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Surfaces of the second order

Remark 8.6. If line l is given by general equation, then it is useful to switch to
parametric one.

Lecture 9. Surfaces of the second order

9.1. Canonical equations of surfaces of the second order. Method of

parallel cuts

Surfaces of the second order are the surfaces, which are determined by means of algebraic equa-
tions of the second order in the system of coordinates Oxyz. Geometrical study of these surfaces
is conducted according to their equations by means of method of parallel cuts, which consists of
studying the form of the surface according to the cuts of the surface with coordinate axes. Let’s
consider some surfaces.

9.2. Ellipsoid. Sphere

Definition 9.1. The surface with the equation

x 2 y 2 z 2
+ + = 1 (9.1)
a 2 b 2 c 2
is called the ellipsoid. An equation (9.1) is the canonical equation of the ellipsoid.✓

Let’s study its geometrical shape. In order to do that let’s consider cuts of the given ellipsoid by
coordinate axes x = 0, y = 0, z = 0.
{
x = 0,
1. y 2 z 2 We will get the ellipse with half-axes b and c in plane Oyz.
b 2 + c 2 = 1.
{
y = 0,
2. We will get the ellipse with half-axes a and c in plane Oxz.
x 2 z 2
a 2 + c 2 = 1.
{
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
So, the surface of the ellipsoid with half-axes a, b and c is closed and oval (fig. 9.1). If a = b
or a = c or b = c, then the ellipsoid is called the ellipsoid of rotation around axis Oz or Oy or
Ox. If a = b = c = R, then the ellipsoid transforms into the sphere with radius R and center in
the beginning of coordinates. Such a sphere has the following equation:

2
2
2
2
x + y + z = R . (9.2)
9.3. One-sheet hyperboloid

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