Game Development Reference
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a
b
c
.
(B.24)
=
=
sin
α
sin
β
sin
γ
Now observe the following Pythagorean relationships in the triangle shown
in Figure B.3.
2
2
2
+=
++=
x
yb
(
)
2
2
2
ax
y c
(B.25)
Solving the first equation for
y and substituting into the second equation gives
us
(
)
c
2
=+ +−
=++
ax b x
ab x
2
2
2
2
2
2.
(B.26)
The value of x can be replaced by observing
x
πγ b
(
)
cos
−=
.
(B.27)
(
)
Since
cos
πγ
−=−
cos
γ
, we have
=−
b γ
cos
.
(B.28)
x
Plugging this into Equation (B.26) produces the law of cosines :
2
2
2
c
=+−
ab bγ
2 s
.
(B.29)
Of course, this reduces to the Pythagorean theorem when γ is a right angle since
cos
π
2
=
0
.
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