Graphics Reference
In-Depth Information
z
y
(x,y,f(x,y))
y
(x,x
2
)
x
x
x
(x,y)
f(x) = x
2
f(x,y) = x
2
+ y
2
(a)
(b)
Figure 8.1.
Parameterizations of graphs.
z
y
q
x
x
Figure 8.2.
A parameterization for a sphere.
number, an angle, it is therefore clear that one could tell someone how to get to a
point on the sphere by telling that person two numbers, x and q. The x-value speci-
fies a circle and its radius and the q-value a point on that circle. This leads to the para-
meterization of
S
2
defined by
(
)
2
2
(
)
=
F x
,
q
x
,
1
-
x
cos ,
q
1
-
x
sin
q
,
-£ £
1
x
1 0
,
£ £
q
p
.
(8.2)
See Figure 8.2. Note that
2
is the radius of the circle at x.
1
- x
8.2.3. Example.
A slightly more complicated example is a parameterization of the
Moebius strip. See Figure 8.3. One can think of the Moebius strip as a union of line
segments parameterized by [-1,1], one for each point on the circle of radius 2 about
the origin. Now an ordinary vertical cylinder centered on the z-axis of radius 2 could
be parameterized by
