Game Development Reference
In-Depth Information
Applying Equation (6.19) and solving for cos 3 θ gives us
p
q
q
2
q
cos 3
θ
=
=
=
.
(6.22)
3
2
m
p
3
27
3
2
3
Since
, thereby guaranteeing that
the right side of Equation (6.22) is always less than 1 in absolute value. The in-
verse cosine is thus defined, and we can solve for θ to arrive at
′ >
0
, Equation (6.16) implies that
q
<−
p
D
1 cos
3
q
1
θ
=
.
(6.23)
p
3
Therefore, one solution to Equation (6.9) is given by
x
=
2
cos
=
2
p θ
cos
.
(6.24)
1
(
)
(
)
Since
cos 3
θ k
+
2
=
cos 3
θ
for any integer k , we can write
1
q
2
π
1
θ
=
cos
k
.
(6.25)
k
3
3
3
p
Distinct values of cos θ are generated by choosing three values for k that are
congruent to 0, 1, and 2 modulo 3. Using
1
, we can express the remaining
k
two solutions to Equation (6.9) as
2
π
x
=−
2
p θ
cos
+
2
3
2
π
.
(6.26)
x
=−
2
p θ
cos
3
3
6.1.3 Quartic Polynomials
A quartic equation having the form
4
3
2
t
++++=
at
bt
ct
d
0
(6.27)
(where again we have performed any necessary division to produce a leading
coefficient of 1) can be shifted to eliminate the cubic term by making the
substitution
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