Graphics Reference
In-Depth Information
Figure 9.25.
Defining geodesic curvature.
n(g(s))
g ¢¢(s)
T(s)
p = g(s)
S
n
S
(s)
T
p
(S)
parameterized by
arc-length
. See Figure 9.25. As a curve in
R
3
we know that g(s) can
be described in terms of its Frenet frame (T(s),N(s),B(s)), its curvature k(s), and
its torsion t(s). Define a unit vector n
S
(s) so that (T(s),n
S
(s),
n
(g(s))) is a frame that
defines the standard orientation in
R
3
. The vector n
S
(s) will be tangent to
S
at g(s)
and is defined by the equation
()
=
(
()
)
¥
()
ns
n
g
s
Ts
.
(9.63)
S
Recall that the second derivative g≤(s) is closely related to the curvature of the
space
curve g(s). Since the tangent space to
R
3
at g(s) is the direct sum of the tangent space
of
S
at g(s) and the one-dimensional orthogonal subspace with basis
n
(g(s)), we can
write any vector, in particular g≤(s), in the form
g≤
()
=
()
+
()
sWsVs
(9.64)
where V(s) lies in the tangent space of
S
at g(s) and W(s) is a multiple of
n
(g(s)). The
vector V(s) can also be described more explicitly. Since T(s) is tangent to
S
, W(s) • T(s)
= 0. But g(s) is arc-length parameterization, so that g≤(s) • T(s) = 0. It follows from this
and equation (9.64) that V(s) • T(s) = 0, that is, V(s) is orthogonal to T(s). In other
words, the vector V(s) is orthogonal to both
n
(g(s)) and T(s). Equation (9.63) now
implies that V(s) is a multiple of n
S
(s). Putting all these facts together implies that
there are unique functions k
n
(s) and k
g
(s), so that
≤
()
=
() ()
=
()
(
()
)
+
()
()
g
s
k
s Ns
k
s
ns
g
k
s ns
.
(9.65)
n
g
S
The value k
n
(s) =g≤(s) •
n
(g(s)) is of course just the normal curvature of the curve g(s)
at s.
Definition.
The value k
g
(s) is called the
geodesic curvature
of the
arc-length
para-
meterized curve g(s) at s.
The geodesic curvature function was introduced by F. Minding in 1830. Because
we have orthonormal vectors, the following relation holds between k, k
n
, and k
g
:
2
2
2
kkk
=
ng
.
(9.66)
