Non-uniform subdivision surfaces

目前用于航空航天、造船、汽车设计以及越来越多的消费品中的自由曲面的标准表示是非均匀有理B样条（NURBS），它根据控制点的规则网格来指定形状。一种较新的表示法，细分曲面，承诺提供显着的优势：特别是，他们不坚持总的规则性的网格，从而允许一个形状被定义为更自然的网格，局部遵循的形状特征。一个简单的例子是汽车挡风玻璃支柱的底部，其中沿着支柱的形状的流动需要与汽车侧面形状的前后流动混合。基于NURBS的建模系统要求将几何体指定为一组单独的NURBS片段。这些必须费力地连接在一起，以确保适当的数学性质（例如曲率的连续性）跨越连接，并且这些性质通常只能近似地实现。相比之下，基于细分的系统将处理此类情况作为基本机制的一部分。阻碍它们在计算机辅助设计行业中广泛使用的问题是它们不是基于均匀网格理论的NURBS的真正超集。我们打算解决细分的非统一的情况下，从而开辟了接受这种概括为下一个标准的方式。本文的研究工作主要有三个方面：（1）将细分扩展到非均匀有理的情况，使细分成为NURBS的真正超集（特别是广泛使用的立方体）用于封闭表面和开放表面的内部;（B）扩展细分，以类似于NURBS的方式处理网格片边缘的边缘条件，以便可以在其完整性;（c）扩大细分范围，以处理某些计算机辅助设计系统所使用的高阶，特别是五次曲线。

The current standard representation for free-form surfaces, as used in aerospace, shipbuilding, automotive design and increasingly in consumer items, is the non-uniform rational B-spline (NURBS), which specifies a shape in terms of a regular grid of control points. A newer representation, subdivision surfaces, promises to provide significant advantages: in particular, they do not insist on total regularity of grid, thus allowing a shape to be defined more naturally by grids which locally follow the features of the shape. A simple example is the bottom of the windscreen pillar of a car, where the flow of the shape down the pillar needs to be blended with the fore-and-aft flow of the shape of the side of the car. A NURBS-based modelling system requires that the geometry be specified as a set of separate NURBS pieces. These must be laboriously joined together to ensure appropriate mathematical properties (e.g. continuity of curvature) across the joins and such properties can, in general, only be achieved approximately. A subdivision-based system, by contrast, handles such situations as part of the basic mechanism. The problem that prevents their wide use in the computer-aided design industry is that they are not a true superset of NURBS, being based on uniform grid theory. We intend to address the generalisation of subdivision to the non-uniform case, thus opening the way for the acceptance of this generalisation as the next standard. There are three strands to the proposed research: (a) extending subdivision to the non-uniform, rational case, which will make subdivision a true superset of NURBS for low orders (in particular the widely used cubics) for closed surfaces and the interiors of open surfaces; (b) extending subdivision to handle edge conditions at the edges of mesh pieces in analagous ways to NURBS so that open surfaces can be handled in their entiriety; (c) extending subdivision to handle the higher orders which are used in some computer-aided design systems, in particular to quintics.