Slice a double cone.

A rotatable double cone (semi-vertical angle α) cut by a plane at angle β to the axis; the section curve is drawn and named — circle, ellipse, parabola, hyperbola — or a point, line or line-pair when the plane passes through the vertex.

Make a parabola (β = α), then move the plane through the vertex to get the degenerate line.

• 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.

Teaching this idea? Put the simulation on your own page or LMS with this code.