Spherical image

In differential geometry, the spherical image of a unit-speed curve is given by taking the curve's tangent vectors as points, all of which must lie on the unit sphere. The movement of the spherical image describes the changes in the original curve's direction If α {displaystyle alpha } is a unit-speed curve, that is ‖ α ′ ‖ = 1 {displaystyle |alpha ^{prime }|=1} , and T {displaystyle T} is the unit tangent vector field along α {displaystyle alpha } , then the curve σ = T {displaystyle sigma =T} is the spherical image of α {displaystyle alpha } . All points of σ {displaystyle sigma } must lie on the unit sphere because ‖ σ ‖ = ‖ T ‖ = 1 {displaystyle |sigma |=|T|=1} . In differential geometry, the spherical image of a unit-speed curve is given by taking the curve's tangent vectors as points, all of which must lie on the unit sphere. The movement of the spherical image describes the changes in the original curve's direction If α {displaystyle alpha } is a unit-speed curve, that is ‖ α ′ ‖ = 1 {displaystyle |alpha ^{prime }|=1} , and T {displaystyle T} is the unit tangent vector field along α {displaystyle alpha } , then the curve σ = T {displaystyle sigma =T} is the spherical image of α {displaystyle alpha } . All points of σ {displaystyle sigma } must lie on the unit sphere because ‖ σ ‖ = ‖ T ‖ = 1 {displaystyle |sigma |=|T|=1} .