Choose coordinates for the affine plane such that the area of the parallelogram spanned by two vectors a = (a1, a2) and b = (b1, b2) is given by the determinant In particular, the determinant is a well-defined invariant of the special affine group, and gives the signed area of the parallelogram spanned by the velocity and acceleration of the curve β.
This determinant undergoes then a transformation of the following sort, by the chain rule: The reparameterization can be chosen so that provided the velocity and acceleration, dβ/dt and d2β/dt2 are linearly independent.
An alternative approach, rooted in the theory of moving frames, is to apply the existence of a primitive for the Darboux derivative.
The special affine group acts on the Cartesian plane via transformations of the form with ad − bc = 1.
These vector fields can be determined by the following two requirements: Similarly, the action of the group can be extended to the space of any number of derivatives (x, y, y′, y″,…, y(k)).
The prolonged vector fields generating the action of the special affine group must then inductively satisfy, for each generator X ∈ {T1, T2, X1, X2, H}: Carrying out the inductive construction up to order 4 gives The special affine curvature does not depend explicitly on x, y, or y′, and so satisfies The vector field H acts diagonally as a modified homogeneity operator, and it is readily verified that H(4)k = 0.
[6] Namely, where v is the speed of the hand, κ is the Euclidean curvature and γ is a constant termed the velocity gain factor.