PARAMETRIZATION AND SHAPE RECONSTRUCTION TECHNIQUES FOR DOO-SABIN SUBDIVISION SURFACES

This thesis presents a new technique for the reconstruction of a smooth surface from a set of 3D data points. The reconstructed surface is represented by an everywhere -continuous subdivision surface which interpolates all the given data points. And the topological structure of the reconstructed surface is exactly the same as that of the data points. The new technique consists of two major steps. First, use an efficient surface reconstruction method to produce a polyhedral approximation to the given data points. Second, construct a Doo-Sabin subdivision surface that smoothly passes through all the data points in the given data set. A new technique is presented for the second step in this thesis. The new technique iteratively modifies the vertices of the polyhedral approximation 1CM until a new control meshM, whose Doo-Sabin subdivision surface interpolatesM, is reached. It is proved that, for any mesh M with any size and any topology, the iterative process is always convergent with Doo-Sabin subdivision scheme. The new technique has the advantages of both a local method and a global method, and the surface reconstruction process can reproduce special features such as edges and corners faithfully

Optimizing the geometrical accuracy of curvilinear meshes

This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high order elements. An optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh. Standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two-dimensional meshes, but they prove computationally too costly for optimization purposes. A fast estimate of the geometrical accuracy, based on Taylor expansions of the curves, is introduced. An unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes, as measured by the standard Haudorff distance between the geometrical model and the mesh. Several examples illustrate the beneficial impact of this method on CFD solutions, with a particular role of the enhanced mesh boundary smoothness.Comment: Submitted to JC

New strategies for curve and arbitrary-topology surface constructions for design

This dissertation presents some novel constructions for curves and surfaces with arbitrary topology in the context of geometric modeling. In particular, it deals mainly with three intimately connected topics that are of interest in both theoretical and applied research: subdivision surfaces, non-uniform local interpolation (in both univariate and bivariate cases), and spaces of generalized splines. Specifically, we describe a strategy for the integration of subdivision surfaces in computer-aided design systems and provide examples to show the effectiveness of its implementation. Moreover, we present a construction of locally supported, non-uniform, piecewise polynomial univariate interpolants of minimum degree with respect to other prescribed design parameters (such as support width, order of continuity and order of approximation). Still in the setting of non-uniform local interpolation, but in the case of surfaces, we devise a novel parameterization strategy that, together with a suitable patching technique, allows us to define composite surfaces that interpolate given arbitrary-topology meshes or curve networks and satisfy both requirements of regularity and aesthetic shape quality usually needed in the CAD modeling framework. Finally, in the context of generalized splines, we propose an approach for the construction of the optimal normalized totally positive (B-spline) basis, acknowledged as the best basis of representation for design purposes, as well as a numerical procedure for checking the existence of such a basis in a given generalized spline space. All the constructions presented here have been devised keeping in mind also the importance of application and implementation, and of the related requirements that numerical procedures must satisfy, in particular in the CAD context

Quad Dominant 2-Manifold Mesh Modeling

In this dissertation, I present a modeling framework that provides modeling of 2D smooth meshes in arbitrary topology without any need for subdivision. In the framework, each edge of a quad face is represented by a smooth spline curve, which can be manipulated using edge vertices and additional tangential points. The overall smoothness is achieved by interpolating all four edges of any given quad across the quad surface. The framework consists of simple quad preserving operations that manipulate the principal curves of the smooth model. These operations are all variants of a generic “Curve Split" and its inverse, “Region Collapse". By only using these sets of simple operations, it is possibly to model any desired shape conveniently. I also provide implementation guidelines for these operations. In the results of this dissertation, I present three main applications for this modeling framework. The major application is modeling Mock3D shapes; shapes with well defined interior normals by interpolating the normals at the boundaries of the shape across its surface which can serve as a mock 3D model to mimic a 3D CGI look. As a second application, the framework can be used in origami modeling by allowing assignment of crease patterns across the surface of 2D shapes modelled. Finally, vectorization of reference photos via modeling figures by following their contours is presented as a third application

Conceptual free-form styling in virtual environments

This dissertation introduces the tools for designing complete models from scratch directly in a head-tracked, table-like virtual work environment. The models consist of free-form surfaces, and are constructed by drawing a network of curves directly in space. This is accomplished by using a tracked pen-like input device. Interactive deformation tools for curves and surfaces are proposed and are based on variational methods. By aligning the model with the left hand, editing is made possible with the right hand, corresponding to a natural distribution of tasks using both hands. Furthermore, in the emerging field of 3D interaction in virtual environments, particularly with regard to system control, this work uses novel methods to integrate system control tasks, such as selecting tools, and workflow of shape design. The aim of this work is to propose more suitable user interfaces to computersupported conceptual shape design applications. This would be beneficial since it is a field that lacks adequate support from standard desktop systems.Diese Dissertation beschreibtWerkzeuge zum Entwurf kompletter virtueller Modelle von Grund auf. Dies geschieht direkt in einer tischartigen, virtuellen Arbeitsumge-bung mit Hilfe von Tracking der Hände und der Kopfposition. Die Modelle sind aus Freiformlächen aufgebaut und werden als Netz von Kurven mit Hilfe eines getrack-ten, stiftartigen Eingabegerätes direkt im Raum gezeichnet. Es werden interaktive Deformationswerkzeuge für Kurven und Flächen vorgestellt, die auf Methoden des Variational Modeling basieren. Durch das Ausrichten des Modells mit der linken Hand wird das Editieren mit der rechten Hand erleichtert. Dies entspricht einer natürlichen Aufteilung von Aufgaben auf beide Hände. Zusätzlich stellt diese Arbeit neue Techniken für die 3D-Interaktion in virtuellen Umgebungen, insbesondere im Bereich Anwendungskontrolle, vor, die die Aufgabe der Werkzeugauswahl in den Arbeitsablauf der Formgestaltung integrieren. Das Ziel dieser Arbeit ist es, besser geeignete Schnittstellen für den computer-unterstützten, konzeptionellen Formentwurf zur Verfügung zu stellen; ein Gebiet, für das Standard-Desktop-Systeme wenig geeignete Unterstützung bieten