Graphics Reference
In-Depth Information
be a curve parallel to p(t), where n(t) are the unit vectors pointing out of the region
enclosed by p(t). Show that
()
=
()
+ 2p .
Length
of
p
t
Length of
p t
d
d
Section 9.8
Let F : [a,b] Æ
C
be a regular parameterization of a curve
C
in
R
2
. Show that the volume
V(F) of the parameterization F as defined in this section agrees with the definition of
the length of a curve given in Section 9.2.
9.8.1.
9.8.2.
Show by direct computation that Theorem 9.8.5 computes the same length of a curve
in the plane as the definition in Section 9.2.
9.8.3.
Find the coefficients of the first fundamental form for the surface of revolution that is
parameterized by F(x,q) = (f(x) cos q,f(x) sin q,g(x)).
9.8.4.
Consider the parameterization
[
]
¥
(
)
Æ
R0
2
Y :
02
,
p
0
,
p
defined by
cos
qf
f
sin
cos
,
sin
qf
f
sin
cos
Ê
Á
ˆ
˜
(
)
=
Yqf
,
,
1
-
1
-
which is the composite of the spherical coordinate parameterization of
S
2
in Example
9.8.4 and the stereographic projection. Compute the coefficients E, F, and G with respect
to Y. Deduce from this and the computations in Example 9.8.4 that the stereographic
projection does not preserve area.
Section 9.9
Show that (0,0,0) is a planar point of the surface of revolution where the curve z = x
4
is revolved about the z-axis.
9.9.1.
9.9.2.
Prove that a curve in a surface is an asymptotic line if and only if every point of
the curve is an inflection point or the osculating plane at the point is tangent to the
surface.
9.9.3.
Let F(u,v) be a regular parameterization of a neighborhood of a point
p
in a surface
S
. Prove that the u- and v-parameter curves of the patch, that is, F(u,0) and F(0,v),
respectively, are asymptotic lines for the surface if and only if L = N = 0 at each
point.
9.9.4.
Let
S
be the surface of revolution obtained by revolving a curve
[
Æ
2
()
=
(
() ()
)
g
:
01
,
R
,
g
t
xt
,
yt
,
about the x-axis. Assume that y(t) > 0. Let
(
)
[]
¥
[
]
Æ
3
()
=
() ()
()
j
:
01
,
0 2
,
p
R
,
j
t
,
q
x t
,
y t
cos
q
,
y t
sin
q
,
