In
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 direction
[O'Neill, B. ''Elementary Differential Geometry'', 1961, pg 71.] 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
.
References
Differential geometry
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