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Preface to Computer Graphics: Principles and Practice, 3rd Edition

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The authors of Computer Graphics: Principles and Practice introduce the long-anticipated third edition of their well regarded textbook.
From the book

This book presents many of the important ideas of computer graphics to students, researchers, and practitioners. Several of these ideas are not new: They have already appeared in widely available scholarly publications, technical reports, textbooks, and lay-press articles. The advantage of writing a textbook sometime after the appearance of an idea is that its long-term impact can be understood better and placed in a larger context. Our aim has been to treat ideas with as much sophistication as possible (which includes omitting ideas that are no longer as important as they once were), while still introducing beginning students to the subject lucidly and gracefully.

This is a second-generation graphics book: Rather than treating all prior work as implicitly valid, we evaluate it in the context of today’s understanding, and update the presentation as appropriate.

Even the most elementary issues can turn out to be remarkably subtle. Suppose, for instance, that you’re designing a program that must run in a low-light environment—a darkened movie theatre, for instance. Obviously you cannot use a bright display, and so using brightness contrast to distinguish among different items in your program display would be inappropriate. You decide to use color instead. Unfortunately, color perception in low-light environments is not nearly as good as in high-light environments, and some text colors are easier to read than others in low light. Is your cursor still easy to see? Maybe to make that simpler, you should make the cursor constantly jitter, exploiting the motion sensitivity of the eye. So what seemed like a simple question turns out to involve issues of interface design, color theory, and human perception.

This example, simple as it is, also makes some unspoken assumptions: that the application uses graphics (rather than, say, tactile output or a well-isolated audio earpiece), that it does not use the regular theatre screen, and that it does not use a head-worn display. It makes explicit assumptions as well—for instance, that a cursor will be used (some UIs intentionally don’t use a cursor). Each of these assumptions reflects a user-interface choice as well.

Unfortunately, this interrelatedness of things makes it impossible to present topics in a completely ordered fashion and still motivate them well; the subject is simply no longer linearizable. We could have covered all the mathematics, theory of perception, and other, more abstract, topics first, and only then moved on to their graphics applications. Although this might produce a better reference work (you know just where to look to learn about generalized cross products, for instance), it doesn’t work well for a textbook, since the motivating applications would all come at the end. Alternatively, we could have taken a case-study approach, in which we try to complete various increasingly difficult tasks, and introduce the necessary material as the need arises. This makes for a natural progression in some cases, but makes it difficult to give a broad organizational view of the subject. Our approach is a compromise: We start with some widely used mathematics and notational conventions, and then work from topic to topic, introducing supporting mathematics as needed. Readers already familiar with the mathematics can safely skip this material without missing any computer graphics; others may learn a good deal by reading these sections. Teachers may choose to include or omit them as needed. The topic-based organization of the book entails some redundancy. We discuss the graphics pipeline multiple times at varying levels of detail, for instance. Rather than referring the reader back to a previous chapter, sometimes we redescribe things, believing that this introduces a more natural flow. Flipping back 500 pages to review a figure can be a substantial distraction.

The other challenge for a textbook author is to decide how encyclopedic to make the text. The first edition of this book really did cover a very large fraction of the published work in computer graphics; the second edition at least made passing references to much of the work. This edition abandons any pretense of being encyclopedic, for a very good reason: When the second edition was written, a single person could carry, under one arm, all of the proceedings of SIGGRAPH, the largest annual computer graphics conference, and these constituted a fair representation of all technical writings on the subject. Now the SIGGRAPH proceedings (which are just one of many publication venues) occupy several cubic feet. Even a telegraphic textbook cannot cram all that information into a thousand pages. Our goal in this book is therefore to lead the reader to the point where he or she can read and reproduce many of today’s SIGGRAPH papers, albeit with some caveats:

  • First, computer graphics and computer vision are overlapping more and more, but there is no excuse for us to write a computer vision textbook; others with far greater knowledge have already done so.
  • Second, computer graphics involves programming; many graphics applications are quite large, but we do not attempt to teach either programming or software engineering in this book.We do briefly discuss programming (and especially debugging) approaches that are unique to graphics, however.
  • Third, most graphics applications have a user interface. At the time of this writing, most of these interfaces are based on windows with menus, and mouse interaction, although touch-based interfaces are becoming commonplace as well. There was a time when user-interface research was a part of graphics, but it’s now an independent community—albeit with substantial overlap with graphics—and we therefore assume that the student has some experience in creating programs with user interfaces, and don’t discuss these in any depth, except for some 3D interfaces whose implementations are more closely related to graphics.

Of course, research papers in graphics differ. Some are highly mathematical, others describe large-scale systems with complex engineering tradeoffs, and still others involve a knowledge of physics, color theory, typography, photography, chemistry, zoology. . . the list goes on and on. Our goal is to prepare the reader to understand the computer graphics in these papers; the other material may require considerable external study as well.

Historical Approaches

The history of computer graphics is largely one of ad hoc approaches to the immediate problems at hand. Saying this is in no way an indictment of the people who took those approaches: They had jobs to do, and found ways to do them. Sometimes their solutions had important ideas wrapped up within them; at other times they were merely ways to get things done, and their adoption has interfered with progress in later years. For instance, the image-compositing model used in most graphics systems assumes that color values stored in images can be blended linearly. In actual practice, the color values stored in images are nonlinearly related to light intensity; taking linear combinations of these does not correspond to taking linear combinations of intensity. The difference between the two approaches began to be noticed when studios tried to combine real-world and computer-generated imagery; this compositing technology produced unacceptable results. In addition, some early approaches were deeply principled, but the associated programs made assumptions about hardware that were no longer valid a few years later; readers, looking first at the details of implementation, said, “Oh, this is old stuff—it’s not relevant to us at all,” and missed the still important ideas of the research. All too frequently, too, researchers have simply reinvented things known in other disciplines for years.

We therefore do not follow the chronological development of computer graphics. Just as physics courses do not begin with Aristotle’s description of dynamics, but instead work directly with Newton’s (and the better ones describe the limitations of even that system, setting the stage for quantum approaches, etc.), we try to start directly from the best current understanding of issues, while still presenting various older ideas when relevant. We also try to point out sources for ideas that may not be familiar ones: Newell’s formula for the normal vector to a polygon in 3-space was known to Grassmann in the 1800s, for instance. Our hope in referencing these sources is to increase the reader’s awareness of the variety of already developed ideas that are waiting to be applied to graphics.


The most striking aspect of graphics in our everyday lives is the 3D imagery being used in video games and special effects in the entertainment industry and advertisements. But our day-to-day interactions with home computers, cell phones, etc., are also based on computer graphics. Perhaps they are less visible in part because they are more successful: The best interfaces are the ones you don’t notice. It’s tempting to say that “2D graphics” is simpler—that 3D graphics is just a more complicated version of the same thing. But many of the issues in 2D graphics—how best to display images on a screen made of a rectangular grid of light-emitting elements, for instance, or how to construct effective and powerful interfaces—are just as difficult as those found in making pictures of three-dimensional scenes. And the simple models conventionally used in 2D graphics can lead the student into false assumptions about how best to represent things like color or shape. We therefore have largely integrated the presentation of 2D and 3D graphics so as to address simultaneously the subtle issues common to both.

This book is unusual in the level at which we put the “black box.” Almost every computer science book has to decide at what level to abstract something about the computers that the reader will be familiar with. In a graphics book, we have to decide what graphics system the reader will be encountering as well. It’s not hard (after writing a first program or two) to believe that some combination of hardware and software inside your computer can make a colored triangle appear on your display when you issue certain instructions. The details of how this happens are not relevant to a surprisingly large part of graphics. For instance, what happens if you ask the graphics system to draw a red triangle that’s below the displayable area of your screen? Are the pixel locations that need to be made red computed and then ignored because they’re off-screen? Or does the graphics system realize, before computing any pixel values, that the triangle is off-screen and just quit? In some sense, unless you’re designing a graphics card, it just doesn’t matter all that much; indeed, it’s something you, as a user of a graphics system, can’t really control. In much of the book, therefore, we treat the graphics system as something that can display certain pixel values, or draw triangles and lines, without worrying too much about the “how” of this part. The details are included in the chapters on rasterization and on graphics hardware. But because they are mostly beyond our control, topics like clipping, antialiasing of lines, and rasterization algorithms are all postponed to later chapters.

Another aspect of the book’s pedagogy is that we generally try to show how ideas or techniques arise. This can lead to long explanations, but helps, we hope, when students need to derive something for themselves: The approaches they’ve encountered may suggest an approach to their current problem.

We believe that the best way to learn graphics is to first learn the mathematics behind it. The drawback of this approach compared to jumping to applications is that learning the abstract math increases the amount of time it takes to learn your first few techniques. But you only pay that overhead once. By the time you’re learning the tenth related technique, your investment will pay off because you’ll recognize that the new method combines elements you’ve already studied.

Of course, you’re reading this book because you are motivated to write programs that make pictures. So we try to start many topics by diving straight into a solution before stepping back to deeply consider the broader mathematical issues. Most of this book is concerned with that stepping-back process. Having investigated the mathematics, we’ll then close out topics by sketching other related problems and some solutions to them. Because we’ve focused on the underlying principles, you won’t need us to tell you the details for these sketches. From your understanding of the principles, the approach of each solution should be clear, and you’ll have enough knowledge to be able to read and understand the original cited publication in its author’s own words, rather than relying on us to translate it for you. What we can do is present some older ideas in a slightly more modern form so that when you go back to read the original paper, you’ll have some idea how its vocabulary matches your own.

Current Practice

Graphics is a hands-on discipline. And since the business of graphics is the presentation of visual information to a viewer, and the subsequent interaction with it, graphical tools can often be used effectively to debug new graphical algorithms. But doing this requires the ability to write graphics programs. There are many alternative ways to produce graphical imagery on today’s computers, and for much of the material in this book, one method is as good as another. The conversion between one programming language and its libraries and another is routine. But for teaching the subject, it seems best to work in a single language so that the student can concentrate on the deeper ideas. Throughout this book, we’ll suggest exercises to be written using Windows Presentation Foundation (WPF), a widely available graphics system, for which we’ve written a basic and easily modified program we call a “test bed” in which the student can work. For situations where WPF is not appropriate, we’ve often used G3D, a publicly available graphics library maintained by one of the authors. And in many situations, we’ve written pseudocode. It provides a compact way to express ideas, and for most algorithms, actual code (in the language of your choice) can be downloaded from the Web; it seldom makes sense to include it in the text. The formatting of code varies; in cases where programs are developed from an informal sketch to a nearly complete program in some language, things like syntax highlighting make no sense until quite late versions, and may be omitted entirely. Sometimes it’s nice to have the code match the mathematics, leading us to use variables with names like xR, which get typeset as math rather than code. In general, we italicize pseudocode, and use indentation rather than braces in pseudocode to indicate code blocks. In general, our pseudocode is very informal; we use it to convey the broad ideas rather than the details.

This is not a book about writing graphics programs, nor is it about using them. Readers will find no hints about the best ways to store images in Adobe’s latest image-editing program, for instance. But we hope that, having understood the concepts in this book and being competent programmers already, they will both be able to write graphics programs and understand how to use those that are already written.


Throughout the book we have identified certain computer graphics principles that will help the reader in future work; we’ve also included sections on current practice—sections that discuss, for example, how to approximate your ideal solution on today’s hardware, or how to compute your actual ideal solution more rapidly. Even practices that are tuned to today’s hardware can prove useful tomorrow, so although in a decade the practices described may no longer be directly applicable, they show approaches that we believe will still be valuable for years.


Much of this book assumes little more preparation than what a technically savvy undergraduate student may have: the ability to write object-oriented programs; a working knowledge of calculus; some familiarity with vectors, perhaps from a math class or physics class or even a computer science class; and at least some encounter with linear transformations. We also expect that the typical student has written a program or two containing 2D graphical objects like buttons or checkboxes or icons.

Some parts of this book, however, depend on far more mathematics, and attempting to teach that mathematics within the limits of this text is impossible. Generally, however, this sophisticated mathematics is carefully limited to a few sections, and these sections are more appropriate for a graduate course than an introductory one. Both they and certain mathematically sophisticated exercises are marked with a “math road-sign” symbol thus: Correspondingly, topics that use deeper notions from computer science are marked with a “computer science road-sign,”

Some mathematical aspects of the text may seem strange to those who have met vectors in other contexts; the first author, whose Ph.D. is in mathematics, certainly was baffled by some of his first encounters with how graphics researchers do things. We attempt to explain these variations from standard mathematical approaches clearly and thoroughly.

Paths through This Book

This book can be used for a semester-long or yearlong undergraduate course, or as a general reference in a graduate course. In an undergraduate course, the advanced mathematical topics can safely be omitted (e.g., the discussions of analogs to barycentric coordinates, manifold meshes, spherical harmonics, etc.) while concentrating on the basic ideas of creating and displaying geometric models, understanding the mathematics of transformations, camera specifications, and the standard models used in representing light, color, reflectance, etc., along with some hints of the limitations of these models. It should also cover basic graphics applications and the user-interface concerns, design tradeoffs, and compromises necessary to make them efficient, possibly ending with some special topic like creating simple animations, or writing a basic ray tracer. Even this is too much for a single semester, and even a yearlong course will leave many sections of the book untouched, as future reading for interested students.

An aggressive semester-long (14-week) course could cover the following.

  1. Introduction and a simple 2D program: Chapters 1, 2, and 3.
  2. Introduction to the geometry of rendering, and further 2D and 3D programs: Chapters 3 and 4. Visual perception and the human visual system: Chapter 5.
  3. Modeling of geometry in 2D and 3D: meshes, splines, and implicit models. Sections 7.1–7.9, Chapters 8 and 9, Sections 22.1–22.4, 23.1–23.3, and 24.1–24.5.
  4. Images, part 1: Chapter 17, Sections 18.1–18.11.
  5. Images, part 2: Sections 18.12–18.20, Chapter 19.
  6. 2D and 3D transformations: Sections 10.1–10.12, Sections 11.1–11.3, Chapter 12.
  7. Viewing, cameras, and post-homogeneous interpolation. Sections 13.1–13.7, 15.6.4.
  8. Standard approximations in graphics: Chapter 14, selected sections.
  9. Rasterization and ray casting: Chapter 15.
  10. Light and reflection: Sections 26.1–26.7 (Section 26.5 optional); Section 26.10.
  11. Color: Sections 28.1–28.12.
  12. Basic reflectance models, light transport: Sections 27.1–27.5, 29.1–29.2, 29.6, 29.8.
  13. Recursive ray-tracing details, texture: Sections 24.9, 31.16, 20.1–20.6.
  14. Visible surface determination and acceleration data structures; overview of more advanced rendering techniques: selections from Chapters 31, 36, and 37.

However, not all the material in every section would be appropriate for a first course.

Alternatively, consider the syllabus for a 12-week undergraduate course on physically based rendering that takes first principles from offline to real-time rendering. It could dive into the core mathematics and radiometry behind ray tracing, and then cycle back to pick up the computer science ideas needed for scalability and performance.

  1. Introduction: Chapter 1
  2. Light: Chapter 26
  3. Perception; light transport: Chapters 5 and 29
  4. A brief overview of meshes and scene graphs: Sections 6.6, 14.1–5
  5. Transformations: Chapters 10 and 13, briefly.
  6. Ray casting: Sections 15.1–4, 7.6–9
  7. Acceleration data structures: Chapter 37; Sections 36.1–36.3, 36.5–36.6, 36.9
  8. Rendering theory: Chapters 30 and 31
  9. Rendering practice: Chapter 32
  10. Color and material: Sections 14.6–14.11, 28, and 27
  11. Rasterization: Sections 15.5–9
  12. Shaders and hardware: Sections 16.3–5, Chapters 33 and 38

Note that these paths touch chapters out of numerical order. We’ve intentionally written this book in a style where most chapters are self-contained, with cross-references instead of prerequisites, to support such traversal.

Differences from the Previous Edition

This edition is almost completely new, although many of the topics covered in the previous edition appear here. With the advent of the GPU, triangles are converted to pixels (or samples) by radically different approaches than the old scan-conversion algorithms. We no longer discuss those. In discussing light, we strongly favor physical units of measurement, which adds some complexity to discussions of older techniques that did not concern themselves with units. Rather than preparing two graphics packages for 2D and 3D graphics, as we did for the previous editions, we’ve chosen to use widely available systems, and provide tools to help the student get started using them.


Often in this book you’ll see references to the book’s website. It’s at http://cgpp.net and contains not only the testbed software and several examples derived from it, but additional material for many chapters, and the interactive experiments in WPF for Chapters 2 and 6.


A book like this is written by the authors, but it’s enormously enhanced by the contributions of others.

Support and encouragement from Microsoft, especially from Eric Rudder and S. Somasegur, helped to both initiate and complete this project.

The 3D test bed evolved from code written by Dan Leventhal; we also thank Mike Hodnick at kindohm.com, who graciously agreed to let us use his code as a starting point for an earlier draft, and Jordan Parker and Anthony Hodsdon for assisting with WPF.

Two students from Williams College worked very hard in supporting the book: Guedis Cardenas on the bibliography, and Michael Mara on the G3D Innovation Engine used in several chapters; Corey Taylor of Electronic Arts also helped with G3D.

Nancy Pollard of CMU and Liz Marai of the University of Pittsburgh both used early drafts of several chapters in their graphics courses, and provided excellent feedback.

Jim Arvo served not only as an oracle on everything relating to rendering, but helped to reframe the first author’s understanding of the field.

Many others, in addition to some of those just mentioned, read chapter drafts, prepared images or figures, suggested topics or ways to present them, or helped out in other ways. In alphabetical order, they are John Anderson, Jim Arvo, Tom Banchoff, Pascal Barla, Connelly Barnes, Brian Barsky, Ronen Barzel, Melissa Byun, Marie-Paule Cani, Lauren Clarke, Elaine Cohen, Doug DeCarlo, Patrick Doran, Kayvon Fatahalian, Adam Finkelstein, Travis Fischer, Roger Fong, Mike Fredrickson, Yudi Fu, Andrew Glassner, Bernie Gordon, Don Greenberg, Pat Hanrahan, Ben Herila, Alex Hills, Ken Joy, Olga Karpenko, Donnie Kendall, Justin Kim, Philip Klein, Joe LaViola, Kefei Lei, Nong Li, Lisa Manekofsky, Bill Mark, John Montrym, Henry Moreton, Tomer Moscovich, Jacopo Pantaleoni, Jill Pipher, Charles Poynton, Rich Riesenfeld, Alyn Rockwood, Peter Schroeder, François Sillion, David Simons, Alvy Ray Smith, Stephen Spencer, Erik Sudderth, Joelle Thollot, Ken Torrance, Jim Valles, Daniel Wigdor, Dan Wilk, Brian Wyvill, and Silvia Zuffi. Despite our best efforts, we have probably forgotten some people, and apologize to them.

It’s a sign of the general goodness of the field that we got a lot of support in writing from authors of competing books. Eric Haines, Greg Humphreys, Steve Marschner, Matt Pharr, and Pete Shirley all contributed to making this a better book. It’s wonderful to work in a field with folks like this.

We’d never had managed to produce this book without the support, tolerance, indulgence, and vision of our editor, Peter Gordon. And we all appreciate the enormous support of our families throughout this project.

For the Student

Your professor will probably choose some route through this book, selecting topics that fit well together, perhaps following one of the suggested trails mentioned earlier. Don’t let that constrain you. If you want to know about something, use the index and start reading. Sometimes you’ll find yourself lacking background, and you won’t be able to make sense of what you read. When that happens, read the background material. It’ll be easier than reading it at some other time, because right now you have a reason to learn it. If you stall out, search the Web for someone’s implementation and download and run it. When you notice it doesn’t look quite right, you can start examining the implementation, and trying to reverse engineer it. Sometimes this is a great way to understand something. Follow the practice-theory-practice model of learning: Try something, see whether you can make it work, and if you can’t, read up on how others did it, and then try again. The first attempt may be frustrating, but it sets you up to better understand the theory when you get to it. If you can’t bring yourself to follow the practice-theory-practice model, at the very least you should take the time to do the inline exercises for any chapter you read.

Graphics is a young field, so young that undergraduates are routinely coauthors on SIGGRAPH papers. In a year you can learn enough to start contributing new ideas.

Graphics also uses a lot of mathematics. If mathematics has always seemed abstract and theoretical to you, graphics can be really helpful: The uses of mathematics in graphics are practical, and you can often see the consequences of a theorem in the pictures you make. If mathematics has always come easily to you, you can gain some enjoyment from trying to take the ideas we present and extend them further. While this book contains a lot of mathematics, it only scratches the surface of what gets used in modern research papers.

Finally, doubt everything. We’ve done our best to tell the truth in this book, as we understand it. We think we’ve done pretty well, and the great bulk of what we’ve said is true. In a few places, we’ve deliberately told partial truths when we introduced a notion, and then amplified these in a later section when we’re discussing details. But aside from that, we’ve surely failed to tell the truth in other places as well. In some cases, we’ve simply made errors, leaving out a minus sign, or making an off-by-one error in a loop. In other cases, the current understanding of the graphics community is just inadequate, and we’ve believed what others have said, and will have to adjust our beliefs later. These errors are opportunities for you. Martin Gardner said that the true sound of scientific discovery is not “Aha!” but “Hey, that’s odd. . . .” So if every now and then something seems odd to you, go ahead and doubt it. Look into it more closely. If it turns out to be true, you’ll have cleared some cobwebs from your understanding. If it’s false, it’s a chance for you to advance the field.

For the Teacher

If you’re like us, you probably read the “For the Student” section even though it wasn’t for you. (And your students are probably reading this part, too.) You know that we’ve advised them to graze through the book at random, and to doubt everything.

We recommend to you (aside from the suggestions in the remainder of this preface) two things. The first is that you encourage, or even require, that your students answer the inline exercises in the book. To the student who says, “I’ve got too much to do! I can’t waste time stopping to do some exercise,” just say, “We don’t have time to stop for gas . . . we’re already late.” The second is that you assign your students projects or homeworks that have both a fixed goal and an open-ended component. The steady students will complete the fixed-goal parts and learn the material you want to cover. The others, given the chance to do something fun, may do things with the open-ended exercises that will amaze you. And in doing so, they’ll find that they need to learn things that might seem just out of reach, until they suddenly master them, and become empowered. Graphics is a terrific medium for this: Successes are instantly visible and rewarding, and this sets up a feedback loop for advancement. The combination of visible feedback with the ideas of scalability that they’ve encountered elsewhere in computer science can be revelatory.

Discussion and Further Reading

Most chapters of this book contain a “Discussion and Further Reading” section like this one, pointing to either background references or advanced applications of the ideas in the chapter. For this preface, the only suitable further reading is very general: We recommend that you immediately begin to look at the proceedings of ACM SIGGRAPH conferences, and of other graphics conferences like Eurographics and Computer Graphics International, and, depending on your evolving interest, some of the more specialized venues like the Eurographics Symposium on Rendering, I3D, and the Symposium on Computer Animation. While at first the papers in these conferences will seem to rely on a great deal of prior knowledge, you’ll find that you rapidly get a sense of what things are possible (if only by looking at the pictures), and what sorts of skills are necessary to achieve them. You’ll also rapidly discover ideas that keep reappearing in the areas that most interest you, and this can help guide your further reading as you learn graphics.

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  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020