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