<s>
In	O
geometry	O
,	O
a	O
polygon	B-General_Concept
P	O
in	O
the	O
plane	O
is	O
called	O
monotone	O
with	O
respect	O
to	O
a	O
straight	O
line	O
L	O
,	O
if	O
every	O
line	O
orthogonal	O
to	O
L	O
intersects	O
the	O
boundary	O
of	O
P	O
at	O
most	O
twice	O
.	O
</s>
<s>
For	O
many	O
practical	O
purposes	O
this	O
definition	O
may	O
be	O
extended	O
to	O
allow	O
cases	O
when	O
some	O
edges	O
of	O
P	O
are	O
orthogonal	O
to	O
L	O
,	O
and	O
a	O
simple	O
polygon	B-General_Concept
may	O
be	O
called	O
monotone	O
if	O
a	O
line	O
segment	O
that	O
connects	O
two	O
points	O
in	O
P	O
and	O
is	O
orthogonal	O
to	O
L	O
lies	O
completely	O
in	O
P	O
.	O
</s>
<s>
Following	O
the	O
terminology	O
for	O
monotone	O
functions	O
,	O
the	O
former	O
definition	O
describes	O
polygons	B-General_Concept
strictly	O
monotone	O
with	O
respect	O
to	O
L	O
.	O
</s>
<s>
Then	O
the	O
leftmost	O
and	O
rightmost	O
vertices	O
of	O
a	O
monotone	B-Algorithm
polygon	I-Algorithm
decompose	O
its	O
boundary	O
into	O
two	O
monotone	O
polygonal	O
chains	O
such	O
that	O
when	O
the	O
vertices	O
of	O
any	O
chain	O
are	O
being	O
traversed	O
in	O
their	O
natural	O
order	O
,	O
their	O
X-coordinates	O
are	O
monotonically	O
increasing	O
or	O
decreasing	O
.	O
</s>
<s>
In	O
fact	O
,	O
this	O
property	O
may	O
be	O
taken	O
for	O
the	O
definition	O
of	O
monotone	B-Algorithm
polygon	I-Algorithm
and	O
it	O
gives	O
the	O
polygon	B-General_Concept
its	O
name	O
.	O
</s>
<s>
A	O
convex	O
polygon	B-General_Concept
is	O
monotone	O
with	O
respect	O
to	O
any	O
straight	O
line	O
and	O
a	O
polygon	B-General_Concept
which	O
is	O
monotone	O
with	O
respect	O
to	O
every	O
straight	O
line	O
is	O
convex	O
.	O
</s>
<s>
A	O
linear	O
time	O
algorithm	O
is	O
known	O
to	O
report	O
all	O
directions	O
in	O
which	O
a	O
given	O
simple	O
polygon	B-General_Concept
is	O
monotone	O
.	O
</s>
<s>
It	O
was	O
generalized	O
to	O
report	O
all	O
ways	O
to	O
decompose	O
a	O
simple	O
polygon	B-General_Concept
into	O
two	O
monotone	O
chains	O
(	O
possibly	O
monotone	O
in	O
different	O
directions	O
.	O
)	O
</s>
<s>
Point	O
in	O
polygon	B-General_Concept
queries	O
with	O
respect	O
to	O
a	O
monotone	B-Algorithm
polygon	I-Algorithm
may	O
be	O
answered	O
in	O
logarithmic	O
time	O
after	O
linear	O
time	O
preprocessing	O
(	O
to	O
find	O
the	O
leftmost	O
and	O
rightmost	O
vertices	O
)	O
.	O
</s>
<s>
A	O
monotone	B-Algorithm
polygon	I-Algorithm
may	O
be	O
easily	O
triangulated	B-Algorithm
in	O
linear	O
time	O
.	O
</s>
<s>
For	O
a	O
given	O
set	O
of	O
points	O
in	O
the	O
plane	O
,	O
a	O
bitonic	B-Algorithm
tour	I-Algorithm
is	O
a	O
monotone	B-Algorithm
polygon	I-Algorithm
that	O
connects	O
the	O
points	O
.	O
</s>
<s>
The	O
minimum	O
perimeter	B-Application
bitonic	B-Algorithm
tour	I-Algorithm
for	O
a	O
given	O
point	O
set	O
with	O
respect	O
to	O
a	O
fixed	O
direction	O
may	O
be	O
found	O
in	O
polynomial	O
time	O
using	O
dynamic	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
It	O
is	O
easily	O
shown	O
that	O
such	O
a	O
minimal	O
bitonic	B-Algorithm
tour	I-Algorithm
is	O
a	O
simple	O
polygon	B-General_Concept
:	O
a	O
pair	O
of	O
crossing	O
edges	O
may	O
be	O
replaced	O
with	O
a	O
shorter	O
non-crossing	O
pair	O
while	O
preserving	O
the	O
bitonicity	O
of	O
the	O
new	O
tour	O
.	O
</s>
<s>
A	O
simple	O
polygon	B-General_Concept
may	O
be	O
easily	O
cut	O
into	O
monotone	B-Algorithm
polygons	I-Algorithm
in	O
O(nlogn )	O
time	O
.	O
</s>
<s>
However	O
,	O
since	O
a	O
triangle	O
is	O
a	O
monotone	B-Algorithm
polygon	I-Algorithm
,	O
polygon	B-Algorithm
triangulation	I-Algorithm
is	O
in	O
fact	O
cutting	O
a	O
polygon	B-General_Concept
into	O
monotone	O
ones	O
,	O
and	O
it	O
may	O
be	O
performed	O
for	O
simple	O
polygons	B-General_Concept
in	O
O(n )	O
time	O
with	O
a	O
complex	O
algorithm	O
.	O
</s>
<s>
Cutting	O
a	O
simple	O
polygon	B-General_Concept
into	O
the	O
minimal	O
number	O
of	O
uniformly	O
monotone	B-Algorithm
polygons	I-Algorithm
(	O
i.e.	O
,	O
monotone	O
with	O
respect	O
to	O
the	O
same	O
line	O
)	O
can	O
be	O
performed	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
motion	O
planning	O
,	O
two	O
nonintersecting	O
monotone	B-Algorithm
polygons	I-Algorithm
are	O
separable	O
by	O
a	O
single	O
translation	O
(	O
i.e.	O
,	O
there	O
exists	O
a	O
translation	O
of	O
one	O
polygon	B-General_Concept
such	O
that	O
the	O
two	O
become	O
separated	O
by	O
a	O
straight	O
line	O
into	O
different	O
halfplanes	O
)	O
and	O
this	O
separation	O
may	O
be	O
found	O
in	O
linear	O
time	O
.	O
</s>
<s>
A	O
polygon	B-General_Concept
is	O
called	O
sweepable	O
,	O
if	O
a	O
straight	O
line	O
may	O
be	O
continuously	O
moved	O
over	O
the	O
whole	O
polygon	B-General_Concept
in	O
such	O
a	O
way	O
that	O
at	O
any	O
moment	O
its	O
intersection	O
with	O
the	O
polygonal	O
area	O
is	O
a	O
convex	O
set	O
.	O
</s>
<s>
A	O
monotone	B-Algorithm
polygon	I-Algorithm
is	O
sweepable	O
by	O
a	O
line	O
which	O
does	O
not	O
change	O
its	O
orientation	O
during	O
the	O
sweep	O
.	O
</s>
<s>
A	O
polygon	B-General_Concept
is	O
strictly	O
sweepable	O
if	O
no	O
portion	O
of	O
its	O
area	O
is	O
swept	O
more	O
than	O
once	O
.	O
</s>
<s>
There	O
is	O
no	O
single	O
straightforward	O
generalization	O
of	O
polygon	B-General_Concept
monotonicity	O
to	O
higher	O
dimensions	O
.	O
</s>
<s>
In	O
one	O
approach	O
the	O
preserved	O
monotonicity	O
trait	O
is	O
the	O
line	O
L	O
.	O
A	O
three-dimensional	O
polyhedron	O
is	O
called	O
weakly	O
monotonic	O
in	O
direction	O
L	O
if	O
all	O
cross-sections	O
orthogonal	O
to	O
L	O
are	O
simple	O
polygons	B-General_Concept
.	O
</s>
