Game Development Reference

In-Depth Information

Since it passes through the origin in camera space, the new plane has a
w
coordi-

nate of 0.

The situation demonstrated in Figure 8.16 can be avoided by constructing an

extra plane whenever two adjacent frustum planes having normals

N
and

N
sat-

isfy the condition

N N
, where
α
represents an acuteness threshold. The

extra planes do not actually contribute to the shape of the view frustum since they

are coincident with the lines at which previously existing planes intersect. They

should be used only for visibility testing within a single zone and should not par-

ticipate in the clipping of any portals leading to other zones.

⋅

2
α

<

1

Chapter 8 Summary

Principal Components

The principal axes
R
,
S
, and
T
of a set of
N
vertices
12

PP

,

,

are given by the

,

N

eigenvectors of the covariance matrix
C
defined by

1

N

(

)(

)

T

C

=

PmPm
,

−

−

i

i

N

i

=

1

where the mean position
m
is given by

1

N

m

=

P
.

i

N

i

=

1

If
λ
,
λ
, and
λ
are the eigenvalues corresponding to the vectors
R
,
S
, and
T
, re-

spectively, then
1

λλλ

≥≥

.

2

3

Bounding Boxes

The two planes perpendicular to the principal axis
A
that bound the set of verti-

ces
12

PP

,

,

are given by

,

N

{

}

{

}

A

,min

−

P

⋅

A

−

A

,max

P

⋅

A

.

i

i

1

≤≤

iN

1

≤≤

iN

The center
Q
of a bounding box is given by

=

+

+

Q

k

AAA
,

k

k

11

2 2

3 3

where

min

PA

⋅

+

max

PA

⋅

{

}

{

}

i

j

i

j

k

=

1

≤≤

iN

1

≤≤

iN

,

j

2

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