Spring09
Math 231-Sec 13.6
Instructor: Yanxiang Zhao
Section 14.3: Cylinders And Quadric Surfaces
In this section, we will discuss all the possible 3-d algebraic surfaces of order 2. All the
pictures in this section are cited from http://en.wikipedia.org/wiki/Quadric.
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Cylinders
The cylinders, for the algebraic point of view, are the quadratic equations involving 2
variables. Since 3-d quadratic equations, generally, should contain 3 variables (x, y, z),
if there are only 2 variables appearing in the equations, it means along the missing
variable, the surface will repeat itself infinitely many times.
x2
y2
Example
In 2-d space,
+
= 1 stands for a ellipse. But when we extend
3
2
the dimension into 3-d, it is called elliptic cylinder The fact that z is missing in the
x2 y 2
equation implies that if we cut
+ = 1 by z = C, the intersection(horizontal trace)
3
2
will be always a ellipse. And the vertical traces are lines. The following figure is a 3-d
elliptic cylinder.
Example
cylinder,
The following figures are examples of parabolic cylinder and hyperbolic
which corresponds to (from left to right)
y = ax2 ,
and ax2 − by 2 = 1
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Spring09
2
Math 231-Sec 13.6
Instructor: Yanxiang Zhao
Quadric Surfaces
For the quadric surfaces, the main goal is to figure out the correspondence between the
graphs and the equations.
Example
to it?
Consider the equation z = 4x2 + y 2 , which one of the graphs corresponds
If we set z = C, which means, geometrically, we cut the graph by z = C plane, then
based on the equation z = 4x2 + y 2 , the horizontal trace will be a ellipse when z > 0,
a point when z = 0, or nothing when z < 0. If we set x = C, which means we cut the
graph by x = C plane, then the perp-to-x-axis vertical trace is a parabola. Similarly,
the perp-to-y-axis vertical trace is a parabola as well. So the graph of z = 4x2 + y 2 is
the left one on top, which we call elliptic paraboloid.
I list all the possible quadric surfaces here to help you recognize them. Example
• Ellipsoid
x2 y 2 z 2
+ 2 + 2 =1
a2
b
c
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Spring09
• Cone
• Elliptic Paraboloid
• Hyperbolic Paraboloid
Math 231-Sec 13.6
x2 y 2
z2
+
=
a2
b2
c2
z
x2 y 2
= 2+ 2
c
a
b
z
x2 y 2
= 2− 2
c
a
b
• Hyperboloid of One Sheet
x2 y 2 z 2
+ 2 − 2 =1
a2
b
c
3
Instructor: Yanxiang Zhao
Spring09
Math 231-Sec 13.6
• Hyperboloid of two Sheets
x2 y 2 z 2
+ 2 − 2 = −1
a2
b
c
4
Instructor: Yanxiang Zhao