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 $\gamma(t)\,$ be a closed curve with nowhere-vanishing tangent vector $\dot{\gamma}$ . Then the tangent indicatrix $T(t)\,$ of $\gamma\,$ is the closed curve on the unit sphere given by $T = \frac{\dot{\gamma}}{|\dot{\gamma}|}$ .

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

References

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

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