Tangent indicatrix

tantrix

The tangent indicatrix of a regular curve is the curve of oriented unit vectors tangent to .

More precisely, if is a differentiable curve whose velocity vector never vanishes, then

The tangent indicatrix of any regular curve in thus traces out a curve on the unit sphere which, as a point set, is independent of the parametrization of . A direct computation shows that the "speeds" of and relate via the curvature function of (cf. also Curvature):

It follows immediately that the length of the tangent indicatrix on gives the total curvature (the integral of with respect to arc-length; cf. also Complete curvature) of the original curve . Because of this, the tangent indicatrix has proven useful, among other things, in studying total curvature of closed space curves (see [a1], p. 29 ff).

References

[a1]	S.S. Chern, "Studies in global analysis and geometry" , Studies in Mathematics , 4 , Math. Assoc. America (1967)
[a2]	B. Solomon, "Tantrices of spherical curves" Amer. Math. Monthly , 103 : 1 (1996) pp. 30–39
[a3]	V.I. Arnol'd, "The geometry of spherical curves and the algebra of quaternions" Russian Math. Surveys , 50 : 1 (1995) pp. 1–68 (In Russian)