Sec.10.6
Cylinders and Quadric Surfaces
Sketches:
To sketch a surface it is useful to find the traces of cross-sections where the planes passing
through the surface are parallel to the coordinated planes.
A cylinder is a surface that consist of all lines (rulings) parallel to a given line and passing through a
given plane curve (generating curve).
If one of the variables x y, or z is missing from the equation of a
surface, then the surface is a cylinder.
EX1 Sketch the graph of the surfaces below.
A.
z = y
2
B.
x
2
+ z = 1
A quadric surface is the graph of a second degree equation in 3 variables say x, y, and z.
Quadratic
surfaces are the 3D counter part of conic sections in 2D. (See table on page 655 of text.)
Ellipsoid
All three traces are ellipses.
2
2
2
2
2
2
1
x
y
z
a
b
c
+
+
=
If a = b = c > 0, we have a sphere.
EX 2 Sketch
4x
2
+ 9y
2
+ 36z
2
= 36

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