Hierarchical G1 smooth surface interpolation with local control

Piecewise Smooth Surfaces: Piecewise smooth surfaces have innumerable applications in art, entertainment, industrial design, industrial inspection, scientific visualization, medical diagnosis, surgical planning, education, and computer simulation of physical and biological phenomena, special effects and computer animation, etc. Manipulating piecewise smooth surface models efficiently and intuitively, maintaining desired degrees of continuity across patches, including creases ( G continuity), continuity of surface normals ( G), and of curvatures ( G), has been a challenging problem in special effects and animation, where subdivision surfaces have become the surface representation of choice. A very attractive modeling paradigm is the ability to prescribe surface points and normal vectors specified at given vertices of a control polygon mesh. Still, performing normal control for subdivision surfaces remains challenging. Several deformations methods using normal control have been proposed, however most rely on approximation schemes [Nasri 1991; Biermann et al. 2000; Salomon et al. 2002]. In addition, in current subdivision schemes supporting higher orders of continuity across patch boundaries the geometry of each surface patch depends not only on the data associated with boundary vertices of the corresponding face in the control mesh, but also on all the data specified at vertices of neighboring faces. Closely related problems are how to impose boundary constraints, the creation and control of creases, and the construction of smooth surfaces interpolating networks of smooth curves [Biermann et al. 2000; Levin 1999].