Game Development Reference
In-Depth Information
The discriminant is then given by
(
)
3
2
D
=−
108
pq
+
.
(6.15)
Setting
D
(
)
3
2
D
=
−+
pq
(6.16)
108
lets us express r and s as
r
=−+−
3
q
D
3
s
=−−−
q
D
.
(6.17)
As with quadratic equations, the discriminant gives us information about how
many real roots exist. In the case that
, the value of x given in Equation
(6.13) represents the only real solution of Equation (6.9).
<
D
0
In the case that
=
, we have rs
=
, so there are two real solutions, one of
D
0
which is a double root:
x r
xx ρ ρ r
=
=+
2
1
(
)
2
,
=−
r
.
(6.18)
23
>
, Equation (6.13) yields three distinct real
solutions. Unfortunately, we still have to use complex numbers to calculate these
solutions. An alternative method can be applied in this case that does not require
complex arithmetic. The method relies on the trigonometric identity
In the remaining case that
D
0
3
4cos
θ
3cos
θ
=
cos 3
θ
,
(6.19)
which can be verified using the Euler formula (see Exercise 1 at the end of this
chapter). Making the substitution
=
2
m θ
cos
in Equation (6.9) with
mp
=−
3
,
x
gives us
8
m
3
cos
3
θ pm θ q
+
2
cos
+
=
0
.
(6.20)
(Note that p must be negative in order for D
to be positive.) Replacing p with
3 m
2
2 m out of the first two terms yields
3
and factoring
(
)
2
m
3
4 cos
3
θ
3cos
θ q
+
=
0
.
(6.21)
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