Geometric processing of compound T-spline surfaces - Li, Yusha (2016)
This thesis is concerned with geometric processing algorithms for compound T-spline surfaces. A compound T-spline surface model is usually formed by a collection of T-spline surfaces. It is suitable for representing complicated shapes in computer-aided geometric design and solid modeling. Geometric processing refers to the theory and algorithms for analyzing and manipulating geometric objects. While T-spline technology is becoming popular,there is need to develop a rich family of geometric processes for T-splines. This research investigates four fundamental geometric processes of compound T-spline surfaces: rational Bezier extraction from T-splines, adaptive tessellation of compound T-spline models, T-spline knot removal, and T-spline surface extension, which are relatively less explored. Our objectives are to gain deep understanding of these processes and develop effective and efﬁcient algorithms on GPU to handle compound T-spline surface models accurately, interactively and seamlessly. First, we present a method for extracting Bezier patches from T-spline surfaces, which is a useful process for T-spline tessellation, T-spline surface intersection, T-spline based iso-geometric analysis, etc. The difﬁculty of the process lies on the ﬂexibility of T-spline topological structure and the complexity of GPU implementation. The underlying steps of our method contain correctly ﬁguring out C∞ zones, partitioning non-rectangular zones into Bezier patch domains and computation of Bezier representation. We design techniques to implement all these steps on GPU. As a result, our method can correctly extract a minimum number of Bezier patches from a compound T-spline surface model in real time. Second, we propose a novel framework for tessellating compound T-spline surfaces. The underlying techniques involve boundary merging, surface decomposition, tessellation estimation and mesh generation. Except for boundary merging, all others are designed to be GPU-friendly. We improve the tessellation factor estimation for rational Bezier curves and surfaces, and design parallel strategies for curve and patch tessellation and mesh generation. As a result, our method can adaptively tessellate T-spline models into crack-free triangular meshes in real time on GPU. The generated triangular meshes are guaranteed to approximate the T-spline models within the given tolerance. Third, we analyze the characteristics of T-spline knot removal, which is surprisingly much more complicated than B-spline knot removal. Based on typical patterns of T-spline knot insertion and removal, we propose a T-spline multi-point removal algorithm. The algorithm ﬁrst identiﬁes removal groups in the mesh, and then removes all the points in a group simultaneously. Compared to a single-point removal, the multi-point removal is more effective. Moreover, we parallel the removal process and implement it on GPU, which greatly improves the efﬁciency of the algorithm in terms of run time. We also consider the approximate T-spline multi-point removal, which can be used to simplify T-spline models. Fourth, we propose a scheme to represent T-splines with complicated boundaries and develop techniques for extending T-splines from partial boundaries, which results in T-spline models with complicated boundaries. This can alleviate the workload of handling cracks in compound models and reduce the number of surfaces used for modeling complicated shapes. We provide T-spline surface extension and trimming tools for the user to perform interactive design. The user can extend the surface to interpolate a curve or trim part of the surface. These operations keep the original surface unchanged. The complex boundary T-splines enhance the representation power of T-splines in modeling complicated shapes including those with complicated boundaries or holes.