Surface reconstruction with quartic triangular patches

Surface reconstruction is needed in many applications, and very often, the reconstructed surface is required to be of low degree so that the surface can be computed efficiently and robustly. However, reconstructing surfaces with lower degrees are much more difficult than with higher degrees. The C^r continuous Hermite interpolant has to use a rather high polynomial degree, d≥4r+1, unless each triangle in the triangulation is subdivided into several subtriangles. In this paper, we propose a method for constructing a G¹ quartic triangular B-B surface from a 3D triangulation. A sequence of triangulations are initially constructed by continuously removing a boundary vertex and all associated triangles from the 3D triangulation. According to this sequence of triangulations, an efficient algorithm is developed for constructing a smooth quartic surface over the 3D triangulation. The shape parameters in the G¹ condition are determined by a corresponding 2D triangulation of the 3D triangulation. This 2D triangulation has the same topology as the 3D one. The new method has many advantages over the existing methods.