An ellipse can be defined as the locus of points for each of which the sum of the distances to two given foci is a constant.
A circle is the special case of an ellipse in which the two foci coincide with each other. 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 circle can also be defined as the circle of Apollonius, in terms of two different foci, as the set of points having a fixed ratio of distances to the two foci.
A parabola is a limiting case of an ellipse in which one of the foci is a point at infinity.
A hyperbola can be defined as the locus of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
It is also possible to describe all conic sections in terms of a single focus and a single directrix, which is a given line not containing the focus. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If e is between zero and one the conic is an ellipse; if e=1 the conic is a parabola; and if e>1 the conic is a hyperbola. 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.