Solution
As shown in the middle six columns of Table 2.9, the numbers of faces (F), edges (E), vertices (V), bodies (B), inner loops on faces (L), and genuses (G) in a geometry are counted and the valid condition for a simple solid using the Euler–Poincare Law is applied. It shows that all generic objects satisfy the conditions as generic objects.
2.3.3 Euler–Poincare Law for Solids
Objects in the physical world are solid. A solid object consists of solid elements and the process of adding, merging, uniting, or deleting solid elements in a solid object is commonly called Euler operation. To sustain a valid solid object, the second condition in the Euler–Poincare Law, which is usually called the Euler formula (F − E + V = 2), must be satisfied. This leads to the following features of a solid object:
1 Each vertex must have at least three edges to meet together
2 Each edge must be shared by two and only two faces.
3 Any face must be homomorphic to a disk with no holes; it is simply connected and bounded by a single ring of edges.
4 A solid must be simply connected with no through hole.
2.4 Basic Modelling Methods
The methods of geometric modelling differ from one to another based on the level of information completeness of the features in objects. Figure 2.22 shows the evolution of modelling methods. Wireframe modelling is historically developed for computer aided drawing but only for the representation of key vertices and boundary edges. Solid modelling is the most advanced modelling technique since it is capable of representing the complete information of solids and beyond. Other methods such as surface modelling, boundary surface modelling (also called B‐Rep or Mantle modelling), and space decomposition are between the extremes of wireframe modelling and solid modelling in terms of the completeness of information about solids.
Figure 2.22 Variety of geometric modelling methods.
2.4.1 Wireframe Modelling
A wireframe model represents the boundary edges of an object; these edges can be of lines, arcs, and curves. A wireframe model does not include the upper‐level information such as boundary surfaces or volumes. In addition, the results from wireframe modelling have the following limitations:
All of the edges are displayed as elements in an image and the visibility caused by overlapping is not identifiable.
No high‐level information related to solids and masses such as surface areas or masses is available.
The primary data from wireframe modelling is the coordinates of vertices; therefore, the preparation, importing, and processing of modelling data are very time‐consuming and error‐prone.
The wireframe modelling method is incapable of designing shapes and specifying more complex forms due to the need for a large number of data points.
In addition, due to incompleteness of solid information, a wireframe model may cause the ambiguity of a represented geometry. Figure 2.23 shows two examples whose wireframe models in the left column are not able to distinguish the actual geometrics of objects since their corresponding wireframe models are the same.
Figure 2.23 Ambiguity examples of wireframe models. (a) Wireframe model. (b) Possible solid geometries.
On the other hand, a wireframe model is very concise and fundamental. It can be used as the supporting technique for other modelling methods.
2.4.2 Surface Modelling
Surface modelling aims to create the models for finite, non‐open surface patches of free forms. Note that the boundary surfaces of an object are formed by positioning and joining surface patches with the specified continuity restrictions. A surface modelling method does not represent the topological information of geometric elements such as knitting of boundary surfaces into a watertight volume. Figure 2.24 shows an example of a modelled surface from the surface modelling method. A surface model differs from a solid model since there is no information of the thickness.
Figure 2.24 Surface model example.
Differing from a wireframe model, a surface model has sufficient information to determine hide and shade displays when multiple surfaces are involved; however, a surface model does not include information of volumes or masses. Therefore, it is usually unsuitable to be used in simulations for engineering analysis unless the object is a sheet part with uniform thickness perpendicular to surfaces.
2.4.3 Boundary Surface Modelling (B‐Rep)
Boundary surface modelling (B‐Rep) is used to define the finite and closed cover of an object (the mantle) upon a surface model. In B‐Rep modelling, it is assumed that each physical object has an unambiguously determinable boundary surface, which is a continuous closing set of surface patches. Since a finite volume is defined, the B‐Rep method provides a comprehensive topological characterization of solids.
In defining the volume of solids, B‐Rep modelling utilizes a surface model with all of its determined boundary surface patches, and the normal vector of each surface patch is then determined. A watertight volume can be finally determined by collecting all spatial points at the internal sides of all the surrounding boundary surfaces. This is implemented by the half‐space concept.
Mathematically,
