One of the fundamental problems in representing a curved surface is how to define an intrinsic, i.e., viewer independent, coordinate system over the surface. More precisely, in order to establish point matching between model and observed feature distributions over curved surfaces, we need to set up a coordinate system that maps a point on a curved surface to a point on a standard coordinate system. This mapping should be independent of the viewing direction. Since the boundary of a 3-D object forms a closed surface, a coordinate system defined on the sphere is preferred. We have been exploring several intrinsic mappings from an object surface to a spherical surface. We have investigated several representations including: the EGI (Extended Gaussian Image), the DEGI (Distributed Extended Gaussian Image), the CEGI (Complex Extended Gaussian Image), and the SAI (Spherical Attribute Image). This paper describes each representation and the lessons that we have learned by using those representations in recognition systems.