Conic Sections

Conic sections are the curves that result from the intersection of a plane and a double cone. The three types of curve so produced are:
  • the parabola – when the plane is parallel to the side of the cone,
  • the ellipse – when the plane cuts through one nappe of the cone,
  • the hyperbola – when the plane cuts through both nappes.
A parabola has a single focus, while the ellipse and hyperbola have two foci. These points, plus a particular length property, define the shape in the same way as a centre and radius define a circle.

For all points on an ellipse, the sum of the distances from each focus is constant. You can imagine fixing a string loosely between two points, then tightening the string by pushing a pencil against it. The curve you trace with this pencil while keeping the string taut will be an ellipse. If the foci are moved closer together the ellipse becomes more circular, until they eventually become a single point and the traced out curve is a circle.

For a hyperbola it is the difference of these distances that is constant, while for all points on a parabola, it is the distance from the only focus to a line (called the directrix) that is constant.

It is also possible to produce circles, intersecting diagonal lines, and a single point via the intersection of a plane and a double cone. These are called degenerate cases. Can you see how they arise?

Because conic sections arise naturally, have many useful properties, and are symmetric, they are important curves in mathematics, science, engineering, architecture, and design. In the following activity, we are interested in their reflective properties, i.e. in how light or sound might bounce off them.

Play the game below, and see what properties you can discover.



Conic Billiards
Reflections and conic sections
The game of Conic Billiards is played on a billiard table with a single ball and a parabolic, elliptical or hyperbolic cushion. The object of the game is to pot the ball, but only after it has bounced off the cushion at least once. The pocket is at a focus of the cushion curve.

You start with 100 points, and points are subsequently awarded or deducted as follows:

  • Score 40 points if you pot the ball after bouncing off the cushion two or more times.
  • Score 20 points if you pot the ball after bouncing off the cushion once.
  • Lose 10 points if you fail to pot the ball.
  • Lose 20 points if you play the ball out of bounds.
  • Lose 50 points if you pot the ball without bouncing off the cushion.

To play, choose a cushion shape, and then drag the cue into the desired position and hit the Strike button or click the ball to play your shot. When playing with an ellipse or hyperbola, you can choose for the other focus to be shown, and also you can optionally show a guide extending from the tip of the cue to aid with aiming.

If Show path is selected, the path followed by the ball, plus the tangent and normal at each point of reflection are shown after the shot is completed.

About the activity

The law of reflection states that the angle of incidence equals the angle of reflection. This is relatively easy to see and estimate when an incoming ball collides with a straight line, but can be confusing when the line (or surface) is curved. In such cases, the tangent at the point of impact is the "straight" line to consider.

The reflective properties of conics lead to many important applications with light, sound, and radio waves. Use your experience from the Conic Billiards game to think about what might happen if you put a transmitter or receiver at a focus of a conic.

A parabolic reflector, made by rotating a parabola in 3D about the y-axis, with a light source at the focus projects the light as a beam—such as a headlight or search light, since all rays are emitted parallel. Similarly, if a receiver is at the focus, all directly incoming rays are focussed at that point (hence the name focus). This is why satellite dishes have a parabolic cross-section. Similarly, these properties can be exploited for sending and receiving sound waves in a directionally controlled manner. This application of parabolas was first described by Diocles in 200 BC.

As recognised by Kepler in 1605, all planets move in ellipses, with the sun at one focus. This is known as Kepler's first law of planetary motion. Fresnel zones describe important aspects of interference in wave transmission, and arise from the reflective properties of ellipses. Similarly, a whisper room employs an elliptical ceiling so that someone standing at one focus can whisper something to his friend standing at the other, who will clearly hear what was said since the dispersed sound waves are reflected by the ceiling and concentrated at the other focus. In the game, suppose the cue ball is sitting exactly on the non-pocket focus. What can we say about the next shot?

Hyperbolic mirrors are important in optics. In particular, a Cassegrain reflector is a compact telescope design that has a parabolic primary mirror to collect the light, and a hyperbolic secondary mirror that reflects it back down through a hole in the primary to the eyepiece. Hyperbolic reflection was also a fundamental aspect of Rutherford's analysis of his alpha scattering experiments in the early 1900s, that led directly to the discovery of the atomic nucleus.