Motivation

A conic section is, basically by definition, the image of a circle under a perspectivity whose center is the vertex of the cone (Fig. 1).

Figure 1.

By moving and tilting the plane, we may obtain an ellipse, a parabola, and a hyperbola, but also a point, a line, and a pair of intersecting lines. These last conic sections are usually termed degenerate. Since the nondegenerate conics are all equivalent to the circle under projective (and even perspective) transformations, we may think that the projective geometry of conics is rather dull. Far from that, by forgetting the differences between circles, ellipses, hyperbolas, and parabolas, we are able to concentrate on their common properties and gain a deeper understanding of them.

In simple terms, what projective geometry allows us to do is to concentrate on the cone and not on its sections. Indeed, the cone $x^2 + y^2 - z^2$ represented in figure 1 is made up of lines passing through the origin. This one-parameter family of lines can be interpreted as a curve ${\cal C}$ on the projective plane. Notice that on identifying ${\Bbb R}P^2$ with the plane $z = 1$ plus the line at infinity, the curve ${\cal C}$ is represented as a circle.

Exercise 1.1 (10)   Let $L \subset {\Bbb R}P^2$ be a projective line and let $\varphi_{L} : {\Bbb R}P^2 \setminus L \rightarrow {\Bbb R}^2$ be the associated affine chart. Show that the curve $\varphi_{L}({\cal C})$ is an ellipse, a parabola, or a hyperbola depending on whether the number of points in the intersection of $L$ and ${\cal C}$ is zero, one, or two.

In what follows, we prefer to fix the chart and transform the curve ${\cal C}$ by different projective transformations.

Exercise 1.2 (10)   Transform the cone $x^2 + y^2 - z^2 = 0$ under the following linear transformations and write the equations of the intersection of the resulting cone with the plane $z = 1$.

$$1. \ \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \e... .... \ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix} .$$

In the next two sections we will introduce the algebraic concepts necessary to study conics and projective quadrics in higher dimensional projective spaces.