Indicatrix of tangents

of a curve $\Gamma$ in a Euclidean space $E^n$

The curve $\Gamma^*$ on the sphere $S^{n-1}\subset E^n$ which assigns to the parameter value $\tau$ the point whose position vector is parallel to the tangent vectors at $\tau$ to $\Gamma$. In order that a spherical curve $L$ be the indicatrix of tangents of some closed curve in $E^n$ it is necessary and sufficient that $L$ is not confined to some open half-sphere (Krein's theorem).

References

[1]	M.Ya. Vygodskii, "Differential geometry" , Moscow-Leningrad (1949) (In Russian)

Comments

The indicatrix of tangents is also called the spherical tangent image. See also Spherical indicatrix.