We first need a virtual world to contain the geometric models. For our purposes, it is enough to have a 3D Euclidean space with Cartesian coordinates. Therefore, let denote the virtual world, in which every point is represented as a triple of real-valued coordinates: . The coordinate axes of our virtual world are shown in Figure 3.1. We will consistently use right-handed coordinate systems in this book because they represent the predominant choice throughout physics and engineering; however, left-handed systems appear in some places, with the most notable being Microsoft's DirectX graphical rendering library. In these cases, one of the three axes points in the opposite direction in comparison to its direction in a right-handed system. This inconsistency can lead to hours of madness when writing software; therefore, be aware of the differences and their required conversions if you mix software or models that use both. If possible, avoid mixing right-handed and left-handed systems altogether.
Geometric models are made of surfaces or solid regions in and contain an infinite number of points. Because representations in a computer must be finite, models are defined in terms of primitives in which each represents an infinite set of points. The simplest and most useful primitive is a 3D triangle, as shown in Figure 3.1. A planar surface patch that corresponds to all points ``inside'' and on the boundary of the triangle is fully specified by the coordinates of the triangle vertices:
To model a complicated object or body in the virtual world, numerous triangles can be arranged into a mesh, as shown in Figure 3.2. This provokes many important questions: