With the rapid advancement of modern 3D scanning technologies, CAD-based digital prototypes are routinely acquired in forms of raw points and/or triangular meshes. In order to enable geometric design and downstream product development processes (e.g., accurate shape analysis, finite element simulation, and e-manufacturing) in CAE environments, discrete data inputs must be converted into continuous, compact representations for scientific computing and engineering applications. In this dissertation, we present a novel spline-based data modeling framework to directly define tensor-product splines over any manifolds (serving as parametric domains). Since tensor-product B-splines and NURBS are current standards in CAD software industry, our entire mesh-to-spline data transformation pipeline enables and expedites the manifold surface design over existing CAD software platform industry (without any trimming), and thus, has great potential in shape modeling and reverse engineering applications of complicated real-world objects. Tensor-product spline schemes require the parametric domains have the regular (rectangular) structures, and constructing the domain manifold with regular structures in an efficient way still remains a challenge. In this dissertation, we study and present efficient regular domain construction methods, and demonstrate their applications in modeling 3D objects of arbitrary topology.

First, we propose the novel concept of polycube splines by defining splines directly upon the polycube map, serving as its parametric domain. We present a systematic way to construct polycube maps for surfaces of arbitrary topology based on global conformal parameterization, and demonstrate the modeling efficacy of the proposed polycube splines in solid modeling and shape computing.

We then further improve the stage of the polycube map construction by introducing the user-controllable polycube map, which allows users to directly select the corner points of the polycubes on the original 3D surfaces, then construct the polycube maps by using the discrete Euclidean Ricci flow. The location of singularities can be interactively placed where no important geometric features exist, which makes the entire hole-filling process much easier to accomplish.

We also develop an effective method to construct polycube maps in an automatic fashion. The proposed algorithm can both construct a similar polycube of high geometric fidelity and compute a high-quality polycube map. In addition, it is theoretically guaranteed to output a one-to-one map.

Finally, we propose a geometry-aware domain decomposition algorithm for T-spline-based manifold modeling by which objects with arbitrary topology (especially objects with long branches) can be modeled elegantly. The segmentation process simultaneously respects local geometric features and global topological structures.

Through our experiments, we demonstrate that the proposed framework is very flexible and can potentially serve as a geometric standard for product data representation and model conversion in shape design and geometric processing. The great potential of our geometric modeling framework will be highlighted through many valuable applications such as shape modeling, remeshing, texture synthesis, finite element analysis, deformation editing, animation morphing, and physics-based modeling. Furthermore, we envision broader application scopes including computer vision, data-driven information retrieval, digital medicine, virtual environments, etc.