In differential geometry, the spherical image of a unit-speed curve is given by taking the curve's tangent vector

Tangent vector

A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....

s as points, all of which must lie on the unit sphere

Unit sphere

In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...

. The movement of the spherical image describes the changes in the original curve's direction If is a unit-speed curve, that is , and is the unit tangent vector field along , then the curve is the spherical image of . All points of must lie on the unit sphere because .