Graphics Reference
In-Depth Information
But y π 0 because the curve h(t) does not cross the x-axis, and so this expression will
vanish if and only if y¢=0. This is equivalent to saying that F
t
(t,q) = (x¢(t),0,0), which
is what we were trying to show.
As an application, consider the torus and its parameterization as defined in
Example 9.9.18. We can think of this torus as a surface of revolution where the curve
()
=
(
)
+
(
)
=
(
)
=
(
() ()
)
h t
0,
Rr
cos ,
t r
sin
t
r
cos ,
t Rr
+
sin
t
xt
,
yt
gets revolved about the x-axis. It is clear that the meridians, which are circles in this
case, are geodesics. On the other hand, y¢(t) = cos t = 0 implies that t =p/2 or 3p/2. In
other words, the only circles of latitude that are geodesics are the inner circle with
radius R - r and the outer circle with radius R + r.
Our definition of geodesic applied only to surfaces in
R
3
because we made use
of the normal vectors to the surface. Obviously, the normal curvature of a curve
depends on having a normal vector, but it turns out that its geodesic curvature does
not.
9.10.10. Theorem.
(Minding) The geodesic curvature of a curve in a surface is a
metric invariant.
Proof.
Specifically, what we want to show is that the geodesic curvature depends
only on the curve and the metric coefficients of the surface. How the surface is
imbedded in
R
3
plays no role.
Let
F :
US
Æ
be a regular one-to-one parameterization of a surface
S
in
R
3
and let
n
(
p
) be the unit
normal vector to the surface
S
at the point
p
. Let
[
]
Æ
S
g :
ab
be a curve parameterized by arc-length. Express g in the form g(s) =F(m(s)),
where
[
]
Æ
R
2
()
=
(
() ()
)
m
:
a b
,
and
m
s
u s v s
,
.
Now
g¢ =
FF
u
u
¢ +
v
¢
(9.69)
v
and
2
2
g≤=
F
u
¢ +
2
F
u v
¢ ¢ +
F
v
¢ +
F
u
≤+
F
v
≤
.
(9.70)
uu
uv
vv
u
v
Substituting the right-hand side of equations (9.58) for the F
uu
, F
uv
, and F
vv
in equa-
tion (9.70) gives
