166
J. Benson and F. Valiquette
Theorem 1 Let I n be an invariant. Under the geometric curve flow (8), the
invariant I n evolves according to the differential-difference equation
∂I n
∂t
= A I n
α n
β n
,
where A I n is the invariant linearization of I n .
To preserve the fact that our curves are parametrized by arc-length, we consider
flows that are arc-length preserving. Thus, we require that
∂K n
∂t
= A
x
K n
(α n ) + A
y
K n
(β n ) = 0.
This leads to the finite difference equation
α n +
I n+1 α n+1 + 2
J n+1 β n+1 = 0.
(9)
If 2 = 0, then
β n = −
1
2
J n
(
I n α n + α n−1 ),
(10)
where α n is arbitrary. If 2 = 0, Eq. (9) reduces to
α n+1 = −
1
I n+1
α n
whose solution is
α n = (−1)
n π k
1
I k+1
, 0, n
α 0 ,
(11)
where
π k (f k , n 0 , n) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
n−1
k=n 0
f k n > n 0
1
n = n 0
n 0 −1
k=n
1
f k
n < n 0
.
In this case, we note that β n is arbitrary. We observe that both constraints (10)
and (11) are satisfied when the components of the flow are