Game Development Reference

In-Depth Information

a

.

(6.8)

t

=−

x

3

This gives us the equation

x

3

++=

px

q

0

,

(6.9)

where

1

3

2

p

=−

a

+

b

2

1

q

=

a

3

−

ab

+

c

.

(6.10)

27

3

Once a solution
x
to Equation (6.9) is found, we subtract

a

3

to obtain the solu-

tion
t
to Equation (6.7).

The discriminant
D
of a cubic polynomial is given by

3

2

.

(6.11)

D

=−

4

p

−

27

q

By setting

1

1

r

=−

q

+−

D

3

2

108

1

1

s

=−

q

−−

D

,

(6.12)

3

2

108

we can express the three complex roots of Equation (6.9) as

x rs

x ρr ρ s

x ρ r ρs

=+

=+

=+

1

2

2

2

,

(6.13)

3

where
ρ
is the primitive cube root of unity given by

ρ

=− +

1

2

i

3

. (Note that

2

2

ρ

=− −

1

2

i

3

.)

2

We can simplify our arithmetic significantly by making the substitutions

p

1 1

39 3

1

p

′ ==−

a

2

+

b

q

1

1

′
==

3

q

a

−

ab

+

c

.

(6.14)

2 7

6

2

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