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

QUADRIC SURFACES

551

The Hyperboloid of Two Sheets. This surface, also known
as the biparted hyperboloid, is defined by the equation
(x2 y2 2 X2 y2 Z2
(3) f or - — 1.
a2 b2 C2 a2 b2 2
A particular case, when a = b, is the hyperboloid of revolution
of two sheets.
In the general case, a t b, the sections by the vertical coordinate planes are the hyperbolas conjugate to the hyperbolas
(2). The (x, y)-plane, z = 0, does not
intersect the surface. This is true
of all the planes z =k, for which
k2 < c2. The planes z= c meet the
surface in the points (0, 0, ~ c), and
the planes z = I, where k2 > C2, meet'
it in ellipses, which increase in size
as k increases in numerical value;
cf. Fig. 9, Ch. XXII.
Center, Axes, Principal Planes.
Each hyperboloid is symmetric in F 4
the origin 0 and in the coordinate
axes and coordinate planes; 0 is the center, the coordinate
axes, the axes, and the coordinate planes the principal planes
for each surface. The sections by the principal planes are the
principal sections.
The Asymptotic Cone. The hyperboloids (1) and (3) are
called conjugate hyperboloids. We have seen that each vertical
coordinate plane intersects them in conjugate hyperbolas whose
common asymptotes pass through the origin. This is true also
of any vertical plane,
(4) y = mx,
which passes through the axis of z. For, the sections of (1)
and (3) by the plane (4) are also the sections by this plane of
the cylinders,
m2 2)- 2(1 2)- 2 -
XI + = 1),Xi- - +
\a~z b2 c2 \a2 b2