Projective Geometry

Preview

At about the same time as the algebraic revolution in classical geometry, a new kind of geometry also came to light: projective geometry. Based on the idea of projecting a figure from one plane to another, projective geometry was initially the concern of artists. In the 17th century, only a handful of mathematicians were interested in it, and their discoveries were not seen to be important until the 19th century. The fundamental quantities of classical geometry, such as length and angle, are not preserved by projection, so they have no meaning in projective geometry. Projective geometry can discuss only things that are preserved by projection, such a points and lines. Surprisingly, there are nontrivial theorems about points and lines. One of them was discovered by the Greek geometer Pappus around 300 ce, and another by the French mathematician Desargues around 1640. Even more surprisingly, there is a numerical quantity preserved by projection. It is a “ratio of ratios” of lengths called the cross-ratio. In projective geometry, the cross-ratio plays a role similar to that played by length in classical geometry. One of the virtues of projective geometry is that it simplifies the classification of curves. All conic sections, for example, are “projectively the same,” and there are only five types of cubic curve. The projective viewpoint also removes some apparent exceptions to the theorem of B´ezout. For example, a line (curve of degree 1) always meets another line in exactly one point, because in projective geometry even parallel lines meet.