Parametric Surfaces On Manifold

Geometric modeling is one of the key techniques for modern industrial production design and manufacture. As the main part of geometric modeling, the design of parametric curves and surfaces plays an important role for long time in computer aided design.NURBS has become a representation standard for geometric design systems since 1990. However, it is difficult to model complicated objects with only one NURBS surface patch due to its rectangular domain, and it still suffers from serious limitations in surface stitching. In the past decade, a lot of new work had been proposed to overcome the limitations of NURBS, such as triangular NURBS, T-spline, G-spline, rational Gauss surface, and so on. Unfortunately, some of those methods are very complex and difficult to use in practice; and others are unable to be used in mechanical design since they can not describe objects precisely.In the dissertation, a novel theoretical framework is proposed to represent and construct parametric curves and surfaces on manifold, named as generalized rational parametric curve and surface (abbreviated as GRPCS). In the framework, a control mesh may be arbitrary one-dimensional or two-dimensional orientated topological manifold, and the curve or surface is defined on differential manifold homeomorphic to the control mesh with a potential function as its basis functions. This method is an extension of NURBS, which efficiently overcomes the limitations of NURBS. The basic idea to construct GRPCS is to establish object topology first, then use geometry to change the shape of differential manifold.In chapter 2, we discuss the theoretical framework of GRPCS that includes some relative idea about differential manifold. Firstly, the definition of potential function on manifold is given. By employing the potential function, we construct the unit partition on differential manifold. And then, by regarding the curve and surface as a map from differential manifold to topological manifold, we present the framework of GRPCS and discussed its basic properties in detail. GRPCS provides a unified framework for parametric curve and surface. It does not only inherit a lot of good properties from NURBS such as locality, convex hull, affine and perspective invariance etc., but alsohas the ability to directly represent trimmed surfaces and closed surfaces.In chapter 3, we discuss a novel parametric curve representation scheme on 1D manifold in-depth. All construction and design techniques for parametric curves, such as basis function design, parameterization, and local features control etc. are discussed in detail. A new construction method for conic curve is also presented. Our study shows that it is a good choice to apply generalized rational parametric curve to interpolate and fit unorganized data, because it provides higher numerical stability by appropriately adjusting the support of basis functions.According to topological structure of control mesh, we generate generalized rational parametric surfaces in two manners. For a control mesh with consistent parameterization, the surface can be easily constructed because it can be mapped to differential manifold directly. Its properties and design method is discussed in chapter 4. For control meshes with arbitrary topology, we present a universal method in chapter 5 to construct parametric curves and surfaces. Generalized rational parametric surface can be controlled precisely and flexible, and it is easy to model local features and 3D primitives. At last, several experimental results are given, and some future works are discussed.GRPCS can be widely applied in computer graphics, computer animation, digital geometry processing and all kinds of industry production design. It has significantly theoretical meaning and practical worthiness in computer aided design.