Conic sections

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:

where .

Likewise, a parabola which is aligned along the -axis, and passes through the origin (see Figure A.5), satisfies

where .

Finally, a hyperbola which is aligned along the -axis, and whose asymptotes intersect at the origin (see Figure A.6), satisfies

where . Here, is the distance of closest approach to the origin. The asymptotes subtend an angle with the -axis.

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:

where is a constant. When expressed in terms of polar coordinates, the preceding equation can be rearranged to give

where .

When written in terms of Cartesian coordinates, Equation (A.104) can be rearranged to give

for . Here,

Equation (A.106) can be recognized as the equation of an ellipse whose center lies at ( ,

When again written in terms of Cartesian coordinates, Equation (A.104) can be rearranged to give

(A.110) |

for . Here, . This is the equation of a parabola that passes through the point ( ,

Finally, when written in terms of Cartesian coordinates, Equation (A.104) can be rearranged to give

for . Here,

(A.112) | ||||||

(A.113) | ||||||

and | (A.114) |

Equation (A.111) can be recognized as the equation of a hyperbola whose asymptotes intersect at ( ,

(A.115) |

with the -axis [see Equation (A.103)].

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.