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." American Mathematical Monthly 103, 30–39, 1996.