164
J. Benson and F. Valiquette
A
x
I n
= −
(1 + 1 K 2
n−1 )I n
K n−1
S
−1
+ 1 (( 2 J
2
n − I
2
n ) − 1 +
2 J n J n+1 (1 + 1 K 2
n )
K 2
n
S,
A
y
I n
=
2 (1 + 1 K 2
n−1 )J n
K n−1
S
−1
+ 1 2 J n (K n − 2I n ) −
2 J n
K n
−
2 J n I n+1 (1 + 1 K 2
n )
K 2
n
S,
A
x
J n
= −
(1 + 1 K 2
n−1 )J n
K n−1
S
−1
− 2 1 I n J n −
I n J n+1 (1 + 1 K 2
n )
K 2
n
S,
A
y
J n
= −
(1 + 1 K 2
n−1 )I n
K n−1
S
−1
+
I n
K n
− 1 + 1 (I
2
n − 2 J
2
n − I n K n )
+
I n I n+1 (1 + 1 K 2
n )
K 2
n
S,
A
x
K n
= −
I n+1 (1 + 1 K 2
n )
K n
S − (1 + 1 K
2
n ),
A
y
K n
= −
2 J n+1 (1 + 1 K 2
n )
K n
S.
(6)
6 Geometric Flows
In the Euclidean plane, where 1 = 0 and 2 = 1, the authors of [1–3] define a
discrete curve to be parametrized by arc-length if K n = |Δz n | = 1 for all n ∈ Z.
We extend the notion of arc-length parametrized discrete curves to the remaining
Cayley–Klein planes as follows.
Definition 4 A discrete curve z n is said to be parametrized by arc-length if
K n = K
is constant for all n ∈ Z.
In light of the Maurer–Cartan invariants (5), we notice that when 1 = −1, we
cannot set K = 1. This explains why the value of the constant in Definition 4
remains unspecified. From now on, we restrict our considerations to arc-length
parametrized discrete curves. For such curves, we introduce the discrete curvature
κ n =
J n
I n − K
=
J n
I n − 1
,