A curve on the unit sphere is an eversion if it has no corners or cusps (but it may be self-intersecting). These
properties are guaranteed by requiring that the curve's velocity never vanishes. A mapping
immersion of the Circle into the Sphere Iff, for all
See also Sphere Eversion
Smale, S. ``A Classification of Immersions of the Two-Sphere.'' Trans. Amer. Math. Soc. 90, 281-290, 1958.