Tangent indicatrix

In differential geometry, the tangent indicatrix of a closed space curve is a curve on the unit sphere intimately related to the curvature of the original curve. Let \( T \gamma(t)\, \) be a closed curve with nowhere-vanishing tangent vector \( T\dot{\gamma} \) . Then the tangent indicatrix \( T T(t)\ \) , of \( T \gamma\, \) is the closed curve on the unit sphere given by \( T = \frac{\dot{\gamma}}{|\dot{\gamma}|}. \)

The total curvature of \( T\gamma\, \) (the integral of curvature with respect to arc length along the curve) is equal to the arc length of \( TT\,. \)
References

Solomon, B. "Tantrices of Spherical Curves." Amer. Math. Monthly 103, 30-39, 1996.

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