Computer graphics has grown phenomenally in recent decades, progressing from simple 2-D graphics to complex, high-quality, three-dimensional environments. In entertainment, computer graphics is used extensively in movies and computer games. Animated movies are increasingly being made entirely with computers. Even nonanimated movies depend heavily on computer graphics to develop special effects: witness, for instance, the success of the Star Wars movies beginning in the mid-1970s. The capabilities of computer graphics in personal computers and home game consoles have now improved to the extent that low-cost systems are able to display millions of polygons per second.

There are also significant uses of computer graphics in nonentertainment applications. For example, virtual reality systems are often used in training. Computer graphics is an indispensable tool for scientific visualization and for computer-aided design (CAD). We need good methods for displaying large data sets comprehensibly and for showing the results of large-scale scientific simulations.

The art and science of computer graphics have been evolving since the advent of computers and started in earnest in the early 1960s. Since then, computer graphics has developed into a rich, deep, and coherent field. The aim of this book is to present the mathematical foundations of computer graphics along with a practical introduction to programming using OpenGL. I believe that understanding the mathematical basis is important for any advanced use of computer graphics. For this reason, this book attempts to cover the underlying mathematics thoroughly.

This chapter discusses some of the basic concepts behind computer graphics with particular emphasis on how to get started with simple drawing in OpenGL. A major portion of the chapter explains the simplest methods of drawing in OpenGL and various rendering modes. If this is your first encounter with OpenGL, it is highly suggested that you look at the included sample code and experiment with some of the OpenGL commands while reading this chapter.

The first topic considered is the different models for graphics displays. Of particular importance for the topics covered later in the book is the idea that an arbitrary three-dimensional geometrical shape can be approximated by a set of polygons – more specifically as a set of triangles. Second, we discuss some of the basic methods for programming in OpenGL to display simple two- and three-dimensional models made from points, lines, triangles, and other polygons. We also describe how to set colors and polygonal orientations, how to enable hidden surface removal, and how to make animation work with double buffering. The included sample OpenGL code illustrates all these capabilities. Later chapters will discuss how to use transformations, how to set the viewpoint, how to add lighting and shading, how to add textures, and other topics.

Display Models

We start by describing three models for graphics display modes: (1) drawing points, (2) drawing lines, and (3) drawing triangles and other polygonal patches. These three modes correspond to different hardware architectures for graphics display.

This chapter briefly discusses some of the issues in color perception and color representation that are important for computer graphics. Color perception and color representation are complicated topics, and more in-depth information can be found in references such as (Berns, Billmeyer, and Saltzman, 2000); (Jackson, MacDonald, and Freeman, 1994); (Foley et al., 1990); Volume I of (Glassner, 1995); or (Hall, 1989). Also recommended is the short, readable introduction to the physics of color and the physiological aspects of color perception in (Feynman, 1989). Some more detailed recommendations for further reading are given at the end of this chapter.

The first section of this chapter discusses the physiology of color perception and its implications for computer graphics. The second, more applied section discusses some of the common methods for representing color in computers.

Color Perception

The basic theories of how humans perceive color were formulated already in the nineteenth century. There were two competing theories of color perception: the trichromatic theory and the opponent color theory. These two theories will appear contradictory at first glance, but in fact they are both correct in that they are grounded in different aspects of human color perception.

The Trichromatic Theory of Vision. The trichromatic theory was formulated by G. Palmer, in 1777 and then again by T. Young in 1801; it was extended later by Helmholtz. This theory states that humans perceive color in three components: red, green, and blue.

This chapter covers uniform and nonuniform B-splines, including rational B-splines (NURBS). B-splines are widely used in computer-aided design and manufacturing and are supported by OpenGL. B-splines are a powerful tool for generating curves with many control points and provide many advantages over Bézier curves – especially because a long, complicated curve can be specified as a single B-spline. Furthermore, a curve designer has much flexibility in adjusting the curvature of a B-spline curve, and B-splines can be designed with sharp bends and even “corners.” In addition, it is possible to translate piecewise Bézier curves into B-splines and vice versa. B-splines do not usually interpolate their control points, but it is possible to define interpolating B-splines. Our presentation of B-splines is based on the Cox–de Boor definition of blending functions, but the blossoming approach to B-splines is also presented.

The reader is warned that this chapter is a mix of introductory topics and more advanced, specialized topics. You should read at least the first parts of Chapter VII before this chapter. Sections VIII.1–VIII.4 give a basic introduction to B-splines. The next four sections cover the de Boor algorithm, blossoming, smoothness properties, and knot insertion; these sections are fairly mathematical and should be read in order. If you wish, you may skip these mathematical sections at first, for the remainder of the chapter can be read largely independently. Section VIII.9 discusses how to convert a piecewise Bézier curves into a B-spline. The very short Section VIII.10 discusses degree elevation.

Overview

The term animation refers to the process of specifying or controlling the movement of objects. Of particular concern to us in this chapter is the use of mathematical techniques to aid in the programming of animation.

Traditional animation techniques, used notably in movies, substantially predate computerized animation. Traditionally, animation requires drawing a series of pictures, each picture showing an instantaneous snapshot of objects in motion. These are then displayed in rapid succession, giving the visual effect of smooth motion. The same general idea applies to computer animation: a software program repeatedly renders a three-dimensional scene with each scene showing the objects at a particular point in time. These scenes are then displayed at a sufficiently fast frame rate to give the visual illusion of motion.

Nowadays, much of the work of animation is being taken over by computers. An artist will create objects, people, creatures, and so on as three-dimensional models in a CAD program. These models can include information about the formation and permissible motions of the objects. For instance, an object may have a skeleton of rigid links connected by joints, and the joints can be given characteristics that control their motion. The rest of the object can be controlled to move in synchronization with the skeleton. One such technique is called skinning and permits the “skin,” or surface, of a creature to move in conjunction with the movement of the skeleton.

Lighting and shading are important tools for making graphics images appear more realistic and more understandable. Lighting and shading can provide crucial visual cues about the curvature and orientation of surfaces and are important in making three-dimensionality apparent in a graphics image. Indeed, good lighting and shading are probably more important than correct perspective in making a scene understandable.

Lighting and illumination models in computer graphics are based on a modular approach wherein the artist or programmer specifies the positions and properties of light sources, and, independently, specifies the surface properties of materials. The properties of the lights and the materials interact to create the illumination, color, and shading seen from a given viewpoint.

For an example of the importance of lighting and shading for rendering three-dimensional images, refer to Figure III.1. Figure III.1(b) shows a teapot rendered with a solid color with no shading. This flat, featureless teapot is just a silhouette with no three-dimensionality. Figure III.1(c) shows the same teapot but now rendered with the Phong lighting model. This teapot now looks three-dimensional, but the individual polygons are clearly visible. Figure III.1(d) further improves the teapot by using Gouraud interpolation to create a smooth, rounded appearance. Finally, Figures III.1(e) and (f) show the teapot with specular lighting added; the brightly reflecting spot shown in (e) and (f) is called a specular highlight.

Ray tracing is a technique that performs, by a single unified technique, global calculations of lighting and shading, hidden surface elimination, reflection and transmission of light, casting of shadows, and other effects. As such, it significantly extends the local lighting models such as the Phong and Cook–Torrance lighting models from Chapter III. Ray tracing also eliminates the use of a depth buffer for hidden surface determination. In addition, it allows for many special effects and can create images that are more realistic looking than those that can be easily obtained by the methods we have discussed so far.

With all these advantages, ray tracing sounds too wonderful to be true; however, it has the big disadvantage of being computationally very expensive. Indeed, a single ray-traced image may take minutes, hours, or occasionally even days to render. For example, modern computer-animated movies routinely use ray tracing to render scenes; it is not unusual for an average frame of a movie to require an hour of computation time to render, and individual frames might require 10 hours or more to render. A quick calculation shows that this means that a movie with 24 frames per second, lasting for 100 minutes, may require 6,000 CPU days to render, which is over 16 CPU years! It is fortunate that individual frames can be ray traced independently in parallel, and it is common for animated movies to be developed with the aid of hundreds of computers dedicated to rendering images.

This chapter discusses issues in intersection testing and describes algorithms for intersecting rays with geometric objects. Intersection testing is one of the fundamental algorithms used in ray tracing, which usually requires intersecting a large number of rays against a large set of geometric objects. For this reason, it is often the case that the bulk of the computational time required for ray tracing is spent on intersection testing. Intersection testing has other applications besides ray tracing (for e.g., in motion planning and collision detection).

Intersection testing is a large and diverse field. In this chapter, we discuss only some of the first topics, namely, intersecting rays against simple geometric objects. We only briefly discuss methods such as bounding boxes and bounding spheres and global pruning methods such as octtrees and binary space partitioning. More information on intersection testing can be found in the book (Möller, and, Haines, 1999), which has two chapters devoted to this topic, including numerous references to the literature. The Graphics Gems volumes also contain many articles on intersection testing. For additional reading suggestions, see the end of this chapter.

Two important goals need be kept to in mind when designing software for intersection testing. The first design goal concerns accuracy and robustness. Clearly, it is important that the algorithms be accurate; even more so, the algorithms need to be free from occasional errors that may make visible artifacts, such as holes, or cause an occasional inconsistency in intersection status.

Introduction to the Ray Tracing Package

I have written a ray tracing package that implements basic recursive ray tracing. This software and its source code are freely available and can be downloaded from this book's Web site.

The ray tracing software uses an object-oriented approach to rendering and ray tracing. The object-oriented design includes base classes for materials, for geometric shapes, for lights, and for textures. This provides considerable flexibility in adding features since it allows the addition of new geometric shapes without affecting the functionality of older code; similarly, new kinds of lighting models, new kinds of textures, and so on, can be added without needing to change the structure of the software.

The material and lighting classes supported by the software include the usual material properties such as ambient, diffuse, and specular color and specular exponents. In addition, the material classes include reflection and transmission color coefficients for use in recursive ray tracing. The complete Phong model for local lighting is supported, including all the OpenGL-type features such as spotlights and attenuation. A version of the Cook–Torrance model is also supported.

The ray tracing software supports a range of geometric shapes, including spheres, triangles, parallelograms, cones, cylinders, tori, ellipsoids, parallelepipeds, and Bézier patches. Collectively, these geometric shapes are called viewable objects. The viewable object classes are responsible for detecting intersections of rays against a particular geometric shape. The viewable object classes also calculate normals and keep track of the material of an object.

This chapter discusses the mathematics of linear, affine, and perspective transformations and their uses in OpenGL. The basic purpose of these transformations is to provide methods of changing the shape and position of objects, but the use of these transformations is pervasive throughout computer graphics. In fact, affine transformations are arguably the most fundamental mathematical tool for computer graphics.

An obvious use of transformations is to help simplify the task of geometric modeling. For example, suppose an artist is designing a computerized geometric model of a Ferris wheel. A Ferris wheel has considerable symmetry and includes many repeated elements such as multiple cars and struts. The artist could design a single model of the car and then place multiple instances of the car around the Ferris wheel attached at the proper points. Similarly, the artist could build the main structure of the Ferris wheel by designing one radial “slice” of the wheel and using multiple rotated copies of this slice to form the entire structure. Affine transformations are used to describe how the parts are placed and oriented.

A second important use of transformations is to describe animation. Continuing with the Ferris wheel example, if the Ferris wheel is animated, then the positions and orientations of its individual geometric components are constantly changing. Thus, for animation, it is necessary to compute time-varying affine transformations to simulate the motion of the Ferris wheel.

This chapter takes up the subject of interpolation. For the purposes of the present chapter, the term “interpolation” means the process of finding intermediate values of a function by averaging its values at extreme points. Interpolation was already studied in Section II.4, where it was used for Gouraud and Phong interpolation to average colors or normals to create smooth lighting and shading effects. In Chapter V, interpolation is used to apply texture maps. More sophisticated kinds of interpolation will be important in the study of Bézier curves and B-splines in Chapters VII and VIII. Interpolation is also very important for ianimation, where both positions and orientations of objects may need to be interpolated.

The first three sections below address the simplest forms of interpolation; namely, linear interpolation on lines and triangles. This includes studying weighted averages, affine combinations, extrapolation, and barycentric coordinates. Then we turn to the topics of bilinear and trilinear interpolation with an emphasis on bilinear interpolation, including an algorithm for inverting bilinear interpolation. The next section has a short, abstract discussion on convex sets, convex hulls, and the definition of convex hulls in terms of weighted averages. After that, we take up the topic of weighted averages performed on points represented in homogeneous coordinates. It is shown that the effect of the homogeneous coordinate is similar to an extra weighting coefficient, and as a corollary, we derive the formulas for hyperbolic interpolation that are important for accurate interpolation in screen-space coordinates.

Texture Mapping an Image

Texture mapping, in its simplest form, consists of applying a graphics image, a picture, or a pattern to a surface. A texture map can, for example, apply an actual picture to a surface such as a label on a can or a picture on a billboard or can apply semirepetitive patterns such as wood grain or stone surfaces. More generally, a texture map can hold any kind of information that affects the appearance of a surface: the texture map serves as a precomputed table, and the texture mapping then consists simply of table lookup to retrieve the information affecting a particular point on the surface as it is rendered. If you do not use texture maps, your surfaces will either be rendered as very smooth, uniform surfaces or will need to be rendered with very small polygons so that you can explicitly specify surface properties on a fine scale.

Texture maps are often used to very good effect in real-time rendering settings such as computer games since they give good results with a minimum of computational load. In addition, texture maps are widely supported by graphics hardware such as graphics boards for PCs so that they can be used without needing much computation from a central processor.

Texture maps can be applied at essentially three different points in the graphics rendering process, which we list more or less in order of increasing generality and flexibility:

• A texture map can hold colors that are applied to a surface in “replace” or “decal” mode: the texture map colors just overwrite whatever surface colors are otherwise present. In this case, no lighting calculations should be performed, as the results of the lighting calculations would just be overwritten.

• […]

A spline curve is a smooth curve specified succinctly in terms of a few points. These two aspects of splines, that they are smooth and that they are specified succinctly in terms of only a few points, are both important. First, the ability to specify a curve with only a few points reduces storage requirements. In addition, it facilitates the computer-aided design of curves and surfaces because the designer or artist can control an entire curve by varying only a few points. Second, the commonly used methods for generating splines give curves with good smoothness properties and without undesired oscillations. Furthermore, these splines also allow for isolated points where the curve is not smooth, such as points where the spline has a “corner.” A third important property of splines is that there are simple algorithms for finding points on the spline curve or surface and simple criteria for deciding how finely a spline must be approximated by linear segments to obtain a sufficiently faithful representation of the spline. The main classes of splines discussed in this book are the Bézier curves and the B-spline curves. Bézier curves and patches are covered in this chapter, and B-splines in the next chapter.

Historically, splines were specified mechanically by systems such as flexible strips of wood or metal that were tied into position to record a desired curve. These mechanical systems were awkward and difficult to work with, and they could not be used to give a permanent, reproducible description of a curve.