Section 12.6: Cylinders and Quadric Surfaces
MATH251 c Justin Cantu
Section 12.6: Cylinders and Quadric Surfaces
Cylinders: A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve. We will focus on two-variable equations, where rulings are parallel to a coordinate axis. Example 1: Sketch the following surfaces.
y2 z2 (a) elliptic cylinder: + = 1
4 16
(b) parabolic cylinder: z = x2
(c) hyperbolic cylinder: y2 - x2 = 1 9
1
MATH251 c Justin Cantu
Quadric Surfaces: A quadric surface is the graph of a second-degree equation in three variables. They are characterized by the curves of intersection of the surface with planes of the form x = k, y = k, and z = k. These curves are called traces (or cross-sections) of the surface.
Ellipsoid: The quadric surface defined by x2 y2 z2 a2 + b2 + c2 = 1
is called an ellipsoid. All traces are ellipses.
x2 y2 z2 + + =1
9 25 4
Hyperboloid of one sheet: A quadric surface defined by
x2 y2 z2 + - = 1 or
a2 b2 c2
x2 y2 z2 - + = 1 or
a2 b2 c2
x2 y2 z2 - + + =1
a2 b2 c2
is called a hyperboloid of one sheet. The variable with a negative coefficient determines the axis that the hyperboloid opens up along. Traces are hyperbolas and ellipses.
x2 y2 z2 a2 + b2 - c2 = 1
x2 y2 z2 - a2 + b2 + c2 = 1
2
MATH251 c Justin Cantu
Hyperboloid of two sheets: A quadric surface defined by
x2 y2 z2 - - + = 1 or
a2 b2 c2
x2 y2 z2 - + - = 1 or
a2 b2 c2
x2 y2 z2 - - =1
a2 b2 c2
is called a hyperboloid of two sheets. The variable with a positive coefficient determines the axis that the hyperboloid opens up along. Traces are hyperbolas and ellipses.
x2 y2 z2 - a2 - b2 + c2 = 1
-x2 + y2 - z2 = 1 42
(Elliptic) Cone: A quadric surface defined by
z2 x2 y2
y2 z2 x2
x2 y2 z2
=+
or
=+
or
=+
c2 a2 b2
b2 c2 a2
a2 b2 c2
is called an (elliptic) cone. The variable by itself on one side of the equal sign determines the axis that the cone opens up along. Traces are hyperbolas/lines and ellipses.
z2 x2 y2 c2 = a2 + b2
x2 y2 z2 a2 = b2 + c2
3
MATH251 c Justin Cantu
Elliptic Paraboloid: A quadric surface defined by
z x2 y2
y x2 z2
x y2 z2
=+
or
=+
or
=+
c a2 b2
b a2 c2
a b2 c2
is called an elliptic paraboloid. The linear variable determines the axis that the graph opens up along, and its coefficient's sign determines the direction. Traces are ellipses and parabolas.
z x2 y2 c = a2 + b2 , c > 0
z x2 y2 c = a2 + b2 , c < 0
Hyperbolic Paraboloid: A quadric surface defined by
z x2 y2
y x2 z2
x y2 z2
c = a2 - b2
or
b = a2 - c2
or
a = b2 - c2
is called a hyperbolic paraboloid. The surface is "saddle-shaped" and "opens up" according to the two variables with the same sign (one always linear). Traces are hyperbolas and parabolas.
z = y2 - x2
-y = x2 - z2
4
MATH251 c Justin Cantu
Example 2: Sketch the following quadric surfaces. Label any intercepts with the coordinate axes. (a) 36x2 - 9y2 - 4z2 = 36
(b)
x2 y2 +
= 1 + z2
16 4
(c) y = x2 + z2
5
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