<s>
In	O
computational	O
geometry	O
,	O
the	O
smallest	B-Algorithm
enclosing	I-Algorithm
box	I-Algorithm
problem	O
is	O
that	O
of	O
finding	O
the	O
oriented	O
minimum	O
bounding	O
box	O
enclosing	O
a	O
set	O
of	O
points	O
.	O
</s>
<s>
It	O
is	O
a	O
type	O
of	O
bounding	B-Algorithm
volume	I-Algorithm
.	O
</s>
<s>
"	O
Smallest	O
"	O
may	O
refer	O
to	O
volume	O
,	O
area	O
,	O
perimeter	B-Application
,	O
etc	O
.	O
</s>
<s>
It	O
is	O
sufficient	O
to	O
find	O
the	O
smallest	B-Algorithm
enclosing	I-Algorithm
box	I-Algorithm
for	O
the	O
convex	O
hull	O
of	O
the	O
objects	O
in	O
question	O
.	O
</s>
<s>
It	O
is	O
straightforward	O
to	O
find	O
the	O
smallest	B-Algorithm
enclosing	I-Algorithm
box	I-Algorithm
that	O
has	O
sides	O
parallel	O
to	O
the	O
coordinate	O
axes	O
;	O
the	O
difficult	O
part	O
of	O
the	O
problem	O
is	O
to	O
determine	O
the	O
orientation	O
of	O
the	O
box	O
.	O
</s>
<s>
For	O
the	O
convex	O
polygon	O
,	O
a	O
linear	O
time	O
algorithm	O
for	O
the	O
minimum-area	B-Algorithm
enclosing	I-Algorithm
rectangle	I-Algorithm
is	O
known	O
.	O
</s>
<s>
The	O
same	O
approach	O
is	O
applicable	O
for	O
finding	O
the	O
minimum-perimeter	B-Algorithm
enclosing	I-Algorithm
rectangle	I-Algorithm
.	O
</s>
<s>
Rourke	O
's	O
approach	O
uses	O
a	O
3-dimensional	O
rotating	B-Algorithm
calipers	I-Algorithm
technique	O
,	O
and	O
is	O
based	O
on	O
lemmas	O
characterizing	O
the	O
minimum	B-Algorithm
enclosing	I-Algorithm
box	I-Algorithm
:	O
</s>
<s>
It	O
follows	O
in	O
the	O
most	O
general	O
case	O
where	O
no	O
convex	O
hull	O
vertices	O
lie	O
in	O
edges	O
of	O
the	O
minimal	B-Algorithm
enclosing	I-Algorithm
box	I-Algorithm
,	O
that	O
at	O
least	O
8	O
convex	O
hull	O
points	O
must	O
lie	O
within	O
faces	O
of	O
the	O
box	O
:	O
two	O
endpoints	O
of	O
each	O
of	O
the	O
two	O
edges	O
,	O
and	O
four	O
more	O
points	O
,	O
one	O
for	O
each	O
of	O
the	O
remaining	O
four	O
box	O
faces	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
the	O
convex	O
hull	O
consists	O
of	O
7	O
or	O
fewer	O
vertices	O
,	O
at	O
least	O
one	O
of	O
them	O
must	O
lie	O
within	O
an	O
edge	O
of	O
the	O
hull	O
's	O
minimal	B-Algorithm
enclosing	I-Algorithm
box	I-Algorithm
.	O
</s>
<s>
Then	O
,	O
this	O
initial	O
bounding	O
box	O
is	O
partitioned	O
into	O
a	O
grid	O
of	O
smaller	O
cubes	O
,	O
and	O
grid	O
points	O
near	O
the	O
boundary	O
of	O
the	O
convex	O
hull	O
of	O
the	O
input	O
are	O
used	O
as	O
a	O
coreset	B-Algorithm
,	O
a	O
small	O
set	O
of	O
points	O
whose	O
optimum	O
bounding	O
box	O
approximates	O
the	O
optimum	O
bounding	O
box	O
of	O
the	O
original	O
input	O
.	O
</s>
<s>
Finally	O
,	O
Rourke	O
's	O
algorithm	O
is	O
applied	O
to	O
find	O
the	O
exact	O
optimum	O
bounding	O
box	O
of	O
this	O
coreset	B-Algorithm
.	O
</s>
<s>
The	O
minimal	B-Algorithm
enclosing	I-Algorithm
box	I-Algorithm
of	O
the	O
regular	O
tetrahedron	O
is	O
a	O
cube	O
,	O
with	O
side	O
length	O
1/	O
that	O
of	O
the	O
tetrahedron	O
;	O
for	O
instance	O
,	O
a	O
regular	O
tetrahedron	O
with	O
side	O
length	O
fits	O
into	O
a	O
unit	O
cube	O
,	O
with	O
the	O
tetrahedron	O
's	O
vertices	O
lying	O
at	O
the	O
vertices	O
(	O
0	O
,	O
0	O
,	O
0	O
)	O
,	O
(	O
0	O
,	O
1	O
,	O
1	O
)	O
,	O
(	O
1	O
,	O
0	O
,	O
1	O
)	O
and	O
(	O
1	O
,	O
1	O
,	O
0	O
)	O
of	O
the	O
unit	O
cube	O
.	O
</s>
