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

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