Introduction to Projective Geometry

Juan Carlos Álvarez Paiva


Twenty years ago the study of projective geometry would have seemed somewhat like the study of Latin: good for the brain and a means to keep in contact with the ancients. Today, however, projective geometry is a popular subject with mathematicians and computer scientists. This is due to the new applications to computer vision and to the fact that geometry and geometric thinking are again on the upswing. Moreover, computer science has returned the favor by providing powerful tools for the visualization and diffusion of mathematics. Java applets, animated GIF images, and linked HTML pages allow us to present mathematics and, most particularly, geometry in a way that was unthinkable twenty years ago.

In this course, we shall study projective geometry from a modern perspective using modern tools. A lot of effort will be spent in studying general unifying ideas in geometry such as group actions, homogeneous spaces, invariants, duality, compactification, complexification, and metric spaces. It will be an advantage to the student of geometry to be armed with these ideas before he tackles other more complex ideas such as connexions, fiber bundles, spinors, and twistors. On the other hand these abstract concepts will be illustrated by geometric constructions of great simplicity. These constructions become alive through their presentation as applets and animated images. Moreover, as these constructions, and indeed the whole course, is available through internet, the student can make better use of the alloted class time to ask questions, solve problems, and expand his or her horizons.

We now turn to a short description of the subject matter of this course. Today we will deal only with the most elementary and visual ideas, but I think it will be enough to give you an idea of what lies beyond.

Perspective

The architects and painters of the renaissance asked themselves how to represent a three-dimensional object on a two-dimensional surface. Computer scientists today ask how to make a computer recognize that two distinct images


This woodcut by Albrecht Dürer illustrates Alberti's idea.

represent the same object from different points of view. In answer to the first question, Leone Battista Alberti (1404-1472) proposed the following procedure: interpose a glass screen between yourself and the object, close one eye, and mark on the glass the points that appear to be on the image. The resulting image, although two-dimensional, will give a faithful impression of the three-dimensional object.

Alberti's method has been described as projection and section : first we draw the sheaf of light rays entering the eye from the object and then we cut with the glass screen. Since we are free to move our eye and the position of the screen, we have many different two-dimensional representations of the three-dimensional object. An interesting problem, raised by Alberti himself, is to recognize the common properties of all these different representations. This question is the same computer scientists ask today.

To compare Giotto's attempts at perspective in the epiphany with Raphael's the school of Athens click here.


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http://gauss.math.ucl.ac.be/~alvarez/teaching/projective-geometry/Inaugural-Lecture/inaugural.html