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Appendix B
Trigonometry
“Life is good for only two things, discovering mathematics and
teaching mathematics.”
—Sim´eon Poisson
This appendix is intended to be a reference to some simple laws of trigonom-
etry as well as some more sophisticated ones. The laws of trigonometry are
particularly important tools in computer graphics. One example of their
usefulness is that they provide ways to simplify equations and thereby to
increase speed.
B.1 Definitions
According to Figure B.1, where p =(p
x
,p
y
) is a unit vector, i.e., ||p|| =1,
the fundamental trigonometric functions, sin, cos, and tan, are defined by
Equation B.1.
Fundamental trigonometric functions :
sin φ = p
y
cos φ = p
x
tan φ =
sin φ
cos φ
=
p
y
p
x
(B.1)
The sin, cos, and tan functions can be expanded into MacLaurin se-
ries, as shown in Equation B.2. MacLaurin series are a special case of the
more general Taylor series. A Taylor series is an expansion about an ar-
bitrary point, while a MacLaurin series always is developed around x =0.
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