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 $$T$$.