Focus (geometry)

For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola.

Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus.

A hyperbola can be defined as the locus of points for which the absolute value of the difference between the distances to two given foci is constant.

If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero, then the conic is a circle.

Thus, for instance, the minor planet Pluto's largest moon Charon has an elliptical orbit which has one focus at the Pluto-Charon system's barycenter, which is a point that is in space between the two bodies; and Pluto also moves in an ellipse with one of its foci at that same barycenter between the bodies.

Two binary stars also move in ellipses sharing a focus at their barycenter; for an animation, see here.

A Cartesian oval is the set of points for each of which the weighted sum of the distances to two given foci is constant.

An n-ellipse is the set of points all having the same sum of distances to n foci (the n = 2 case being the conventional ellipse).

The case H = 0 can be eliminated as degenerate, so the tangential equation of C can be written as P + fQ = 0 where f is an arbitrary polynomial of degree 2m.