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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