Game Development Reference

In-Depth Information

where
P
is the projection matrix and
M
is the transformation from world space to

camera space. The components of each vertex of the view frustum in clip space

are 1

K
is the one

having component signs that match the signs of the
x
,
y
, and
z
components of

. The vertex producing the greatest dot product with the plane

±

′

K
.

′

The vertex producing the least dot product with

K
is the one having component

′

signs opposite those of the components of

K
. The greatest dot product
max

d

and

′

the least dot product

d

are thus given by

min

′

′

′

′

d

=+++

′

KKKK

max

x

y

z

w

d

=−

K

−

K

′

−

K

′

+

K

′

.

(8.51)

min

x

y

z

w

, then the view frustum lies entirely on

the negative side of the plane
K
. This means that nothing on the
positive
side of

the plane is visible. Similarly, if
min

As shown in Figure 8.12, if

d

≤

0

max

, then the view frustum lies entirely on

the positive side of the plane
K
, and thus nothing on the
negative
side of the

plane is visible. If neither of the conditions

d

≥

0

is satisfied, then

the plane
K
intersects the view frustum, and we cannot cull either halfspace.

d

≤

0

or
min

d

≥

0

max

d

max

K

d

min

Figure 8.12.
Let
max

d
and
mi
d
be the greatest dot product and least dot product of any

frustum vertex with the plane
K
. If
max

d

≤

0

or
min

d

≥

0

, then the view frustum lies com-

pletely on one side of
K
, so the other side is not visible.

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