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

SPHERES AND OTHER SURFACES

543

axis of the common asymptotes of the hyperbolas, namely, the
cone
X 2 z2 —0
(6) + 0,
~(6) ~ a2 a2 b2
is called the common asymptotic cone of the conjugate hyperboloids. See Ch. XXIII, Fig. 5.
Since a parabola has but one axis, there can be obtained
from it but one quadric surface, or paraboloid, of revolution,
namely, that which results from rotating
it about its axis. If the equation of the
parabola, in the (y, z)-plane, is
y2 = 2 mz, I
the equation of the paraboloid of revolution \-
is
(7T) x2+~y2= 2mz.
An ellipsoid or hyperboloid of revolu-
tion is symmetric in the center, 0, of the
ellipse or hyperbola which generates the Y
surface. Accordingly, we call 0 the center
of the surface.
FIG. 10
Every surface of revolution is symmetric
in its axis and in every plane passing through the axis. An
ellipsoid or hyperboloid of revolution is also symmetric in the
plane through the center perpendicular to the axis, and in
every line through the center lying in this plane. Thus the
surface (1) is symmetric, not only in the axis of z and in all
planes through this axis, but also in the (x, y)-plane and all
lines lying in this plane and passing through 0.
EXERCISES
Establish each of the following equations.
1. Equation (2). 2. Equation (4). 3. Equation (5).
What surface does each of the following equations represent?
Construct the surface.