Affine Geometry of Curves - The Affine Frame

The Affine Frame

Let x(t) be a curve in Rn. Assume, as one does in the Euclidean case, that the first n derivatives of x(t) are linearly independent so that, in particular, x(t) does not lie in any lower-dimensional affine subspace of Rn. Then the curve parameter t can be normalized by setting determinant

Such a curve is said to be parametrized by its affine arclength. For such a parameterization,

determines a mapping into the special affine group, known as a special affine frame for the curve. That is, at each point of the, the quantities define a special affine frame for the affine space Rn, consisting of a point x of the space and a special linear basis attached to the point at x. The pullback of the Maurer–Cartan form along this map gives a complete set of affine structural invariants of the curve. In the plane, this gives a single scalar invariant, the affine curvature of the curve.