The Z-spline: An interpolating subdivision surface scheme for smooth modeling and three-dimensional morphing

3D morphing of smooth surfaces defined over triangular meshes presents a new problem. A change in surface topology requires a change in the connectivity of the domain triangulation, which can drastically alter the surface shape. Traditional surface fitting algorithms are only designed to construct static smooth surfaces and fail to address issues involved in smooth surface fitting of dynamically changing geometry.; This thesis presents an efficient approach for smoothly transforming surfaces into different topologies without causing sudden changes in their shapes. The surface used in this approach is based on a new type of subdivision spline, called the Z-Spline.; The Z-Spline algorithm generates {dollar}Gsp1{dollar} smooth interpolation surfaces over triangulations of arbitrary topology. Each domain triangle is covered with a single Z-Spline patch, or Z-patch. The control points of this Z-patch are constructed locally from the position and the normal vector at each vertex of the triangle, together with a few other shape parameters. The Z-Spline surface is defined as the limit of a linear subdivision process over these control points.; The Z-Spline can be used to represent smooth objects of arbitrary topological type that may include sharp features. The Z-Spline algorithm also overcomes drawbacks of other triangular spline surface fitting schemes. It can naturally represent a subset of extrusions, which traditionally are defined by tensor product surfaces. The design of the Z-Spline surface fitting algorithm provides enough control to avoid sudden changes in surface shape during smooth morphing between objects with different domain topologies.; The Z-Spline surfaces can be rendered efficiently through adaptive subdivision. With such a rendering system developed, morphing animations between basic geometry primitives have been generated in real-time. The Z-Spline surface fitting scheme is therefore shown to be efficient enough for practical applications.