Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

QUADRIC SURFACES

575

9. Poles and Polars. Through a point PO: (Xo, Yo, Zo) not on
the ellipsoid:
x2 y2 z2
(1) X _ _ _ _
an aa2+ b2 C2
an arbitrary line L is drawn meeting the ellipsoid in Qi and
Q2. What is the locus of the point P which with Po divides
Q1Q2 harmonically?
By the method used in solving the corresponding problem
in the plane, Ch. XIV, ~ 9, the locus is found to be the plane
(2) + yoY + Z0Z -1,
(w a 62a b2 c2
or a portion of this plane.
The point P0 (not on the ellipsoid) and the plane (2) are
said to be pole and polar in the ellipsoid: Po is the pole of (2),
and (2) the polar of PO. A point on the ellipsoid and the
tangent plane at the point are defined to be pole and polar.
By the methods of Ch. XIV, ~~ 9-11, the following theorems
can now be proved.
THEOREM 1. Let Q be a central quadric (an ellipsoid or
hyperboloid). Every point in space, except the center of Q, has
a polar with respect to Q.
THEOREM 2. Every point in space has a polar with respect
to a paraboloid.
THEOREM 3. Let Q be a central quadric or a paraboloid.
Every plane in space, which is not a diametral plane of Q, has
a pole with respect to Q.
The poles and polars considered in the following theorems
are taken with respect to an arbitrarily chosen central quadric
or paraboloid.
THEOREM 4a. If one point lies in the polar plane of a second,
the second point lies in the polar plane of the first.
THEOREM 4b. If one plane contains the pole of a second, the
second plane contains the pole of the first.