Game Development Reference

In-Depth Information

′

culate the signed distance
D

from the transformed plane to the origin using the

equation

(

(

)

)

′
=−

−

1T

(

)

D

MNMPT

MNMPMNT

⋅

+

(

(

)

)

(

(

)

)

=−

−

1T

T

−

−

1T

T

T

−

1

T

−

1

=−

NM MP NM T

NM T

−

−

1

=−⋅

D

.

(5.23)

Recall from Equation (4.27) that the inverse of the 4

×

4

matrix
F
constructed

from the 3

×

3

matrix
M
and the 3D translation vector
T
is given by

−

1

−

1

MMT

−

−

1

F

=

.

(5.24)

0

1

−

1

We therefore have for the transpose of

F

(

)

−

1T

M

0

(

)

F

−

1T

=

.

(5.25)

−

1

−

MT

1

−

1

The quantity

D

−⋅

NM T
is exactly the dot product between the fourth row of

(

)

−

1T

NNN
. This shows that we may treat planes as

four-dimensional vectors that transform in the same manner as three-dimensional

normal vectors, except that we use the inverse transpose of the 4

F

and the 4D vector

,

,

,

z
D

x

y

×

4

transfor-

mation matrix. Thus, the plane

=

N

,
D

transforms using the 4

×

4

matrix
F
as

L

(

)

−

1T

L

′ =

F

.

(5.26)

5.3 The View Frustum

Figure 5.8 shows the
view frustum
, the volume of space containing everything

that is visible in a three-dimensional scene. The view frustum is shaped like a

pyramid whose apex lies at the camera position. It has this shape because it rep-

resents the exact volume that would be visible to a camera that is looking through

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