Graphics Reference
In-Depth Information
11.13.2 Lemma.
Let c Π[a,b]. Given any vector
v
, there is a unique relatively par-
allel normal vector field
X
(t) to p(t) with the property that
X
(c) =
v
.
Proof.
The uniqueness follows from the fact that the difference of relatively paral-
lel vector fields is again such a field and the fact that the difference is 0 at c and
hence would have to be constantly equal to 0. The existence part follows by express-
ing
X
(t) in terms of a continuously varying orthonormal basis (T(t),n
1
(t),n
2
(t)) for
R
3
along p(t) (which always exists locally) and showing that a solution depends
on the existence of a solution to Serret-Frenet type differential equations. See
[Bish75].
Definition.
A tangential vector field
X
along p(t) is said to be
relatively parallel
to
p(t) if there is a constant c so that
X
¢(t) = cT(t) for all t. An arbitrary vector field
X
is
said to be
relatively parallel
to p(t) if the tangential and normal parts (that is, the vector
fields consisting of the orthogonal projections of
X
(t) onto the tangent line and normal
plane of the curve at p(t), respectively) are relatively parallel.
Lemma 11.13.2 generalizes to the following:
11.13.3 Theorem.
Let p: [a,b] Æ
R
3
be a C
k
regular space curve, k ≥ 2. Given c Œ
[a,b] and any vector
v
there is a unique C
k-1
relatively parallel vector field
X
(t) along
p(t) with
X
(c) =
v
. The dot product of any two relatively parallel vector fields is
constant.
Proof.
See [Bish75].
Now it is clear that the space of relatively parallel vector fields along p(t) is a vector
space over
R
. Furthermore, it follows from Theorem 11.13.3 that this is a three-
dimensional vector space.
Definition.
A triple (T,n
1
,n
2
) of orthonormal relatively parallel vector fields along p(t)
is called a
relatively parallel adapted frame field
for p(t). (As usual, T denotes the unit
tangent vector field for p(t).) The frames (T(t),n
1
(t),n
2
(t)) are called
parallel transport
frames
.
11.13.4 Theorem.
Let c Π[a,b]. Any frame (T(c),
u
1
,
u
2
) at p(c) defines a unique
relatively parallel adapted frame field (T,n
1
,n
2
) for p(t) so that n
1
(c) =
u
1
and
n
2
(c) =
u
2
.
Proof.
One can show that a relatively parallel adapted frame field (T,n
1
,n
2
) satisfies
(and can be obtained as a solution to) the differential equations
¢
()
=
() () ()
+
()
()
()
Tt
vtk tnt vtk tn t
1
1
2
2
¢
()
=-
() () ()
¢
()
=-
()
nt
vtktTt
1
1
() ()
nt
vtktTt
.
(11.136)
2
2
