Affine Geometry of Curves

Curvature

Suppose that the curve x in Rn is parameterized by affine arclength. Then the affine curvatures, k₁, …, k_n−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,

$\dot{A} = \begin{bmatrix}0&1&0&0&\cdots&0&0\\ 0&0&1&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\cdots&\cdots&\vdots&\vdots\\ 0&0&0&0&\cdots&1&0\\ 0&0&0&0&\cdots&0&1\\ k_1&k_2&k_3&k_4&\cdots&k_{n-1}&0 \end{bmatrix}A = CA.$

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.

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Affine Geometry of Curves - Curvature