Discrete Curve Flows in Two-Dimensional Cayley–Klein Geometries
163
For the invariant one-forms, we have the recurrence relation
S[
n
n ] = ι n+1 (dz n+1 ) = m n · ι n (dz n+1 ) = −
I n+1 + i 2 J n+1
K n (1 + 1 K 2
n )
n+1
n
,
so that
(ω n+1
n
+ i 2 σ n+1
n
) = n+1
n
= −
1
K n
(1 + 1 K 2
n )(I n+1 − i 2 J n+1 )S[ n
n ]
= −
1 + 1 K 2
n
K n
(I n+1 S(ω n
n ) + 2 J n+1 S(σ n
n )) + i 2 (−J n+1 S(ω n
n ) + I n+1 S(σ n
n ))
.
Similarly, for the backward shift,
ω n−1
n
+ i 2 σ n−1
n
= n−1
n
= m n−1 · S −1 [ n
n ]
= −
1 + 1 K 2
n−1
K n−1
(I n + i 2 J n )S −1 [ n
n ]
= −
1 + 1 K 2
n−1
K n−1
(I n S −1 (ω n
n ) − 2 J n S −1 (σ n
n )) + i 2 (J n S −1 (ω n
n ) + I n S −1 (σ n
n ))
.
5 Invariant Linearization Operators
Definition 3 The invariant linearization of an invariant I n , is the invariant difference operator A I n satisfying the equality
dI n = A I n
ω n
n
σ n
n
.
To compute the invariant linearization of I n , compute its differential using the
recurrence relations (4) for the exterior derivative. The result is a linear combination
of the invariant one-forms ω k
n , σ k
n . Then, use the recurrence relation for the shift map
to express ω k
n , σ k
n in terms of ω n
n , σ n
n and their shifts. We note that these computations
can be done symbolically, without requiring the coordinate expressions for the
invariant I n and the one-forms ω k
n , σ k
n .
For the normalized invariants I n = ι n (x n−1 ), J n = ι n (y n−1 ), and K n = ι n (x n+1 ),
the components of the invariant linearization operators are