2. On the Fundamental Theorem for Curves in Space

Supplementary Notes for MM08 Geometry I
2. On the Fundamental Theorem for Curves
in Space
Andrew Swann
These notes are a supplement to Theorem 2.3 in Pressley [1] which gives the
Fundamental Theorem for Curves in Space. The theorem has two parts. The rst
states that two curves in R 3 parameterised by arclength with the same curvature and
torsion , with > 0, are equivalent under rigid motions of R 3 . The second concerns
existence of curves in R 3 with prescribed curvature and torsion. The proof of the
rst part is relatively direct and covered perfectly adequately by Pressley's text. The
second part however relies on results from the theory of linear dierential equations,
in particular long-time existence of solutions, that are not covered by previous courses
here in Odense. These notes will obtain the desired result using only the local existence
result proved in MM04.
We begin by considering a smooth curve : (; ) ! R 3 , s 7! (s), which is
parameterised by arclength and whose curvature is never zero. The latter
assumption implies that the torsion is dened and we may write ft; n; bg for the
Frenet frame of . Then these vectors satisfy the Serret-Frenet equations
9>=
t 0 = n
n 0 >;
= t +b
(2.1)
b 0 = n
where 0 = d=ds. We wish to show that these equations essentially determine . The
rst part of Pressley's Theorem 2.3 shows that is unique up to rigid motions. We
are thus left with:
Theorem 2.1 (Existence part of the Fundamental Theorem). Let k; t : (; ) ! R be
smooth functions with k(s) > 0 for all s 2 (; ). Then there exists a smooth curve
: (; ) ! R 3 which is parameterised by arclength, has curvature (s) = k(s) and
has torsion (s) = t(s), for all s 2 (; ).
2.1

To start the proof, let us look at the Serret-Frenet system (2.1) in matrix form.
Regarding t, n and b
0
as row
1
vectors
0
we have
1 1
B@ t n
b
CA0
We therefore consider the equations
for V : (; ) ! M 3 (R) with
A(s) =
=
B@ 0 CA 0 B@ 0 t n
0
0 0
b
CA : (2.2)
V 0 = AV (2.3)
0
B@ 0 (s) 0
(s) 0 (s)
0 (s) 0
1
CA : (2.4)
One may get local existence of solutions to (2.3) with prescribed initial data,
by applying the following theorem from MM04 (Supplerende noter til punktmængdetopologi,
Sætning 2.2, side 6):
Theorem 2.2. Let B R n+1 be an open set. Suppose f : B ! R n , (s; x) 7! f (s; x)
is a continuous function whose partial derivatives @f=@x j , j = 1; : : : ; n, exist and
are continuous in B.
Fix a point (s 0 ; x 0 ) in B. Then the dierential equation
x 0 (s) = f (s; x(s))
with initial condition
x(s 0 ) = x 0 ;
has a unique solution x(s) with s in an interval (s 0 "; s 0 + ") for some " > 0.
To apply this to (2.3), note that M 3 (R) = R 9 , so take n = 9 and put
f (s; V ) = A(s)V;
B = (; ) ¢ M 3 (R):
As the entries of A are smooth in s, the required partial derivatives of f exist and are
smooth. Thus, we have local solutions to (2.3) around any given point s 0 2 (; )
with any given initial data V 0 2 M 3 (R).
2.2

F
or
two solutions V 1 and V 2 to (2.3) on some subinterval of ( 0 ; 0 ) (; ),
consider the matrix V T
2
V 1
obtained by multiplying the transpose of V 2 onto V 1 .
Dierentiating we have
(V T
2
V 1
) 0 = (V 0
2 )T V 1
+ V T
2
V 0
1 = (AV 2) T V 1
+ V T
2 (AV 1 ) = V T
2 (AT + A)V 1
= 0;
since A in (2.4) skew-symmetric. We have thus proved
Lemma 2.3. Suppose V 1 and V 2 satisfy V 0
i
= AV i
, i = 1; 2 on ( 0 ; 0 ) with A as
in (2.4). Then V T
2
V 1
is constant on ( 0 ; 0 ).
Fix a point s 0 2 (; ) and let V be the maximal solution to (2.3) with initial
condition V (s 0 ) = Id, where Id is the (3 ¢ 3) identity matrix. Here maximal
means that V is dened on some subinterval ( 0 ; 0 ) of (; ) and that there is no
such solution dened on a larger subinterval. Our aim is to prove that ( 0 ; 0 ) = (; ).
Lemma 2.4. V (s) is an orthogonal matrix for each s 2 ( 0 ; 0 ), i.e.,
V T (s)V (s) = Id :
Proof. By Lemma 2.3, V T V is constant on ( 0 ; 0 ). At s 0 it has value Id T Id = Id.
Lemma 2.5. The set
O(3) = f X 2 M 3 (R) : X T X = Id g
of orthogonal matrices is a compact subset of M 3 (R).
Proof. If X is orthogonal, then each column x i of X is a unit vector in R 3 . Thus as
vector in R 9 , X has length kx 1 k 2 + kx 2 k 2 + kx 3 k 2 = 3. So O(3) is bounded in M 3 (R).
The function F : M 3 (R) ! M 3 (R) given by F (X) = X T X is continuous, so O(3) =
F 1 (fIdg) is closed. Therefore, O(3) is closed and bounded and hence compact.
Suppose 0 < . Let (~s i ) 1 i=1 be a sequence of points in ( 0; 0 ) converging to 0 .
Then as V (~s i ) lie in the compact set O(3), there is a subsequence (s n ) 1 n=0
and a point
~V 2 O(3) such that
V (s n ) ! ~ V as n ! 1:
Let W be a solution of the system (2.3) with initial condition W ( 0 ) = V ~ . Then W (s)
is dened on some interval ( 1 ; 1 ) around 0 . We may assume 1 > 0 . Then on
( 1 ; 0 ) we have by Lemma 2.3 that W T V is constant. However, (W T V )(s n ) ! V ~ T ~V =
Id as n ! 1, so W T V = Id on ( 1 ; 0 ). Now V is orthogonal, so W = (V T ) 1 = V
on this interval. Putting
^V (s) =(
V (s) 0 < s < 0 ;
W (s) 0 6 s < 1 ;
2.3

we thus obtain a solution to the equations (2.3), with initial value Id at s = s 0 , on the
larger interval ( 0 ; 1 ). This contradicts the maximality of V and we conclude that
0 = . A similar argument shows 0 = and thus our maximal solution V is dened
on the whole of (; ).
To conclude the proof of Theorem 2.1,
0
write
1
our maximal solution V as
B@ t CA V = n :
b
Put
(s) =
Z
s
s 0
t(u) du:
Then : (; ) ! R 3 is a curve with 0 = t. As V is orthogonal, we have V 1 = V T ,
so V V T = Id showing that ft; n; bg is orthonormal. In particular, t is a unit vector
and is parameterised by arclength. We now have t 0 = kn, so = k and n is the
normal to . Now b is proportional to t ¢ n with ht ¢ n; bi = det(V ). For V 2 O(3),
1 = det(Id) = det(V T V ) = det(V T ) det(V ) = det(V ) 2 , so det V = ¦1. As det V (s)
is continuous on (; ) and det V (s 0 ) = det Id = 1, we have det V (s) = 1 for all s,
b = t ¢ n, and b is the binormal. Finally, b 0 = tn shows = t, as required.
References
[1] A. Pressley, Elementary dierential geometry, Springer Undergraduate Mathematics
Series, Springer-Verlag London Ltd., London, 2001.
2.4