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 [math]\displaystyle{ \gamma(t) }[/math] be a closed curve with nowhere-vanishing tangent vector [math]\displaystyle{ \dot{\gamma} }[/math]. Then the tangent indicatrix [math]\displaystyle{ T(t) }[/math] of [math]\displaystyle{ \gamma }[/math] is the closed curve on the unit sphere given by [math]\displaystyle{ T = \frac{\dot{\gamma}}{|\dot{\gamma}|} }[/math]. The total curvature of [math]\displaystyle{ \gamma }[/math] (the integral of curvature with respect to arc length along the curve) is equal to the arc length of [math]\displaystyle{ T }[/math].

References

• Solomon, B. "Tantrices of Spherical Curves." American Mathematical Monthly 103, 30–39, 1996.