Game Development Reference

In-Depth Information

Chapter 6 Summary

Analytic Root Finding

Solutions to the quadratic equation

at

2

++=

bt

c

0

are given by the quadratic

formula:

2

−±

bb c

−

4

t

=

.

2

a

Cubic and quartic equations can also be solved analytically.

Numerical Root Finding

Roots of a function

()

f x
can be found numerically using Newton's method,

which refines an approximate solution
x
using the formula

()

()

x

f

n

x

=−
′

x

.

n

+

1

n

f

x

n

The refinement formula for the reciprocal
x
of a number
r
is

(

)

x

=

x

2

−

rx

,

n

+

1

n

n

and the refinement formula for the reciprocal square root
x
of a number
r
is

1

(

)

x

=

x

3

−

rx

2

.

n

+

1

n

n

2

Intersection of a Ray and a Sphere

The points where a ray

()

PSV
intersect a sphere of radius
r
are given by

the solutions of the quadratic equation

t

=+

t

22

(

)

2

2

.

Vt

+⋅

2

SV

t S

+−=

r

0

Normal Vector Calculation

The normal vector at a point

x yz
on a surface defined by the function

,,

(

)

(

)

xyz

,,

=

0

is given by

=

∇

xyz

,,

.

f

N

f

Reflection Vector Calculation

The reflection
R
of a vector
L
across the normal vector
N
is given by

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