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Note also that, although k is always nonnegative by definition, no such condition holds
for k n or k g . In fact, changing the direction of n (g(s)) changes the sign of both of these
values. We already discussed the normal curvature of a curve in the last section. In
this section it is the geodesic curvature that is interesting because that is what is
needed to define geodesics.
Here are some facts that describe the geometry behind the function k g a little more.
Let g(s) be a curve in a surface S in R 3
9.10.1. Theorem.
parameterized by
arc-length.
(1) If the surface S is a plane, then the geodesic curvature function k g (s) of the
planar curve g(s) is just the ordinary planar signed curvature function of the
curve g(s).
(2) k g (s) = k(s) cos a(s), where a(s) is the angle between the unit normal n (g(s))
and the binormal B(s).
(3) k g (s) is the signed curvature at g(s) of the planar curve that is the orthogonal
projection of g(s) onto the tangent plane of S at g(s).
Proof. In the case where S is the xy-plane, part (1) is an easy consequence of the
definition of n S (s) and the definition of curvature in a plane. Part (2) follows from the
following string of equalities:
() =≤ ()
() =≤ ()
(
(
()
) ¥
()
)
k
g s
g
s
n
s
g
s
n
g
s
T s
S
() ()
(
(
()
) ¥
()
) =
()
(
()
)
(
() ¥
()
)
=
k
sNs
n
g
s
Ts
k
s
n
g
s
Ts
Ns
()
(
()
)
() =
()
()
=
k
s
n
g
s
Bs
k
s
cos
a
s
.
Part (3) generalizes part (1) and its truth follows intuitively from the formula in Part
(2). The proof is left as Exercise 9.10.1.
We have considered the arc-length parameterization of curves in the discussion
above because the formulas are simpler in that case; however, as usual, we are inter-
ested in geometric concepts associated to paths rather than their parameterizations.
First, we extend the definitions to regular curves in the obvious way.
Definition. If h(t) is a regular curve for which g(s) =h(s(s)) is the arc-length para-
meterization after an orientation-preserving change of parameters t =s(s), then the
normal curvature k n (t) and the geodesic curvature k g (t) of h(t) at t are defined to be the
normal and geodesic curvatures, respectively, of the curve g(s) at the point s =s -1 (t).
Fortunately, we have a formula for computing the geodesic curvature for an arbi-
trary regular curve.
If h(t) be a regular curve in a surface S in R 3 , then
9.10.2. Theorem.
()
(
(
()
) ¥ ()
)
h
t
n
h
t
h
t
() =
k g t
.
3
¢ ()
h
t
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