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One-sheet hyperboloid

z

c

. . . . . . . . b y
0
a
x

Figure 9.1 – Ellipsoid

Definition 9.2. The surface with the equation

x 2 y 2 z 2
+ − = 1 (9.3)
a 2 b 2 c 2

is called the the one-sheet hyperboloid. Equation (9.3) is the canonical equation of
the one-sheet hyperboloid. ✓

In order to study its geometrical shape, let’s consider cuts of the given one-sheet hyperboloid by
coordinate axes:
{
x = 0,
1. y 2 z 2 We will get the hyperbola with real half-axis b and imagi-
b 2 − c 2 = 1.
nary half-axis c in plane Oyz.
{
y = 0,
2. x 2 z 2 We will get the hyperbola with real half-axis a and imagi-
a 2 − c 2 = 1.
nary half-axis c in plane Oxz.
{
z = 0,
3. 2 We will get the ellipse with half-axes a and b in plane Oxy.
x 2 2 + y 2 = 1.
a b
In order to build the surface, let’s consider cuts with planes z = h, which are parallel to
{
{
z = h, z = h,
coordinate plane Oxy: x 2 y 2 h 2 or x 2 y 2 So, plane z = h crosses the
a 2 + b 2 = 1 + c 2 a 2 + b 2 = 1.
0
0
√ √
2
2
hyperboloid on the ellipse with half-axes a = a 1 + h 2 and b = b 1 + h 2 . When increasing
′
′
c c
|h| , values of a and b are increasing as well. Thus, the considered cuts allow us to imagine
′
′
the hyperboloid as an endless tube which broadens on the measure of removal (for both sides)
from plane Oxy (fig. 9.2). Values a, b, c are called the half-axes of the hyperboloid, besides a,
b — real, and c — imaginary. In case a = b, we will get the one-sheet hyperboloid of rotation
around Oz-axis.
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