3D Math Primer for Graphics and Game Development

Fletcher Dunn, Ian Parberry

• Verlag: A K Peters/CRC Press
• Jahr: 2011
• ISBN:
  9781498759892
• Auflage: 2
• Thema: Computer Science
• Thema: Programming Games

This engaging book presents the essential mathematics needed to describe, simulate, and render a 3D world. Reflecting both academic and in-the-trenches practical experience, the authors teach you how to describe objects and their positions, orientations, and trajectories in 3D using mathematics. The text provides an introduction to mathematics for

Chapter 1

Cartesian Coordinate Systems

Before turning to those moral and mental aspects of the matter which present the greatest difficulties, let the inquirer begin by mastering more elementary problems.

— Sherlock Holmes from A Study in Scarlett (1887)

3D math is all about measuring locations, distances, and angles precisely and mathematically in 3D space. The most frequently used framework to perform such calculations using a computer is called the Cartesian coordinate system. Cartesian mathematics was invented by (and is named after) a brilliant French philosopher, physicist, physiologist, and mathematician named René Descartes, who lived from 1596 to 1650. René Descartes is famous not just for inventing Cartesian mathematics, which at the time was a stunning unification of algebra and geometry. He is also well-known for making a pretty good stab of answering the question “How do I know something is true?”—a question that has kept generations of philosophers happily employed and does not necessarily involve dead sheep (which will perhaps disturbingly be a central feature of the next section), unless you really want it to. Descartes rejected the answers proposed by the Ancient Greeks, which are ethos (roughly, “because I told you so”), pathos (“because it would be nice”), and logos (“because it makes sense”), and set about figuring it out for himself with a pencil and paper.

This chapter is divided into four main sections.

• Section 1.1 reviews some basic principles of number systems and the first law of computer graphics.
• Section 1.2 introduces 2D Cartesian mathematics, the mathematics of flat surfaces. It shows how to describe a 2D cartesian coordinate space and how to locate points using that space.
• Section 1.3 extends these ideas into three dimensions. It explains left-and right-handed coordinate spaces and establishes some conventions used in this book.
• Section 1.4 concludes the chapter by quickly reviewing assorted prerequisites.

1.1 1D Mathematics

You’re reading this book because you want to know about 3D mathematics, so you’re probably wondering why we’re bothering to talk about 1D math. Well, there are a couple of issues about number systems and counting that we would like to clear up before we get to 3D.

The natural numbers, often called the counting numbers, were invented millennia ago, probably to keep track of dead sheep. The concept of “one sheep” came easily (see Figure 1.1), then “two sheep,” “three sheep,” but people very quickly became convinced that this was too much work, and gave up counting at some point that they invariably called “many sheep.” Different cultures gave up at different points, depending on their threshold of boredom. Eventually, civilization expanded to the point where we could afford to have people sitting around thinking about numbers instead of doing more survival-oriented tasks such as killing sheep and eating them. These savvy thinkers immortalized the concept of zero (no sheep), and although they didn’t get around to naming all of the natural numbers, they figured out various systems whereby they could name them if they really wanted to using digits such as 1, 2, etc. (or if you were Roman, M, X, I, etc.). Thus, mathematics was born.

Figure 1.1. One dead sheep

The habit of lining sheep up in a row so that they can be easily counted leads to the concept of a number line, that is, a line with the numbers marked off at regular intervals, as in Figure 1.2. This line can in principle go on for as long as we wish, but to avoid boredom we have stopped at five sheep and used an arrowhead to let you know that the line can continue. Clearer thinkers can visualize it going off to infinity, but historical purveyors of dead sh...