Most of the sections in this chapter focus on the plane curves called conics or conic sections. As the name implies, these curves are the sections of a cone (similar to an ice cream cone) formed when a plane intersects the cone.

Euclid defined a cone as a surface generated by rotating a right triangle about one of its legs. However, the following description of a cone given by Apollonius is more appropriate.

If we slice the cone with a plane not passing through the vertex, the intersections of the slicing plane and the cone are conic sections. If the slicing plane is horizontal (parallel to the first plane), the intersection is a circle (Figure 7.2(a)). If the slicing plane is inclined slightly from the horizontal, intersecting only one nappe, the intersection is an oval curve called an ellipse (Figure 7.2(b)). As the angle of the slicing plane increases, more elongated ellipses are formed (Figure 7.2(b)). If the angle of the slicing plane increases still further so that the slicing plane is parallel to one of the lines generating the cone, the intersection is called a parabola (Figure 7.2(c)). If the angle of the slicing plane increases yet further so that the slicing plane intersects both nappes of the cone, the resulting intersection is called a hyperbola (Figure 7.2(d)).

Intersections of the slicing plane through the vertex result in points and lines called degenerate conic sections. See Figure 7.3.

Just as a circle (see page 191) is defined as the set of points in the plane at a fixed distance r (the radius) from a fixed point (the center), we can define each of the conic sections just described as a set of points in the plane that satisfy certain geometric conditions. It can be shown that these alternate definitions give the same family of curves described earlier. This enables us to keep our work in a two-dimensional setting. Using these definitions, we will derive the equations of the conic sections and show that an equation of the form $A{x}^2+C{y}^2+Dx+Ey+F=0$ is (except in degenerate cases represented in Figure 7.3) the equation of a parabola, an ellipse, or a hyperbola. (A circle is a special case of an ellipse.) Germinal Pierre Dandelin (1794–1847) was a Belgian/French mathematician who found a relatively straightforward way to establish the equivalence of the algebraic and geometric descriptions of the conics. A web search on Dandelin spheres will provide several sources for his proof. In fact, you can even find YouTube videos demonstrating this proof.

We study these conic sections because of their practical applications. For example, a satellite dish, a flashlight lens, and a telescope lens have a parabolic shape; the planets travel in elliptical orbits, and comets travel in orbits that are either elliptical or hyperbolic. A comet with an elliptical orbit can be viewed from Earth more than once, while those with a hyperbolic orbit can be viewed only once!