计算几何基础与模型分类
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"计算几何讲义.pdf"
这篇文档是一份关于计算几何的讲义，主要讨论了工程领域中复杂物体的数学建模方法。计算几何是几何建模的基础，涉及算法的开发、分析和计算机实现，这些算法在工程对象如机械零件、交通工具或环境对象的三维表示中起着关键作用。
在工程应用中，我们需要对各种形状和结构进行数学描述，从简单的机器元件到复杂的雕塑形体，如船舶、汽车、飞机、涡轮叶片等。此外，物理环境的描述也包括地形以及如盐度、温度、速度、化学浓度等三维物理属性。
讲义的第一部分介绍了几何建模的形式，并进行了分类：
1. **线框建模**：这是一种基本的建模形式，由线和点构成，通常用于表示物体的轮廓或骨架。
2. **曲面建模**：更进阶的建模方式，用于创建连续的二维表面，可以更准确地表现物体的外观。
3. **实体建模**：用于创建封闭的三维物体，通常基于多边形网格或边界表示法（B-Rep）。
4. **非两维流形建模**：处理不规则或非连通的几何形状，例如断裂或穿孔的物体。
接着，讲义进一步深入到固体建模的基本分类：
1. **分解模型**：通过组合简单的形状来构建复杂的物体，比如通过切割和组装基本几何体。
2. **构造性固体几何（CSG）**：使用布尔运算（并集、交集、差集）结合基本几何体来创建复杂的实体。
3. **边界表示（B-Rep）**：描述物体的外表面边界，包含面、边和顶点，是现代CAD系统中最常用的建模方法。
除此之外，还提到了另一种分类方法，包括未评估的表示系统和已评估的表示系统，这涉及到模型是否直接可以用于计算和分析。
未评估的表示系统通常更侧重于几何形状的描述，而不直接提供数值信息。而已评估的表示系统则会计算出几何模型的具体特性，如体积、面积或质量。
这份计算几何讲义涵盖了从基础的几何建模概念到高级的实体建模方法，是理解计算几何和工程建模的重要参考资料。它强调了数学在描述现实世界复杂对象中的核心作用，并探讨了不同的建模策略和技术。
planes (parallel to keel) are the waterlines, while intersections with transverse vertical planes
are called sections. Wireframes are rather incomplete and possibly ambiguous representations
that were superseded by surface models.
1.2.2 Surface modeling
Surface modeling techniques, developed in the late 1960’s, go one step further than wireframe
representations by also providing mathematical descriptions of the shape of the surfaces of
objects, as shown in Figure 1.3.
Surface modeling techniques allow graphic display and numerical control machining of care-
fully constructed models, but usually offer few integrity checking features (e.g. closed volumes).
The surfaces are not necessarily properly connected and there is no explicit connectivity infor-
mation stored. These techniques are still used in areas where only the visual display is required,
e.g. flight simulators.
1.2.3 Solid modeling
Solid modeling, first introduced in the early 1970’s, explicitly or implicitly contains information
about the closure and connectivity of the volumes of solid shapes. Solid modeling offers a
number of advantages over previous wireframe and surface modeling techniques. In principle,
it guarantees closed and bounded objects and provides a fairly complete description of an object
modelled as a rigid solid in 3D space [7, 6, 8].
Figure 1.4 illustrates that for a boundary based solid model of a single homogeneous object,
every surface boundary is always directly adjacent to one other surface boundary, guaranteeing
a closed volume. Solid models, unlike surface models, enable a modeling system to distinguish
the outside of a volume from the inside. This capability, in turn, allows integral property
analysis for the determination of volume, center of volume or gravity, moments of inertia, etc.
An example is Baumgart’s winged edge data structure [1, 2], where every edge has a start
and end point, a face on either side, and at least two edges from each vertex bounding the
faces. This information can be put in tabular form (perhaps using a relational database) or in
a graph like data structure and used to ensure adjacency.
Typical solid modeling systems also offer tools for the creation and manipulation of complete
solid shapes, while maintaining the integrity of the representations.
Solid modeling techniques exclude the two previous modeling forms (wireframe and surface
modeling). The reason is that the solid modeling forms are traditionally constrained to work
only with two-manifold solids.
In a two-manifold solid representation, every point on the surface has a neighborhood on
the surface which is topologically equivalent to a two-dimensional disk. In other words, even
though the surface exists in three dimensional space, it is topologically flat when the surface is
examined closely in a small enough area around any given point, as illustrated by the cube in
Figure 1.5.
1.2.4 Non-two-manifold modeling
Non-two-manifold modeling [1, 9, 5, 10] is a new modeling form which removes constraints
associated with two-manifold solid modeling forms by embodying all of the capabilities of
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
Figure 1.2: Wire frame model of a ship hull.
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