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Proof.
Assume that h(t) =g(a(t)), where g(s) is the arc-length parameterization of
the curve. Differentiating we get
¢
()
=¢
()
(
)
¢
()
≤
()
=≤
()
(
)
¢
()
+¢
()
2
(
)
≤
()
h
t
gaa
t
t
and
h
t
gaa
t
t
gaa
t
t
.
Therefore, |h¢(t)| = |a¢(t)|, (using the fact that |g¢(s)| = 1), and
≤
()
¥
()
=¢
()
≤
3
(
()
)
¥¢
()
(
)
h
t
h
t
a
t
g
a
t
g
a
t
(using the fact that g¢¥g¢=
0
). We now have the following string of equalities:
()
=
(
()
)
=≤
()
(
)
∑
(
()
)
k
t
k
ag a a
ga ga ga
ga g a g a
hh h
t
t
n
t
g
g
S
=≤
()
(
)
∑
(
(
(
()
)
)
¥¢
()
(
)
)
t
n
t
t
(
(
()
)
)
∑≤
()
(
(
)
¥¢
()
(
)
)
=-
n
n
t
t
t
(
()
)
∑≤
()
¥
()
(
)
≤
()
∑
(
(
()
)
¥
()
)
t
t
t
h
t
n
h
t
h
t
=-
=
.
3
3
¢
()
¢
()
a
t
h
t
The theorem is proved.
We are ready to look at the problem of generalizing straight lines. We first gener-
alize property (1) of a straight line, namely, that it is not curved. The tangent plane
at a point of a surface is a good approximation to the surface. This suggests that we
would like the projection of a geodesic onto the tangent plane to be a straight line,
and so we take our cue from Theorem 9.10.1(3).
First definition of a geodesic:
A
geodesic
in a surface
S
in
R
3
is a regular curve in
S
whose geodesic curvature is zero everywhere.
According to the definition, a geodesic is a
function
, but, as usual, we are really
interested in properties of
paths
.
Definition.
Let
S
be a surface in
R
3
. A subset
X
in
S
is called a
geodesic path
, or
simply a
geodesic
, if there is a geodesic s : [a,b] Æ
S
with
X
=s([a,b]). A curve
g : [c,d] Æ
S
is said to
generate a geodesic path
if g([c,d]) is a geodesic path.
The next theorem contains some useful criteria for when regular curves are
geodesics.
Let
S
be a surface in
R
3
.
9.10.3. Theorem.
(1) If h : [a,b] Æ
S
is a regular curve with the property that h≤(t) is orthogonal to
S
at h(t) for all t, then h(t) is a geodesic.
(2) A constant speed regular curve h : [a,b] Æ
S
is a geodesic if and only if h≤(t)
is orthogonal to
S
at h(t) for all t.
Proof.
If we have arc-length parameterization, then both parts of the theorem are
obvious from Equation (9.65). Assume that h(t) is a regular curve but not necessarily
the arc-length parameterization.
