differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, 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 eleme ...

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 unit b ...

. 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