Approximating digital 3D shapes by rational Gaussian surfaces

Abstract

A method for approximating spherical topology digital shapes by rational Gaussian (RaG) surfaces is presented. Points in a shape are parametrized by approximating the shape with a triangular mesh, determining parameter coordinates at mesh vertices, and finding parameter coordinates at shape points from interpolation of parameter coordinates at mesh vertices. Knowing the locations and parameter coordinates of the shape points, the control points of a RaG surface are determined to approximate the shape with a required accuracy. The process starts from a small set of control points and gradually increases the control points until the error between the surface and the digital shape reduces to a required tolerance. Both triangulation and surface approximation proceed from coarse to fine. Therefore, the method is particularly suitable for multiresolution creation and transmission of digital shapes over the Internet. Application of the proposed method in editing of 3D shapes is demonstrated.