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 γ ( t ) be a closed curve with nowhere-vanishing tangent vector γ ˙ . Then the tangent indicatrix T ( t ) of γ is the closed curve on the unit sphere given by T = γ ˙ | γ ˙ | .

The total curvature of γ (the integral of curvature with respect to arc length along the curve) is equal to the arc length of T .