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Projective geometry is a strange form of geometry in which parallel lines sometimes meet, circles are the same as parabolas and ellipses, all angles are right angles, and size has no meaning. When you think about it this way, it sounds as if it can't possibly have much to do with the real world -- and yet we intuitively use projective geometry every day when we interpret what we see. Projective geometry was invented in the fifteenth century by artists trying to understand how to accurately depict what we see in a painting, and since then it has matured into a "universal geometry" that contains most other geometries within it. In this course, we'll start by studying the origins of projective geometry in painting. Then we'll explore several different ways of understanding projective geometry: axiomatically (using logic and proofs), analytically (using coordinates), constructively (using compass and straightedge), and computationally (using computers). Along the way, we'll meet finite projective geometries (in which there are only finitely many points and finitely many lines), and we'll explore the relationships among projective, Euclidean, spherical, and non-Euclidean geometries. At the end, we'll come back to art, and use what we've learned to create accurate perspective drawings. |