Section 11.5: Quadric Surfaces
Definition: A quadric surface is the graph of a second-degree equation in three variables
x, y, and z. These equations can be expressed as
Ax2 + By 2 + Cz 2 + D = 0
or
Ax2 + By 2 + Cz = 0.
Note: Quadric surfaces are the three-dimensional analogues of conic sections in two dimensions. Recall the following equations of conic sections.
Parabola
Hyperbola
Ellipse
𝑦
𝑦
𝑦
𝑥
𝑦 = 𝑎𝑥 2
𝑥
𝑥2 𝑦2
+
=1
𝑎2 𝑏 2
𝑥
𝑥2 𝑦2
−
=1
𝑎2 𝑏 2
Figure 1: Graphs of some conic sections.
Definition: The intersection of a surface with a plane is called the trace of the surface in
that plane. Quadric surfaces are characterized by their traces in vertical planes x = k or
y = k and horizontal planes z = k.
Definition: The quadric surface defined by
x2 y 2 z 2
+ 2 + 2 =1
a2
b
c
is called an ellipsoid. Its traces in vertical or horizontal planes are ellipses.
Figure 2: Graph of the ellipsoid x2 +
y2
+ z 2 = 1.
4
Definition: The quadric surface defined by
x2 y 2 z 2
+ 2 − 2 =1
a2
b
c
is called a hyperboloid of one sheet. Its traces in horizontal planes z = k are ellipses and
its traces in vertical planes x = k or y = k are hyperbolas.
Figure 3: Graph of the hyperboloid of one sheet
x2 y 2
+
− z 2 = 1.
4
9
Definition: The quadric surface defined by
z 2 x2 y 2
− 2 − 2 =1
c2
a
b
is called a hyperboloid of two sheets. Its traces in vertical planes x = k or y = k are
hyperbolas and its traces in horizontal planes z = k for |k| > c are ellipses.
x2 y 2
Figure 4: Graph of the hyperboloid of two sheets z −
−
= 1.
4
9
2
Definition: The quadric surface defined by
x2 y 2
z2
+
=
a2
b2
c2
is called a cone or elliptic cone. Its traces in vertical planes x = k or y = k are hyperbolas
for k 6= 0 and lines for k = 0. Its traces in horizontal planes z = k 6= 0 are ellipses.
x2 y 2
Figure 5: Graph of the cone z =
+ .
4
9
2
Definition: The quadric surface defined by
z
x2 y 2
= 2+ 2
c
a
b
is called an elliptic paraboloid. Its traces in horizontal planes z = k are ellipses and its
traces in vertical planes x = k or y = k are parabolas.
Figure 6: Graph of the elliptic paraboloid z =
x2 y 2
+ .
4
9
Definition: The quadric surface defined by
x2 y 2
z
= 2− 2
c
a
b
is called a hyperbolic paraboloid. Its traces in vertical planes x = k or y = k are parabolas
and its traces in horizontal planes z = k are hyperbolas.
Figure 7: Graph of the hyperbolic paraboloid z =
x2 y 2
− .
4
9
Example: Describe the graph of
3x2 − 12y 2 + z 2 = 12.
In standard form, the equation is
x2
z2
− y2 +
= 1.
4
12
To determine the type of surface, we examine traces of the surface in vertical and horizontal
planes. In the xy-plane (z = 0), the trace is a hyperbola
x2
− y 2 = 1.
4
In the xz-plane (y = 0), the trace is an ellipse
x2 z 2
+
= 1.
4
12
In the yz-plane (x = 0), the trace is a hyperbola
z2
− y 2 = 1.
12
Thus, the surface is a hyperboloid of one sheet.
Example: Describe the graph of
x
y2
=
+ z2.
6
3
In standard form, the equation is
x = 2y 2 + 6z 2 .
In the xy-plane (z = 0), the trace is a parabola x = 2y 2 .
In the xz-plane (y = 0), the trace is a parabola x = 6z 2 .
In the plane x = 6, the trace is an ellipse
y2
+ z 2 = 1.
3
Thus, the surface is an elliptic paraboloid.
Note: If one of the variables x, y, or z is missing from the equation (free variable), then the
surface is a cylinder.
Figure 8: Graph of the parabolic cylinder y = x2 in R3 .
y2
= 1 in R3 .
Figure 9: Graph of the elliptic cylinder x +
4
2
Example: Describe the graph of
x2 y 2
−
= 1.
4
16
Since z is a free variable, the trace in every horizontal plane z = k is a hyperbola. Thus, the
surface is a hyperbolic cylinder centered about the z-axis.
Example: Find an equation for the surface obtained by rotating the line x = 2y about the
x-axis.
The equation of the line is y = x2 . Revolving this line about the x-axis gives a circular
cone centered about the x-axis. The traces in vertical planes x = k are circles centered at
(y, z) = (0, 0) with radius r = y = x2 . Thus, the equation of the surface is
x 2
y2 + z2 =
2
2
2
2
4y + 4z = x .
𝑦
𝑧
𝑟
𝑥
𝑦
𝑟
𝑥
𝑟=𝑦=
𝑥
2
𝑥 2
2
4𝑦 2 + 4𝑧 2 = 𝑥 2
𝑦2 + 𝑧2 =
Figure 10: Graph of the circular cone x2 = 4y 2 + 4z 2 .