Hellenica World

# .

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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