Sections of a cone

Circles, ellipses, parabolas and hyperbolas are called conic sections, or conics, because each one is the curve where a flat plane meets a double napped right circular cone. The names parabola and hyperbola go back to Apollonius.

Build the cone from two lines: a fixed vertical line l and a line m that crosses it at a point V at a fixed angle α. Spin m about l, keeping α unchanged, and m sweeps out a double napped right circular hollow cone that runs on without end both ways.

V is the vertex, l is the axis and the spinning line m is a generator. The vertex splits the cone into two pieces, the upper and lower nappes.

Let β be the angle the cutting plane makes with the vertical axis. When the plane does not pass through V, the curve depends only on how β compares with α.

β = 90° (plane at right angles to the axis): a circle. α < β < 90°: an ellipse. β = α (plane parallel to a generator): a parabola. In these three cases the plane cuts across one nappe only.

0 ≤ β < α: the plane is steeper than the generators, cuts through both nappes, and the curve is a hyperbola, which therefore has two separate branches.

These curves turn up in planetary orbits, telescope and antenna design, and the reflectors of torches and vehicle headlights.