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In the last chapter we have examined a simple model of how light, emanating from light sources, interacts with the surfaces of objects in a scene. This model is purely empirical and bears very little relation to what actually takes place in reality. In this chapter we will introduce a much more sophisticated physically based model of light and examine how it interacts with the surfaces, volume and substance of objects in a computer-generated scene. To do that we must first examine the nature of light and of the materials lit by it. We start by examining how our understanding of light and its perception has evolved over the ages. [ABSTRACT FROM AUTHOR]

Computer image synthesis is the process of generating a two-dimensional pictorial representation of a mathematically defined three-dimensional object or scene. Sometimes this process is referred to as rendering the scene and the computer program that caries out this process is called a renderer. Given the mathematical description of a collection of objects or curved surfaces in a scene, there are three main ways in which we may render this scene. We may render it as a wire-frame drawing, depicting all the edges or curves, as shown in Fig. 9.1a · a hidden-line drawing, depicting only the visible edges or curves (i.e. all the edges or curves not obscured by any faces or surfaces closer to the observer), as shown in Fig. 9.1b; · a shaded image, which attempts to capture the photographic appearance of the scene by simulating how light is reflected off the surface of its objects or curved surfaces, as shown in Fig. 9.1c. [ABSTRACT FROM AUTHOR]

In computer graphics we are often concerned with representing three-dimensional scenes on a two-dimensional surface. In order to generate an image of a 3D object on a 2D display, we must first project this object onto a projection plane and then display this projected image. A point of a 3D object is expressed in terms of the object-space coordinate system (sometimes referred to as the world-space coordinate system), the projection of such a point onto the projection plane is expressed in terms of the projection-space coordinate system and the final image of this point on the display device is expressed in terms of the screen-space coordinate system (sometimes referred to as the image-space coordinate system). The complete transformation of a point from the object-space coordinate system to the screen-space system is achieved through the concatenation of three distinct transformations. The viewing transformation (or the object-space to eye-space coordinate transformation) expresses the location of object-space points relative to the observer's eye, accounting for the observer's position and direction of view.The projection transformation (or the eye-space to projection-space coordinate transformation) expresses the location of eye-space points relative to the projection plane, accounting for the effects of the type of projection used.The viewporting transformation (or the projection-space to screen-space coordinate transformation) expresses the location of projection-space points relative to a viewport of the screen, accounting for the viewport size and displacement. [ABSTRACT FROM AUTHOR]

Geometric transformations play an important part in the visualisation of three-dimensional scenes. The ability to rotate, translate and scale an object is fundamental to the understanding of its shape. This can easily be demonstrated by picking up a relatively complex and unfamiliar object. In an effort to understand its shape one rotates the object and looks at it from close or at arms length. In the generation of different views of a given scene with the computer, transformations are used to achieve the effect of different viewing positions and directions. The techniques we shall develop in this chapter for expressing 3D transformations will be an extension of the 2D techniques that we have developed in the chapter concerning 2D transformations. [ABSTRACT FROM AUTHOR]

Clipping is a process that subdivides each element of a picture to be displayed into its visible and invisible parts, thus allowing us to discard the invisible parts of the picture. In 2D, the clipping process can be applied to a variety of graphics primitives such as points, lines, polygons and curves. Clipping is performed with respect to a clipping boundary, which may be a convex or concave polygonal boundary. Clipping to a convex polygonal boundary is much simpler, thus we will start our discussion by looking at various algorithms that clip graphics primitives against a convex polygonal boundary. In its simplest form, a convex polygonal boundary consists of the four edges of the world window or the screen viewport. In this case, xl (x-left), xr (x-right), yb (y-bottom) and yt (y-top) represent the four extremities of this rectangular clipping boundary. [ABSTRACT FROM AUTHOR]

The introduction of Cartesian coordinates in the 1630s by the French mathematicians Pierre de Fermat (1601–1665) and René Descartes (1596–1650) had a profound influence on the development of mathematics. Cartesian coordinates allowed mathematicians to introduce algebraic methods into the study of geometry. By the middle of the 1800s, however, there was a general dissatisfaction with these coordinate methods and mathematicians began to look for more general and abstract ways by which they could study geometry without referring to coordinate systems. This search leads to the development of vectors and matrices, and eventually to the development of vector spaces or linear spaces. Among the first to introduce a geometry without coordinates was the Prussian mathematician Hermann Günter Grassman (1809–1877). In his 1844 book Die Ausdehnungslehre (Extension Theory) he introduced the notion of abstract quantities and he defined the operations of addition and scalar multiplication on these abstract quantities. It was, however, the Italian mathematician Giuseppe Piano (1858–1932) who introduced the first axiomatic definition of a vector space in his 1888 book Calcolo Geometrico (Geometric Calculus). This book was well in advance of its time and introduced most of the concepts of vector spaces. The complete axiomatic definition of vector spaces appeared in the 1920 doctoral dissertation of the Polish mathematician Stefan Banach (1892–1945). As the theory of vector spaces underpins much of the work in computer graphics and curve and surface theory, we will briefly introduce some of the basic ideas of vector spaces in this chapter. Underlying the concept of a vector space is the concept of a scalar field. Let us start by defining what a scalar field is. [ABSTRACT FROM AUTHOR]

The theory of matrices and their determinants is closely related to the problem of solving a system of linear simultaneous equations. The Babylonians appear to be among the first people to have investigated this problem. In a clay tablet, which has been dated to approximately 400–300 BC, the following problem was formulated. Two fields have a total area of 1800 square yards. One field produces grain at the rate of 2/3 of a bushel per square yard, while the other produces grain at the rate of 1/2 a bushel per square yard. If the total yield is 1100 bushels, what is the size of each field? [ABSTRACT FROM AUTHOR]

Vector algebra is one of the most important topics in modern mathematics. The Czech mathematician Bernard Bolzano first developed the concept of a vector in 1804. This concept was further developed by the French mathematician Jean Argand, the German mathematician August Möbius, the Irish mathematician Sir William Hamilton (who is believed to have coined the term vector) and culminated in the development of the theory of vectors by the Polish mathematician Hermann Grassmann in 1844. The concept of the vector continued to develop through the late nineteenth century and early twentieth century until the American mathematician Josiah Gibbs introduced the theory of vector analysis in 1890 and the theory of vector spaces in 1930. In our study of computer graphics we will frequently use vectors to solve a variety of problems in such diverse areas as geometric modelling, transformations, projections, visibility determination, lighting, shading and texturing, and the development of curves, surfaces and deformations. Thus it is important to gain a thorough understanding of vector algebra. [ABSTRACT FROM AUTHOR]

Unlike many other branches of mathematics, where the formulation of ideas and concepts occurs gradually over time and is developed by many mathematicians before it is formalised into a single theory, the formulation of set theory is almost the single-handed creation of one mathematician, namely Georg Cantor. Georg Ferdinand Ludwig Philipp Cantor (1845–1918) was born in Russia to a Danish father and a Russian mother and spent most of his life in Germany. Between the years 1879 and 1884 Cantor published a six-part treatise on set theory (where he introduced some of the fundamental notions of this theory) followed by the publication of a two-part treatise between the years 1895 and 1897 (where he clarified and systematised what he had introduced in his first cycle of publications). [ABSTRACT FROM AUTHOR]