Game Development Reference
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Notice that the notation (x,y) could refer to an open interval or a 2D
point. Likewise, [x,y] could be a closed interval or a 2D vector (discussed
in the next chapter). The context will always make clear which is the case.
1.4.3 Angles, Degrees, and Radians
An angle measures an amount of rotation in the plane. Variables repre-
senting angles are often assigned the Greek letter θ.
6
The most important
units of measure used to specify angles are degrees (
o
) and radians (rad).
Humans usually measure angles using degrees. One degree measures
1/360th of a revolution, so 360
o
represents a complete revolution.
7
Math-
ematicians, however, prefer to measure angles in radians, which is a unit
of measure based on the properties of a circle. When we specify the angle
between two rays in radians, we are actually measuring the length of the
intercepted arc of a unit circle (a circle centered at the origin with radius 1),
as shown in Figure 1.16.
Figure 1.16
A radian measures arc length on a
unit circle
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One prerequisite that we do not assume in this topic is familiarity with the Greek
alphabet. The symbol θ is the lowercase theta, pronounced “THAY-tuh.”
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The number 360 is a relatively arbitrary choice, which may have had its origin in
primitive calendars, such as the Persian calendar, which divided the year into 360 days.
This error was never corrected to 365 because the number 360 is so darn convenient. The
number 360 has a whopping 22 divisors (not counting itself and 1): 2, 3, 4, 5, 6, 8, 9, 10,
12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180. This means 360 can be divided
evenly in a large number of cases without needing fractions, which was apparently a
good thing to early civilizations. As early as 1750 BCE the Babylonians had devised
a sexagesimal (base 60) number system. The number 360 is also large enough so that
precision to the nearest whole degree is su
cient in many circumstances.
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