Methods for deforming polygonal shapes into smooth spline surface models

Deformation from polygonal shapes into smooth surfaces is an attractive design tool since it relieves the user from the burden of having to first specify a large set of control points and then stitching complex patches together. This thesis presents such a modeling paradigm. Once a polygonal shape with underlying rectangular topology is specified by the user, it is deformed into a default smooth surface that interpolates all the polygonal vertices. The user can then modify the default smooth surface by increasing or decreasing the amount of deformation, either globally or locally. This is accomplished by interactively controlling the shape parameters associated with the polygonal vertices. The mathematical formulation is based on the interpolation method proposed by Loe and Tai [1]. The interpolation technique blends a cubic uniform B-spline curve or surface with a sequence of singularly reparametrized line segments or bilinear patches using a blending function that simulates the tension effect. Uniform B-spline, however, does not produce satisfactory result when the distances between the input points vary substantially; thus, we extend the idea to use the non-uniform rational B-spline (NURBS) and thus introduce an extra shape parameter, called weight. In addition, a roundness parameter is also introduced to adjust the shape of the NURBS curve or surface. This modeling paradigm is conceptually simple, and it allows C2 continuous surfaces to be easily designed, even by novice users.