An ellipse, centered on the origin, of major radius and minor radius , which are aligned along the - and -axes, respectively (see Figure A.4), satisfies the following well-known equation:
Likewise, a parabola which is aligned along the -axis, and passes through the origin (see Figure A.5), satisfies
Finally, a hyperbola which is aligned along the -axis, and whose asymptotes intersect at the origin (see Figure A.6), satisfies
It is not obvious, from the preceding formulae, what the ellipse, the parabola, and the hyperbola have in common. It turns out, in fact, that these three curves can all be represented as the locus of a movable point whose distance from a fixed point is in a constant ratio to its perpendicular distance to some fixed straight line. Let the fixed point--which is termed the focus--lie at the origin, and let the fixed line--which is termed the directrix--correspond to (with ). Thus, the distance of a general point ( , ) (which lies to the left of the directrix) from the focus is , whereas the perpendicular distance of the point from the directrix is . (See Figure A.7.) In polar coordinates, and . Hence, the locus of a point for which and are in a fixed ratio satisfies the following equation:
When written in terms of Cartesian coordinates, Equation (A.104) can be rearranged to give
When again written in terms of Cartesian coordinates, Equation (A.104) can be rearranged to give
Finally, when written in terms of Cartesian coordinates, Equation (A.104) can be rearranged to give
In conclusion, Equation (A.105) is the polar equation of a general conic section that is confocal with the origin (i.e., the origin lies at a focus). For , the conic section is an ellipse. For , the conic section is a parabola. Finally, for , the conic section is a hyperbola.