Tangent Indicatrix

Let the Speed $\boldsymbol{\sigma}$ of a closed curve on the unit sphere $S^2$ never vanish. Then the tangent indicatrix

$$\boldsymbol{\tau}\equiv{\dot\boldsymbol{\sigma}\over \vert\dot\boldsymbol{\sigma}\vert}$$

is another closed curve on $S^2$. It is sometimes called the Tantrix. If $\boldsymbol{\sigma}$ Immerses in $S^2$, then so will $\boldsymbol{\tau}$.

References

Solomon, B. ``Tantrices of Spherical Curves.'' Amer. Math. Monthly 103, 30-39, 1996.