Graphics Reference
In-Depth Information
Figure 12.3.
A surface of revolution tangent
plane problem.
The Curve Crosses the X-axis.
For example, the surface of revolution obtained by
rotating the segment [
A
,
B
] in Figure 12.3 about the x-axis has no tangent plane at
C
where the curve crossed the axis. Even if a tangent plane exists at such points, such
as when one revolves the upper half of the unit circle about the x-axis to get the unit
sphere
S
2
, problems may arise because the standard approach to getting the tangent
plane using the partials of the parameterization may fail. See Exercise 12.2.2.
Choosing a Direction for the “Outward” Normal.
The direction would most likely
depend on the orientation of the curve, but the curve may zigzag.
The main problems are typically caused by partial derivatives vanishing so that it is
messy trying to find the tangent plane at a point of the surface.
Next, we look at a number of important special cases of surfaces of revolution
and work out some concrete examples.
Consider the full surface of revolution
S
obtained by rotating a line segment
X
about the x-axis.
Case 1:
If
X
is parallel to axis, then
S
is a
cylinder
. See Figure 12.4(a).
Case 2:
If
X
is skew to axis, then
S
is a
truncated cone
. See Figure 12.4(b).
Case 3:
If
X
is orthogonal to the axis, then
S
is an
annulus
. See Figure 12.4(c).
12.2.1 Example.
Assume that
S
is the surface of revolution obtained by rotating
the segment
X
= [(0,1),(2,3)] about the x-axis. We want to find the tangent plane to
S
at
p
= (1,
2
,
2
) .
Solution.
The function
(
)
=
(
(
)
(
)
)
px
,
q
x x
,
+
1
cos ,
q
x
+
1
sin
q
parameterizes
S
and
∂
p
x
∂
p
(
)
=
(
)
(
)
=-+
(
(
)
(
)
)
x
,
q
1
,cos ,sin
q
q
and
x
,
q
0
,
x
1
sin ,
q
x
+
1
cos
q
.
∂
∂q
