Game Development Reference

In-Depth Information

The effective radius
ef
r
with respect to a plane having normal direction
N
of a

bounding ellipsoid whose semiaxis lengths and orientations are described by the

vectors
R
,
S
, and
T
is given by

(

)

2

(

)

2

(

)

2

r

=

RN

⋅

+

SN

⋅

+

TN
.

⋅

eff

A bounding ellipsoid having center
Q
is not visible if for any view frustum plane

L
we have

L Q

⋅

≤−

r

.

eff

Bounding Cylinders

A bounding cylinder for the set of vertices
12

PP

,

,

is constructed by first

,

N

calculating the points

H

using the formula

{

}

i

(

)

HP PRR
,

=− ⋅

i

i

i

where
R
is the unit vector parallel to the primary axis. After finding a bounding

circle for the points

H

having center
Q
and radius
r
, the endpoints

Q
and

Q

{

}

i

of the bounding cylinder are given by

QQ

=+

min

PRR

⋅

{

}

1

i

1

≤≤

iN

{

}

Q

=+

Q

max

P

⋅

R R
.

2

i

1

≤≤

iN

The effective radius
ef
r
with respect to a plane having normal direction
N
of a

bounding cylinder is given by

(

AN
,

)

2

r

=−⋅

r

1

eff

where
A
is the unit vector parallel to the axis of the cylinder given by

QQ

−

2

1

A

=

QQ
.

−

2

1

A bounding cylinder is not visible if the line segment connecting the endpoints

Q
and

Q
is completely clipped away by the view frustum planes.

Binary Space Partitioning (BSP) Trees

We can determine whether a world-space plane
K
intersects the view frustum by

transforming the plane into homogeneous clip space using the formula

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