Tangent indicatrix

tantrix

The tangent indicatrix $ T $ of a regular curve $ \gamma : {[ a,b ] } \rightarrow {\mathbf R ^ {n} } $ is the curve of oriented unit vectors tangent to $ \gamma $.

More precisely, if $ \gamma : {[ a,b ] } \rightarrow {\mathbf R ^ {n} } $ is a differentiable curve whose velocity vector $ {\dot \gamma } = { {d \gamma } / {dt } } $ never vanishes, then

$$ T ( t ) = { \frac{ {\dot \gamma } ( t ) }{\left | { {\dot \gamma } ( t ) } \right | } } . $$

The tangent indicatrix $ T $ of any regular curve in $ \mathbf R ^ {n} $ thus traces out a curve on the unit sphere $ S ^ {n - 1 } \in \mathbf R ^ {n} $ which, as a point set, is independent of the parametrization of $ \gamma $. A direct computation shows that the "speeds" of $ T $ and $ \gamma $ relate via the curvature function $ \kappa $ of $ \gamma $( cf. also Curvature):

$$ \left | { { \frac{dT }{dt } } } \right | = \kappa ( t ) \left | { { \frac{d \gamma }{dt } } } \right | . $$

It follows immediately that the length of the tangent indicatrix on $ S ^ {n - 1 } $ gives the total curvature (the integral of $ \kappa $ with respect to arc-length; cf. also Complete curvature) of the original curve $ \gamma $. 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)