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

tangent vector In mathematics, 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 R''n''. More generally, tangent vectors are ele ...

s as points, all of which must lie on the

unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A u ...

. The movement of the spherical image describes the changes in the original curve's directionO'Neill, B. ''Elementary Differential Geometry'', 1961, pg 71. If

\alpha

is a unit-speed curve, that is

\, \alpha^\prime \,  = 1

, and

T

is the unit tangent vector field along

\alpha

, then the curve

\sigma =  T

is the spherical image of

\alpha

. All points of

\sigma

must lie on the unit sphere because

\, \sigma\,  = \,  T\,  = 1

References

Differential geometry {{differential-geometry-stub