Critical-point-based Modeling for Smooth Surfaces

Recent advances in graphics hardware enable us to handle not only simple polyhedral shapes but complicated smooth shapes in CAD systems. Since the CAD systems represent such smooth surfaces by extending conventional simplicial representation, there are several problems in handling the smooth surface shapes that have no intuitive polyhedral approximations. The first problem is that the design of smooth surfaces requires a large amount of user interactions because of the complexity of the shapes. Secondly, none of the design operations characteristic of smooth surfaces are taken into account in contemporary CAD systems. Furthermore, they cannot provide the users with any efficient keys for shape databases due to the lack of information about smooth surface features. To remedy such problems, hierarchical representations of smooth surfaces based on the shape features are necessary. This thesis presents a new feature-based modeling method for smooth surfaces. In particular, the aim of this thesis is to implement bidirectional operations between object shapes and shape features for smooth surfaces, i.e., design by features and feature extraction. As the shape features, critical points such as peaks, pits, and passes are used. The relations among the critical points are represented by the Reeb graph, which is one of the critical point graphs (CPGs). Within a theoretical framework, the smooth surfaces are assumed to be 2-dimensional C-differentiable manifolds. The features such as critical points and CPGs work at the upper levels in the hierarchical representations of smooth surfaces. The shape design process begins with specifying the topological skeletons of an object shape using the Reeb graph. The Reeb graph is constructed by pasting the entities called cells that have one-to-one correspondences with the critical points of a height function. The iconic representation of the Reeb graph is used to visualize the embeddings of the object in 3-dimensional (3D) space. Macro operations are also provided for attaching a branch or a tube to an existing surface. The geometry of the smooth surface shape is outlined by flow curves that run on the object surface. From these flow curves, the system automatically creates a control network that encloses the object shape. Each vertex of the control network has its own local patch and the patches are then glued together using the manifold mappings in order to form the overall surface shape. This thesis also introduces another hierarchical representation called multiresolution surface design that enables us to handle the detailed geometry of the local patches. In this design, the local patches are represented by endpoint-interpolating B-splines and their corresponding wavelets. The shape of the local patch is determined by minimizing the energy function subject to the deformation of the shape while preserving the given constraints. Constraints at a low resolution level are converted to those at a high resolution level using wavelet transforms in order to associate all the constraints with the common basis functions. The constraints of multiresolution levels are then solved recursively from low to high resolution levels. The feature extraction from the polygonal representation of a surface shape, on the other hand, is implemented and used to change the height axis of the designed surface shape. Firstly, the critical points are extracted so that they satisfy the Euler formula which represents a topological invariant of smooth surfaces. The surface network, which is one of the CPGs, is then constructed by tracing ridge and ravine lines on the surface. An algorithm for converting the surface network to the Reeb graph is also presented. Using the obtained Reeb graph, the model of a control network is fit to the surface of the designed object in order to change its height axis. This thesis also presents display examples generated in the system and describes the differences from the conventional shape modeling methods.