Game Development Reference
In-Depth Information
Chapter 1
Cartesian Coordinate Systems
Before turning to those moral and mental aspects of the matter
which present the greatest di culties, 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 coordi-
nate system. Cartesian mathematics was invented by (and is named after)
a brilliant French philosopher, physicist, physiologist, and mathematician
named Rene Descartes, who lived from 1596 to 1650. Rene 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 (“be-
cause 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 topic.
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