Conic SectionsConic 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.
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.