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Cone of the second order

ordinate plane Oyz, the same directed down parabolas are obtained:
{
x = h,
2 h 2
y = −2q(z − .)
2p
z

x . . . . .
0

y

Figure 9.5 – Hyperbolic paraboloid

Let’s consider cuts with planes z = h, which are parallel to coordinate plane Oxy:
{ {
z = h, z = h,
x 2 y 2 = 2h or x 2 y 2 = 1.
p − q 2ph − 2qh
So, in case h > 0 we will get hyperbolas, which cross plane in case h < 0 — hyperbolas,
which cross plane Oyz; in case h = 0 hyperbola degenerates into the pair of lines, crossing
{ {
z = 0, z = 0,
in the beginning of coordinates: x y and x y Thus, the considered
√ − √ = 0 √ + √ = 0.
p q p q
cuts allow us to build the hyperbolic paraboloid as a saddle (fig. 9.5). Point O(0; 0; 0) is called
the top of this paraboloid, digits p and q — its parameters.

9.7. Cone of the second order

Definition 9.6. The surface with the equation

x 2 y 2 z 2
+ − = 0 (9.7)
a 2 b 2 c 2
is called the cone of the second order. Equation (9.7) is the canonical equation of
the cone. ✓

Let’s consider geometrical properties of the cone. When cutting it with plane Oyz we will get
{ { {
x = 0, x = 0, x = 0,
the line y 2 z 2 which is laid on two lines: y z and y z
b 2 − c 2 = 0, b + c = 0 b − c = 0.
{
y = 0,
When cutting cone with plane Oxz we will get the line , which is laid on
x 2 z 2 2 = 0
a 2 − c
{ {
y = 0, y = 0,
two lines: x z and x z Let’s consider cuts with planes z = h, which are
a + c = 0 a − c = 0.
{
{
z = h, z = h,
parallel to coordinate plane Oxy: 2 2 or 2 2 So, in case h > 0 or
x 2 2 + y 2 = h x 2 + y 2 = 1.
a b c 2 a 0 b 0
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