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Unformatted text preview: 3.6 CYLINDERS AND QUADRIC SURFACES ❙❙❙❙ 869 direction of the y-axis. The graph is a surface, called a parabolic cylinder, made up of
inﬁnitely many shifted copies of the same parabola. Here the rulings of the cylinder are
parallel to the y-axis.
z FIGURE 1 0 The surface z=≈ is a parabolic cylinder. y x We noticed that the variable y is missing from the equation of the cylinder in Example 1. This is typical of a surface whose rulings are parallel to one of the coordinate axes.
If one of the variables x, y, or z is missing from the equation of a surface, then the surface
is a cylinder.
EXAMPLE 2 Identify and sketch the surfaces. (a) x 2 y2 (b) y 2 1 z2 1 SOLUTION (a) Since z is missing and the equations x 2 y 2 1, z k represent a circle with
radius 1 in the plane z k, the surface x 2 y 2 1 is a circular cylinder whose axis is
the z-axis (see Figure 2). Here the rulings are vertical lines.
(b) In this case x is missing and the surface is a circular cylinder whose axis is the x-axis
(see Figure 3). It is obtained by taking the circle y 2 z 2 1, x 0 in the y z-plane and
moving it parallel to the x-axis.
z z y
y x FIGURE 2 ≈+¥=1 | NOTE like x 2
x2 y2 ■ FIGURE 3 ¥+z@=1 When you are dealing with surfaces, it is important to recognize that an equation
y 2 1 represents a cylinder and not a circle. The trace of the cylinder
1 in the xy-plane is the circle with equations x 2 y 2 1, z 0. Quadric Surfaces
A quadric surface is the graph of a second-degree equation in three variables x, y, and z.
The most general such equation is
Ax 2 By 2 Cz2 D xy Eyz Fxz Gx Hy Iz J 0 5E-13(pp 868-877) 870 ❙❙❙❙ 1/18/06 11:35 AM Page 870 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE where A, B, C, . . . , J are constants, but by translation and rotation it can be brought into
one of the two standard forms
Ax 2 By 2 Cz2 J 0 Ax 2 or By 2 Iz 0 Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane.
(See Section 11.5 for a review of conic sections.)
EXAMPLE 3 Use traces to sk...
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- Fall '09