In the classical Euclidean geometry of curves, the fundamental tool is the Frenet–Serret frame. In affine geometry, the Frenet–Serret frame is no longer well-defined, but it is possible to define another canonical moving frame along a curve which plays a similar decisive role. The theory was developed in the early 20th century, largely from the efforts of Wilhelm Blaschke and Jean Favard.
Let x(t) be a curve in . 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 . 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 quantities define a special affine frame for the affine space , 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.
The normalization of the curve parameter s was selected above so that
Ifn≡0 (mod 4) or n≡3 (mod 4) then the sign of this determinant is a discrete invariant of the curve. A curve is called dextrorse (right winding, frequently weinwendig in German) if it is +1, and sinistrorse (left winding, frequently hopfenwendig in German) if it is −1.
In three-dimensions, a right-handed helix is dextrorse, and a left-handed helix is sinistrorse.
whose columns are the first n derivatives of x (still parameterized by special affine arclength). Then,
In concrete terms, the matrix C is the pullback of the Maurer–Cartan form of the special linear group along the frame given by the first n derivatives of x.