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.
Read more about this topic: Affine Geometry Of Curves
Famous quotes containing the word frame:
“Human life itself may be almost pure chaos, but the work of the artist—the only thing he’s good for—is to take these handfuls of confusion and disparate things, things that seem to be irreconcilable, and put them together in a frame to give them some kind of shape and meaning. Even if it’s only his view of a meaning. That’s what he’s for—to give his view of life.”
—Katherine Anne Porter (1890–1980)