Recall from sec:TangentPlane that given a parameterization of a regular surface , we can define a field of unique surface normals
The space of all possible normals lies in the unit sphere in , so we can identify with each normal its two-dimensional location in the sphere. This map is called the Gauss map. The Gauss map is differentiable and the differential maps vectors in the tangent plane to vectors in the tangent plane. That is, given a point on the surface and a direction in its tangent plane, gives the change in surface normal as you move from to . The change of the surface normal is a vector again in the tangent plane of at . Hence the differential gives the change in surface normal in the neighbourhood of .
Copyright © 2005 Adrian Secord. All rights reserved.