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

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