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''tantrix''

The tangent indicatrix  $  T $
of a regular [[Curve|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|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|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 [[#References|[a1]]], p. 29 ff).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.S. Chern,   "Studies in global analysis and geometry" , ''Studies in Mathematics'' , '''4''' , Math. Assoc. America  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Solomon,   "Tantrices of spherical curves"  ''Amer. Math. Monthly'' , '''103''' :  1  (1996)  pp. 30–39</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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)</TD></TR></table>