11.5 Quadric Surfaces
Quadric Curves in R2
Ellipse:
x2 y 2
+
=1
a2 b2
Hyperbola:
Parabola:
x2 y 2
−
=1
a2 b2
y = ax2
or
or
−
x = ay 2
x2 y 2
+
=1
a2 b2
Quadric Surfaces
A quadric surface is the graph of a second degree equation in three variables. The
most general such equation is
Ax2 + By 2 + Cz 2 + Dxy + Eyz + F xz + Gx + Hy + Iz + J = 0
where A, B , C , D, E , F , G, H , I , J are constants. By translation and rotation the
equation can be brought into one of two standard forms
Ax2 + By 2 + Cz 2 + J = 0
or
Ax2 + By 2 + Iz = 0
In order to sketch the graph of a quadric surface, it is useful to determine the curves
of intersection of the surface with planes parallel to the coordinate planes. These
curves are called traces or cross-sections of the surface.
Classication of Quadric Surfaces
Quadric surfaces can be classied into 5 categories:
ellipsoids, hyperboloids, cones, paraboloids, quadric cylinders.
x2 y 2 z 2
Ellipsoid: 2 + 2 + 2 = 1
a
b
c
All of its traces are ellipses. The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0)
and (0, 0, ±c). Since the ellipsoid involves only even powers of x, y , and z , the ellipsoid
is symmetric with respect to each coordinate plane.
Example 1. Sketch the surface 4x2 + 9y 2 + 36z 2 = 36.
Hyperboloids of one sheet:
x2 y 2 z 2
• 2 + 2 − 2 =1
a
b
c
2
2
x
y
z2
• 2 − 2 + 2 =1
a
b
c
2
2
y
z2
x
•− 2 + 2 + 2 =1
a
b
c
Two of its traces are hyperbolas and one is an ellipse, and its surface consists of just
one piece.
Example 2. Sketch the surface x2 − y 2 + z 2 = 1.
Hyperboloids of two sheets:
x2 y 2 z 2
• 2 − 2 − 2 =1
a
b
c
2
2
x
y
z2
•− 2 + 2 − 2 =1
a
b
c
2
2
y
z2
x
•− 2 − 2 + 2 =1
a
b
c
Two of its traces are hyperbolas and one is an ellipse, and its surface consists of two
piece.
Example 3. Sketch the surface 9x2 − y 2 − z 2 = 9.
Cones:
z2
x2 y 2
• 2 = 2+ 2
c
a
b
2
2
y
x
z2
• 2 = 2+ 2
b
a
c
2
2
y
z2
x
• 2 = 2+ 2
a
b
c
One of its traces is ellipse and two are hyperbolas or pairs of lines.
z2
Example 4. Sketch the surface y = x + .
9
2
2
Elliptic paraboloids:
z
x2 y 2
• = 2+ 2
c
a
b
2
y
x
z2
• = 2+ 2
b
a
c
2
x y
z2
• = 2+ 2
a
b
c
Two of its traces are parabolas. and one is an ellipse.
z2
Example 5. Sketch the surface x = y + .
9
2
Hyperbolic paraboloids:
z
x2 y 2
• = 2− 2
c
a
b
x2 z 2
y
• = 2− 2
b
a
c
2
z2
x y
• = 2− 2
a
b
c
Two of its traces are parabolas. and one is a hyperbola.
Example 6. Sketch the surface z = x2 − y 2 .
Quadric cylinders:
When one of the variables is missing from the equation of a surface, then the surface
is a cylinder.
Example 7. Sketch the surface x2 + 4z 2 = 1.
Example 8. Describe and sketch the surface z = (x + 4)2 + (y − 2)2 + 5.
Example 9. Classify the quadric surface 4x2 − y 2 + z 2 + 8x + 8z + 24 = 0 and sketch
it.