What is a Hyperbola? Geometry and Equation of a Hyperbola

This animation, created using MATLAB, illustrates how a hyperbola is formed by fixing the difference of the distances between two points, called the foci.

The location of the foci is adjusted to show the viewer that the hyperbola becomes very sharp as c approaches a. However, the hyperbola is wide (or fat) if c approaches infinity.

As all three a, b, and c, approach zero in such a way that keeps the ratio b/a fixed, we find a degenerate hyperbola is obtained. This is in actuality the union of two lines passing through the origin.

From the perspective of a conic section (see video "What is a Conic Section?"), the hyperbola is the intersection of a vertically oriented plane and a vertically oriented cone. In this case, the degenerate hyperbola is obtained when the plane cuts through the cone at precisely the middle, thus making the intersection the two edges or boundaries of the cone.