In the classical Euclidean geometry of curves, the fundamental tool is the Frenet–Serret frame.
The theory was developed in the early 20th century, largely from the efforts of Wilhelm Blaschke and Jean Favard.
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
, consisting of a point x of the space and a special linear basis
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.
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.
Then the affine curvatures, k1, …, kn−1 of x are defined by That such an expression is possible follows by computing the derivative of the determinant so that x(n+1) is a linear combination of x′, …, x(n−1).
Consider the matrix 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.