Affine parameter

affine arc length

A parameter on a curve which is preserved under transformations of the affine group, for the determination of which the derivatives of the position vector of the curve of the lowest order must be known. The best known is the parameter which is invariant with respect to the equi-affine transformations, i.e. with respect to the affine unimodular group. For a plane curve $ \mathbf r = \mathbf r (t) $ this affine parameter is computed by the formula:

$$ s = \int\limits _ {t _ {0} } ^ { t } | ( \dot{\mathbf r} , \dot{\mathbf r} dot ) | ^ {1/3} dt , $$

where $ ( \dot{\mathbf r} , \dot{\mathbf r} dot ) $ is the skew product of the vectors $ \dot{\mathbf r} $ and $ \dot{\mathbf r} dot $. In particular, for the affine length of an arc $ M _ {0} M _ {1} $ of a parabola, the affine parameter is $ s = {2f } ^ {1/3} $, where $ f $ is the area of the triangle formed by the chord $ M _ {0} M _ {1} $ and the tangents to the parabola at the points $ M _ {0} $ and $ M _ {1} $. It is possible to introduce in a similar manner the affine parameter of a space curve in the geometry of the general affine group or any one of its subgroups.

Comments

The arc length given by the formula above is sometimes referred to as the special affine arc length.

The notion of an "affine parameter" is also used in the theory of geodesics. An affine parameter of a geodesic (an autoparallel curve) of an affine connection is a parametrization $ x (t) $ of the geodesic such that for the corresponding covariant derivative $ \nabla $ the equation

$$ \nabla _ {\dot{x} (t) } \dot{x} (t) = 0 $$

is valid (cf. [a3]).

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , 2 , Publish or Perish (1970) pp. 1–5
[a2]	S. Buchin, "Affine differential geometry" , Sci. Press and Gordon & Breach (1983)
[a3]	B. O'Neill, "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press (1983)