Computer Aided Design and Manufacturing. Zhuming Bi

2 Computer Aided Geometric Modelling

      2.1 Introduction

Any matter for an object in the world exists in one of four distinct states, i.e. solid, liquid, gas, and plasma. The solid state is a common state for the majority of products around us. To investigate the behaviours of solids, it is necessary to know its geometry and shape. Similarly, to investigate the behaviours of fluid, gas, or plasma in a certain volume, the geometry and shape of boundary surfaces of a fine volume is also needed to be defined.

      For centuries, geometry has played its crucial role in the development of many scientific and engineering disciplines such as astronomy, geodesy, mechanics, ballistics, civil and mechanical engineering, ship building, and architecture. The importance of the study on geometry has been shown in this century in automobile and aircraft manufacturing. Since geometry is primarily visual, geometry becomes a unique and particularly exciting branch of mathematics. Geometry became a branch of mathematics at the end of the nineteenth century; however, great designs in the history were always inspired by observation and intuition on geometric shapes (Gallier 2008). Geometric modelling is a branch of applied mathematics and computational geometry; it studies the methods and the algorithms for the mathematical representation of geometries and shapes. Geometric modelling serves for the visualization of objects and lays the foundation for computer graphics, which is the construction of models of scenes from the physical world and their visualization as images.

Geometric modelling is as important to computer aided technologies as the governing equilibrium equations are to classical engineering fields such as mechanics and thermal fluids. As shown in Figure 2.1, the design of a part, product, or system usually begins with geometric modelling, so that the physical objects to be designed can be represented virtually in computers. Geometric modelling techniques and algorithms are used to model objects, and the dimensions and spatial constraints of objects are inputs via graphic user interfaces (GUI) of modelling tools. Once the objects are modelled, any information associated with the objects can be utilized in the design processes or decision‐making supports involved in a product lifecycle. For example, virtual geometric models can be used by computer graphics to visualize the designs before physical products are made. Engineering drawings needed in manufacturing processes can be directly generated from solid models of products, and all of the annotations relating to dimensioning and tolerance can be included in the drawings. If a product is a machine with relative motions among system components, a motion study can be defined upon the computer aided design (CAD) model to investigate the relations of driving forces and motions. In addition, engineering analysis can be performed at any stage of the product design process. In the following chapters, you will learn how virtual models of products can be utilized to solve design problems in mould designs, engineering analysis, simulation of manufacturing processes, evaluation of system sustainability, and so on.

Figure 2.1 Role of geometric modelling in computer aided systems (CAD).

2.2 Basic Elements in Geometry

      Geometry relates to the properties and relations of basic geometric elements such as points, lines, surfaces, solids, and higher dimensional analogues.

Figure 2.2 shows an example of basic geometric elements of an object. An object is represented by its geometric elements such as points, edges, surfaces, and solids. In addition, basic geometric elements are at different levels of the topological tree of an object. As shown in Figure 2.2, points and nodes are at the lowest level, edges formed by points or nodes are at the second level, surfaces formed by the boundary edges are at the third level, and solids with a given volume formed by a set of watertight boundary surfaces are at the upper level.

Figure 2.2 Example of basic geometric elements. (a) Points, nodes, lines, edges, axes, and planes. (b) Patches, surfaces, and operations. (c) Volumes, features, solids, and operations.

      Geometry includes the relations of these geometric elements. These relations can be spatial or topological. To represent spatial relations of geometric elements, some references have to be established, and common references can be points, axes, planes, and coordinate systems. To represent topological relations, one has to be familiar with some common logical operations such as union, subtraction, and intersection.

      2.2.1 Coordinate Systems

Any geometric element in a three‐dimensional space has six degrees of freedom (DOF). To describe the spatial relation between two geometric elements in solids, a reference coordinate system must be established. Figure 2.3 shows three most commonly used coordinate systems (CSs), i.e. a Cartesian coordinate system, cylindrical coordinate system, and spherical coordinate system.

Figure 2.3 Three commonly used coordinate systems. (a) Cartesian coordinate system. (b) Cylindrical coordinate system. (c) Spherical coordinate system.

      A Cartesian CS consists of three axes that are mutually perpendicular to each other. A position in the Cartesian CS is defined by its distances to the origin (x, y, z) projected on three axes (X, Y, Z). A cylindrical CS consists of two linear axes (X and Z) and one rotational axis. Correspondingly, a position in the cylindrical CS is defined by two scalar variables and one angular variable, i.e. (r, θ, z). A spherical CS consists of two rotational axes and one translational axis. The position of a point in the spherical CS is specified by three variables: the radial distance (R) of that point from a fixed origin, its polar angle (α) measured from a fixed zenith direction, and the azimuth angle (β) of its orthogonal projection

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