Complex surface modeling : methods, algorithms and applications

Nowadays, three dimensional (3D) movies, such as “Avatar”, “Clash of the Titans” and “How to Train Your Dragon”, are leading the film industry into the 3D era. One supporting element behind the amazing scenes of the 3D movies is the complex surface. Apart from the film industry, complex surfaces have intensive applications in CAD/CAM, Biomedicine, and Interactive & Digital Media (IDM). These applications require efficient and robust methods for modeling of complex surfaces. A complex surface is a surface that is derived from a complex structure of arbitrary topology. A molecular surface is a complex surface geometrically constructed as the boundary of a set of spheres, and a minimal surface can be derived from a single contour by minimizing the surface area. A complex surface can also be built up in free-form interactively. Users can conveniently perform local shape modifications with parameter control. Composite surface can be utilized to locate an area on a complex surface to facilitate the modifications. A hierarchical structure is necessary to represent the final complex surface. This thesis investigates the complex surface modeling techniques covering four topics: molecular surfaces, minimal surfaces, composite surfaces, and hierarchical NURBS (H-NURBS) surfaces. Molecular surfaces fall into three types: van der Waals surfaces (vdWS), solvent accessible surfaces (SAS) and solvent excluded surfaces (SES). A molecule can be geometrically represented as a set of spheres, whose topology can be highly complex. Different solutions for different types of molecular surfaces have been proposed. We design a uniform explicit solution to model all three types of molecular surfaces. In our solution, a molecular surface is decomposed and organized as a set of rational Bézier surfaces. The singularities for an SES are specially treated to avoid self-intersections. The study indicates that rational Bézier representation, more specifically, a bi-cubic or 2× 4 rational Bézier surface, is sufficient for kernel modeling of molecular surfaces and related applications. Minimal surface modeling is to obtain a surface given a closed contour, which is equivalent to the computation for a surface of minimal area. This is an old and complex problem. A numerical method for minimal surfaces is studied to construct i an aesthetically pleasing triangular mesh with a closed polygonal contour given as boundary. From all triangular meshes with the prescribed boundary and the number of triangles, a triangular mesh of minimal area is identified as the solution for this problem. The composition technique is necessary to identify a local area of a surface. We employ the composition technique to derive a triangular sub-patch S from a triangular Bézier surface T of degree n. By assigning three boundary curves on the domain of T, a domain surface P of degree m can first be constructed. The composite surface S, of degree mn, is the composition of P and T. An explicit formula for control points of S is obtained by shifting operators. A robust algorithm is developed in this thesis to calculate the control points. H-NURBS allows a complex surface to be locally modified and globally transplanted. Based on multiresolution and refinement schemes, H-NURBS is investigated to design a mechanism for the purpose of carrying localized geometric information. HNURBS based Monge mapping is developed for detail and local shape modifications. Monge mapping on H-NURBS patch can be easily performed via simple cut-&-paste operations. Parametric control of the local shapes is developed to facilitate easier and better 3D local modeling. With such a technique, a complex surface can be transplanted to any portion of another surface.