Cylinders and Quadric Surfaces

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Section 13.6

• Cylinders and Quadric Surfaces

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SURFACES
The graph of an equation in three variables
(x, y, and z) is normally a surface. Two
examples we have already seen are planes and
spheres. Graphing surfaces can be complicated.
The best way is by finding the intersections of
the surface with well-chosen planes (e.g., the
coordinate planes). The intersections are called
cross sections those intersections with the
coordinate planes are called traces.
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CYLINDERS
In Calculus the term cylinder denotes a much
wider class of surfaces than the familiar right
circular cylinder. 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 (called the generating curve) An
equation of a cylinder is easy to recognize since
it will contain only two of the three variables
x, y, and z.
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QUADRIC SURFACES
If a surface is the graph in three-space of an
equation of second degree, it is called a quadric
surface. Cross sections of quadric surfaces are
conics. Through rotations and translations, any
general second degree equation can be reduced to
either Ax2 By2 Cz2 J 0 or Ax2 By2 Iz
0.
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SIX QUADRIC SURFACES
For graphs of these surfaces, see page 872 in the
text.