<s>
In	O
computer	O
graphics	O
and	O
computational	O
geometry	O
,	O
a	O
bounding	B-Algorithm
volume	I-Algorithm
for	O
a	O
set	O
of	O
objects	O
is	O
a	O
closed	O
volume	O
that	O
completely	O
contains	O
the	O
union	O
of	O
the	O
objects	O
in	O
the	O
set	O
.	O
</s>
<s>
Bounding	B-Algorithm
volumes	I-Algorithm
are	O
used	O
to	O
improve	O
the	O
efficiency	O
of	O
geometrical	O
operations	O
by	O
using	O
simple	O
volumes	O
to	O
contain	O
more	O
complex	O
objects	O
.	O
</s>
<s>
A	O
bounding	B-Algorithm
volume	I-Algorithm
for	O
a	O
set	O
of	O
objects	O
is	O
also	O
a	O
bounding	B-Algorithm
volume	I-Algorithm
for	O
the	O
single	O
object	O
consisting	O
of	O
their	O
union	O
,	O
and	O
the	O
other	O
way	O
around	O
.	O
</s>
<s>
Bounding	B-Algorithm
volumes	I-Algorithm
are	O
most	O
often	O
used	O
to	O
accelerate	O
certain	O
kinds	O
of	O
tests	O
.	O
</s>
<s>
In	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
bounding	B-Algorithm
volumes	I-Algorithm
are	O
used	O
in	O
ray-intersection	B-Algorithm
tests	I-Algorithm
,	O
and	O
in	O
many	O
rendering	O
algorithms	O
,	O
they	O
are	O
used	O
for	O
viewing	B-General_Concept
frustum	I-General_Concept
tests	O
.	O
</s>
<s>
If	O
the	O
ray	O
or	O
viewing	B-General_Concept
frustum	I-General_Concept
does	O
not	O
intersect	O
the	O
bounding	B-Algorithm
volume	I-Algorithm
,	O
it	O
cannot	O
intersect	O
the	O
object	O
contained	O
within	O
,	O
allowing	O
trivial	O
rejection	O
.	O
</s>
<s>
Similarly	O
if	O
the	O
frustum	O
contains	O
the	O
entirety	O
of	O
the	O
bounding	B-Algorithm
volume	I-Algorithm
,	O
the	O
contents	O
may	O
be	O
trivially	O
accepted	O
without	O
further	O
tests	O
.	O
</s>
<s>
In	O
collision	O
detection	O
,	O
when	O
two	O
bounding	B-Algorithm
volumes	I-Algorithm
do	O
not	O
intersect	O
,	O
the	O
contained	O
objects	O
cannot	O
collide	O
.	O
</s>
<s>
Testing	O
against	O
a	O
bounding	B-Algorithm
volume	I-Algorithm
is	O
typically	O
much	O
faster	O
than	O
testing	O
against	O
the	O
object	O
itself	O
,	O
because	O
of	O
the	O
bounding	B-Algorithm
volume	I-Algorithm
's	O
simpler	O
geometry	O
.	O
</s>
<s>
This	O
is	O
because	O
an	O
'	O
object	O
 '	O
is	O
typically	O
composed	O
of	O
polygons	B-General_Concept
or	O
data	O
structures	O
that	O
are	O
reduced	O
to	O
polygonal	O
approximations	O
.	O
</s>
<s>
In	O
either	O
case	O
,	O
it	O
is	O
computationally	O
wasteful	O
to	O
test	O
each	O
polygon	B-General_Concept
against	O
the	O
view	B-General_Concept
volume	I-General_Concept
if	O
the	O
object	O
is	O
not	O
visible	O
.	O
</s>
<s>
To	O
obtain	O
bounding	B-Algorithm
volumes	I-Algorithm
of	O
complex	O
objects	O
,	O
a	O
common	O
way	O
is	O
to	O
break	O
the	O
objects/scene	O
down	O
using	O
a	O
scene	B-Data_Structure
graph	I-Data_Structure
or	O
more	O
specifically	O
a	O
bounding	B-Data_Structure
volume	I-Data_Structure
hierarchy	I-Data_Structure
,	O
like	O
e.g.	O
</s>
<s>
In	O
computer	B-Algorithm
stereo	I-Algorithm
vision	I-Algorithm
,	O
a	O
bounding	B-Algorithm
volume	I-Algorithm
reconstructed	O
from	O
silhouettes	O
of	O
an	O
object	O
is	O
known	O
as	O
a	O
"	O
visual	O
hull.	O
"	O
</s>
<s>
The	O
choice	O
of	O
the	O
type	O
of	O
bounding	B-Algorithm
volume	I-Algorithm
for	O
a	O
given	O
application	O
is	O
determined	O
by	O
a	O
variety	O
of	O
factors	O
:	O
the	O
computational	O
cost	O
of	O
computing	O
a	O
bounding	B-Algorithm
volume	I-Algorithm
for	O
an	O
object	O
,	O
the	O
cost	O
of	O
updating	O
it	O
in	O
applications	O
in	O
which	O
the	O
objects	O
can	O
move	O
or	O
change	O
shape	O
or	O
size	O
,	O
the	O
cost	O
of	O
determining	O
intersections	O
,	O
and	O
the	O
desired	O
precision	O
of	O
the	O
intersection	O
test	O
.	O
</s>
<s>
The	O
precision	O
of	O
the	O
intersection	O
test	O
is	O
related	O
to	O
the	O
amount	O
of	O
space	O
within	O
the	O
bounding	B-Algorithm
volume	I-Algorithm
not	O
associated	O
with	O
the	O
bounded	O
object	O
,	O
called	O
void	O
space	O
.	O
</s>
<s>
Sophisticated	O
bounding	B-Algorithm
volumes	I-Algorithm
generally	O
allow	O
for	O
less	O
void	O
space	O
but	O
are	O
more	O
computationally	O
expensive	O
.	O
</s>
<s>
The	O
types	O
treated	O
here	O
all	O
give	O
convex	O
bounding	B-Algorithm
volumes	I-Algorithm
.	O
</s>
<s>
If	O
non-convex	O
bounding	B-Algorithm
volumes	I-Algorithm
are	O
required	O
,	O
an	O
approach	O
is	O
to	O
represent	O
them	O
as	O
a	O
union	O
of	O
a	O
number	O
of	O
convex	O
bounding	B-Algorithm
volumes	I-Algorithm
.	O
</s>
<s>
Unfortunately	O
,	O
intersection	O
tests	O
become	O
quickly	O
more	O
expensive	O
as	O
the	O
bounding	B-Algorithm
boxes	I-Algorithm
become	O
more	O
sophisticated	O
.	O
</s>
<s>
A	O
is	O
a	O
cuboid	B-General_Concept
,	O
or	O
in	O
2-D	O
a	O
rectangle	O
,	O
containing	O
the	O
object	O
.	O
</s>
<s>
In	O
dynamical	B-Algorithm
simulation	I-Algorithm
,	O
bounding	B-Algorithm
boxes	I-Algorithm
are	O
preferred	O
to	O
other	O
shapes	O
of	O
bounding	B-Algorithm
volume	I-Algorithm
such	O
as	O
bounding	B-Algorithm
spheres	I-Algorithm
or	O
cylinders	B-Algorithm
for	O
objects	O
that	O
are	O
roughly	O
cuboid	B-General_Concept
in	O
shape	O
when	O
the	O
intersection	O
test	O
needs	O
to	O
be	O
fairly	O
accurate	O
.	O
</s>
<s>
The	O
benefit	O
is	O
obvious	O
,	O
for	O
example	O
,	O
for	O
objects	O
that	O
rest	O
upon	O
other	O
,	O
such	O
as	O
a	O
car	O
resting	O
on	O
the	O
ground	O
:	O
a	O
bounding	B-Algorithm
sphere	I-Algorithm
would	O
show	O
the	O
car	O
as	O
possibly	O
intersecting	O
with	O
the	O
ground	O
,	O
which	O
then	O
would	O
need	O
to	O
be	O
rejected	O
by	O
a	O
more	O
expensive	O
test	O
of	O
the	O
actual	O
model	O
of	O
the	O
car	O
;	O
a	O
bounding	B-Algorithm
box	I-Algorithm
immediately	O
shows	O
the	O
car	O
as	O
not	O
intersecting	O
with	O
the	O
ground	O
,	O
saving	O
the	O
more	O
expensive	O
test	O
.	O
</s>
<s>
In	O
many	O
applications	O
the	O
bounding	B-Algorithm
box	I-Algorithm
is	O
aligned	O
with	O
the	O
axes	O
of	O
the	O
co-ordinate	O
system	O
,	O
and	O
it	O
is	O
then	O
known	O
as	O
an	O
axis-aligned	B-Algorithm
bounding	I-Algorithm
box	I-Algorithm
(	O
)	O
.	O
</s>
<s>
To	O
distinguish	O
the	O
general	O
case	O
from	O
an	O
AABB	O
,	O
an	O
arbitrary	O
bounding	B-Algorithm
box	I-Algorithm
is	O
sometimes	O
called	O
an	O
oriented	B-Algorithm
bounding	I-Algorithm
box	I-Algorithm
(	O
)	O
,	O
or	O
an	O
when	O
an	O
existing	O
object	O
's	O
local	O
coordinate	O
system	O
is	O
used	O
.	O
</s>
<s>
This	O
holds	O
for	O
arbitrarily	O
rotated	O
capsules	O
,	O
which	O
is	O
why	O
they	O
're	O
more	O
appealing	O
than	O
cylinders	B-Algorithm
in	O
practice	O
.	O
</s>
<s>
Cylinders	B-Algorithm
are	O
appropriate	O
for	O
3-D	O
objects	O
that	O
can	O
only	O
rotate	O
about	O
a	O
vertical	O
axis	O
but	O
not	O
about	O
other	O
axes	O
,	O
and	O
are	O
otherwise	O
constrained	O
to	O
move	O
by	O
translation	O
only	O
.	O
</s>
<s>
Two	O
vertical-axis-aligned	O
cylinders	B-Algorithm
intersect	O
when	O
,	O
simultaneously	O
,	O
their	O
projections	O
on	O
the	O
vertical	O
axis	O
intersect	O
–	O
which	O
are	O
two	O
line	O
segments	O
–	O
as	O
well	O
their	O
projections	O
on	O
the	O
horizontal	O
plane	O
–	O
two	O
circular	O
disks	O
.	O
</s>
<s>
In	O
video	O
games	O
,	O
bounding	B-Algorithm
cylinders	I-Algorithm
are	O
often	O
used	O
as	O
bounding	B-Algorithm
volumes	I-Algorithm
for	O
people	O
standing	O
upright	O
.	O
</s>
<s>
Bounding	B-Algorithm
spheres	I-Algorithm
are	O
represented	O
by	O
centre	O
and	O
radius	O
.	O
</s>
<s>
This	O
makes	O
bounding	B-Algorithm
spheres	I-Algorithm
appropriate	O
for	O
objects	O
that	O
can	O
move	O
in	O
any	O
number	O
of	O
dimensions	O
.	O
</s>
<s>
A	O
bounding	B-Algorithm
box	I-Algorithm
is	O
the	O
intersection	O
of	O
orthogonally	O
oriented	O
bounding	O
slabs	O
.	O
</s>
<s>
A	O
(	O
DOP	O
)	O
generalizes	O
the	O
bounding	B-Algorithm
box	I-Algorithm
.	O
</s>
<s>
A	O
k-DOP	B-Algorithm
is	O
the	O
Boolean	O
intersection	O
of	O
extents	O
along	O
k	O
directions	O
.	O
</s>
<s>
Thus	O
,	O
a	O
k-DOP	B-Algorithm
is	O
the	O
Boolean	O
intersection	O
of	O
k	O
bounding	O
slabs	O
and	O
is	O
a	O
convex	O
polytope	O
containing	O
the	O
object	O
(	O
in	O
2-D	O
a	O
polygon	B-General_Concept
;	O
in	O
3-D	O
a	O
polyhedron	O
)	O
.	O
</s>
<s>
A	O
minimum	B-Algorithm
bounding	I-Algorithm
rectangle	I-Algorithm
or	O
MBR	O
–	O
the	O
least	O
AABB	O
in	O
2-D	O
–	O
is	O
frequently	O
used	O
in	O
the	O
description	O
of	O
geographic	O
(	O
or	O
"	O
geospatial	O
"	O
)	O
data	O
items	O
,	O
serving	O
as	O
a	O
simplified	O
proxy	O
for	O
a	O
dataset	O
's	O
spatial	O
extent	O
(	O
see	O
geospatial	B-General_Concept
metadata	I-General_Concept
)	O
for	O
the	O
purpose	O
of	O
data	O
search	O
(	O
including	O
spatial	O
queries	O
as	O
applicable	O
)	O
and	O
display	O
.	O
</s>
<s>
It	O
is	O
also	O
a	O
basic	O
component	O
of	O
the	O
R-tree	B-Library
method	O
of	O
spatial	O
indexing	O
.	O
</s>
<s>
For	O
some	O
types	O
of	O
bounding	B-Algorithm
volume	I-Algorithm
(	O
OBB	O
and	O
convex	O
polyhedra	O
)	O
,	O
an	O
effective	O
check	O
is	O
that	O
of	O
the	O
separating	O
axis	O
theorem	O
.	O
</s>
<s>
The	O
intersection	O
of	O
two	O
k-DOP	B-Algorithm
'	O
s	O
can	O
be	O
computed	O
very	O
similarly	O
to	O
AABBs	O
:	O
for	O
each	O
orientation	O
,	O
you	O
just	O
check	O
the	O
two	O
corresponding	O
intervals	O
of	O
the	O
two	O
DOP	O
's	O
.	O
</s>
