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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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