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

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