Conics and Their Duals

So far, we have dealt almost exclusively with situations in which only points and lines were involved. Geometry would be quite a pure topic if these were the only objects to be treated. Large parts of classical elementary geometry deal with constructions involving circles. The most elementary drawing tools treated by Euclid (the straightedge and the compass) contain a tool for generating circles. In a sense, so far we have dealt with the straightedge alone. Unfortunately, circles are not a concept of projective geometry. This can easily be seen by observing that the shape of a circle is not invariant under projective transformations. If you look at a sheet of paper on which a circle is drawn from a skew angle, you will see an ellipse. In fact, projective transformations of circles include ellipses, hyperbolas, and parabolas. They are subsumed under the term conic sections, or conics, for short. Conics are the concept of projective geometry that comes closest to the concept of circles in Euclidean geometry. It is the purpose of this section to give a purely projective treatment of conics. Later on, we will see how certain specializations provide interesting insights into the geometry of circles.