Spherical Mapping

A spherical mapping of a surface S is a continuous mapping of S onto the unit sphere such that the tangent planes at corresponding points of the surface and sphere are parallel. Alternatively, a spherical mapping may be defined as a continuous mapping such that the normals at corresponding points are parallel.

The area s’ of the spherical image of a region G of S does not change under bending of S. This fact permits the number s’ to be regarded as an intrinsic measure of the curvature of G; the sign of s’ depends on the direction in which the boundary is traversed. Suppose s is the area of G. If there exists a limit K of the ratio of s’ to s as G is contracted to some point M on S, then K clearly also does not change under bending of S. Consequently, K is an intrinsic measure of the curvature of S at M. K is called the total, or Gaussian, curvature of S at M.

The spherical mapping of a surface plays an important role in the study of the properties of surfaces.