11.6
Surfaces in Space
Definition of a Cylinder
• Let C be a curve in a plane and let L be a line
not in a parallel plane. The set of all lines
parallel to L and intersecting C is called a
cylinder. C is called the generating curve (or
directrix), and the parallel lines are called
rulings.
• Note: If one of the variables is missing from the
equation of a cylinder, its rulings are parallel to
the coordinate axis of the missing variable.
Examples:
y z4
2
z e  0
x
Quadric Surfaces
• The equation of a quadric surface in space is a
second-degree equation of the form
Ax  By  Cz  Dxy  Exz  Fyz  Gx  Hy  Iz  J  0
2
2
2
• There are six basic types of quadric surfaces:
1) Ellipsoid
• Standard Form
2
2
2
x
y
z
 2  2 1
2
a
b
c
2) Hyperboloids of One Sheet
• Standard Equation:
2
2
2
x
y
z
 2  2 1
2
a b c
3) Hyperboloid of Two Sheets
• Standard equation
2
2
2
x
y
z
 2  2  2 1
a
b
c
4) Elliptic Cone
• Standard Equation:
x2 y2 z 2
 2  2 0
2
a
b
c
5) Elliptic Paraboloid
2
• Standard Equation:
2
y
x
z 2  2
b
a
6) Hyperbolic Paraboloid
• Standard Equation:
2
2
y
x
z 2  2
b
a
To Sketch a Quadric Surface
1) Write the surface in standard form.
2) Determine the traces in the coordinate
planes by setting each variable =0
For example: To get the trace in the xyplane, set z=0. To get the trace in the
xz-plane, set y=0, etc.
3) If needed, find the traces in planes that
are parallel to coordinate planes by
holding a variable constant.
Examples:
Identify and Sketch:
1) 4 x 2  y 2  4 z  0
2)
x  y / 4  z 1
2
2
2
#42
• Sketch the region bounded by the graphs of the
equations. X=0, y=0, z=0
z  4 x
2
y  4  x2