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If h≤(t) is orthogonal to
S
, then Theorem 9.10.2 easily implies that the geodesic
curvature vanishes. This proves part (1) of the theorem and the “if” part for (2). To
prove the “only if” part in (2), assume that h(t) is a constant speed geodesic. Since the
geodesic curvature of h(t) vanishes, h≤(t) is a linear combination of the orthogonal
vectors h¢(t) and
n
(h(t)). This is all that we would be able to say in general, but h(t)
has constant speed, so that h¢(t) • h¢(t) = c for some constant c. Differentiating this
equation shows that h≤(t) is orthogonal to h¢(t) and hence must be parallel to
n
(h(t)),
which is what we had to prove.
Note that the converse to Theorem 9.10.3(1) is false. Consider the (nonconstant
speed) planar curve h(t) = (t
2
,t
2
), t Œ [1,2]. Since h≤(t) = (2,2), h≤(t) is not orthogonal
to the plane even though its path, a straight line segment, is obviously a geodesic. This
example shows that all we can say about h≤(t) for a regular curve h(t) that is a geo-
desic is that h≤(t) is a linear combination of the vectors h¢(t) and
n
(h(t)). Even so,
because checking whether h≤ is orthogonal to the surface is such a simple test, some
texts turn the property into a definition.
Second definition of a geodesic:
A
geodesic
in a surface
S
in
R
3
is a regular curve
h(t) in
S
with the property that h≤(t) is orthogonal to the surface at h(t) for all t.
As we just saw, this is actually a stronger condition than necessary since it forces
a geodesic to be a
constant speed
regular curve. It is therefore technically not equiv-
alent to our first definition.
Because constant speed regular curves will appear frequently in this section, we
collect two of their important properties in a theorem.
Let h(t) be a regular curve in a surface
S
in
R
3
.
9.10.4. Theorem.
(1) If h≤(t) is orthogonal to the surface, then h(t) is a constant speed curve.
(2) If h(t) has constant speed, then the parameter t is proportional to the arc-
length parameter s.
Proof.
To prove (1), note that
(
)
¢=
hh
¢∑ ¢
2
hh
¢∑ ≤=
0
implies that h¢ • h¢ is a constant function. To prove (2), assume that |h¢(t)| is equal to
a constant c π 0, and check that g(s) =h(s/c) is the arc-length parameterization, that
is, s = ct.
Here is a criterion that is more directed at the question of when a
path
is a geo-
desic by telling us what kind of parameterization we should seek.
9.10.5. Theorem.
Let
S
be surface in
R
3
. If h : [a,b] Æ
S
is a regular curve with the
property that the vector h≤(t) lies in the plane spanned by h¢(t) and
n
(h(t)) for all t,
then the arc-length parameterization of the curve h(t) is a geodesic in
S
.
Proof.
Let a : [0,L] Æ [a,b] be the reparameterization of h(t), so that g(s) =h(a(s)) is
the arc-length parameterization. We have that
