navigate across the globe; it helped them understand the emerging warfare of ballistic cannons; andithelpedthembuildfortressesthatcouldwithstandballisticattacks.
Indeed, Girard Desargues—the author of one of the central theorems of this book— extolled the subject of geometry because of its usefulness not only in the world at large, but also to his own well-being:
I freely confess that I never had taste for study or research either in physics or geometry except in so far as they could serve as a means of arriving at some sort of knowledge of the proximate causes for the good and convenience of life, in maintaining health, in the practice of some art … having observed that a good part of the arts is based on geometry … that of perspective in particular [33].
That very same geometry that defined an age named for “new birth” is making its own new strides in our modern world. During our lifetime, video games have moved from the two-dimensional mazes of Pac-Man into the immersive full-body experiences. Your parents and grandparents watched the flat worlds of the Flintstones and the Simpsons, but as the calendar ticked over into the most recent millennium, movies like Shrek and Frozen started bringing pixelated characters to life. And just as in the Renaissance, the uses of this projective geometry have spilled out over the edges of art into many areas of practical technology, paving the way for unprecedented progress in medical imaging, in geological exploration, and in 360-degree map views that have made commercial successes of applications like Zillow and Google. Knowing how to see the world is powerful, and this course will help you to harness that power.