160
J. Benson and F. Valiquette
with local coordinates (n, z n−1 , z n , z n+1 ), where the fiber C 3
i 2
consists of three
neighboring points on the curve. We refer to J
[2] as a discrete jet space since the
points (z n−1 , z n , z n+1 ) are sufficient to provide finite difference approximations of
the order 2 jet of a differentiable curve. A moving frame is then an equivariant
map ρ : J
[2]
→ SO 1 ,, 2 (3) from the discrete jet space into the Cayley–Klein group.
Choosing the cross-section
K = {z n = 0, y n+1 = 0} ⊂ J
[2] ,
a moving frame is obtained by requiring that ρ n · (z n−1 , z n , z n+1 ) ∈ K, which yields
the normalization equations Z n = Y n+1 = 0. Solving the normalization equations
for the group parameters, we obtain the moving frame
β = −
1 + |z n | 2 e
i 2 θ z n ,
α =
1 + |z n | 2 e
i 2 θ ,
where e
i 2 θ
= C 2 (θ ) + i 2 S 2 (θ ) is the generalized complex exponential function
with
C 2 (θ ) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
cos
√
x
> 0
1
= 0
cosh
√ −x
< 0
,
S 2 (θ ) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1
√
sin
√
x
> 0
x
x= 0
1
√ −
sinh
√
−x
< 0
,
denoting the generalized cosine and sine functions, and where the angle θ is
determined by the equation
T 2 (2θ) = −
Δx n η + Δy n ξ
Δx n ξ − 2 Δy n η
,
where T 2 (θ ) =
S 2 (θ )
C 2 (θ )
denotes the generalized tangent function, and
Δx n = x n+1 − x n ,
ξ= Re(1 + 1 z n z n+1 ),
Δy n = y n+1 − y n ,
η= Im(1 + 1 z n z n+1 ).
Given a moving frame, there is a systematic procedure, known as invariantization,
for constructing invariant functions. For example, the invariantization of a coordinate function z k is the invariant ι n (z k ) = ρ n · z k . Invariantizing x n+1 and z n−1 , we
obtain the invariants