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指定形状。一种新的表示，细分曲面，承诺提供显著的优势：特别是，它们不坚持网格的完全规则性，从而允许一个形状更自然地由局部遵循形状特征的网格来定义。一个简单的例子是汽车挡风玻璃柱的底部，在那里，形状沿着柱子的流动需要与汽车侧面形状的前后流动混合在一起。基于NURBS的建模系统要求将几何形状指定为一组独立的NURBS块。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），而这些性质通常只能近似地实现。相比之下，基于细分的系统将这种情况作为基本机制的一部分来处理。阻碍它们在计算机辅助设计行业广泛应用的问题是，它们不是基于统一网格理论的NURBS的真正超集。我们打算解决细分到非统一情况的普遍化问题，从而为接受这种普遍化作为下一个标准开辟道路。提出的研究有三个方面：(a)将细分扩展到非均匀有理情况，这将使细分成为低阶（特别是广泛使用的立方体）的NURBS的真正超集，用于封闭表面和开放表面的内部；(b)扩展细分，以类似NURBS的方式处理网格块边缘的边缘条件，从而可以整体处理开放表面；(c)扩展细分，以处理某些计算机辅助设计系统，特别是五项设计系统所使用的更高阶。

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.