<s>
In	O
computational	O
geometry	O
,	O
polygon	B-Algorithm
triangulation	I-Algorithm
is	O
the	O
partition	O
of	O
a	O
polygonal	O
area	O
(	O
simple	O
polygon	B-General_Concept
)	O
into	O
a	O
set	O
of	O
triangles	O
,	O
i.e.	O
,	O
finding	O
a	O
set	O
of	O
triangles	O
with	O
pairwise	O
non-intersecting	O
interiors	O
whose	O
union	O
is	O
.	O
</s>
<s>
Triangulations	B-Algorithm
may	O
be	O
viewed	O
as	O
special	O
cases	O
of	O
planar	O
straight-line	O
graphs	O
.	O
</s>
<s>
When	O
there	O
are	O
no	O
holes	O
or	O
added	O
points	O
,	O
triangulations	B-Algorithm
form	O
maximal	O
outerplanar	O
graphs	O
.	O
</s>
<s>
Over	O
time	O
,	O
a	O
number	O
of	O
algorithms	O
have	O
been	O
proposed	O
to	O
triangulate	O
a	O
polygon	B-General_Concept
.	O
</s>
<s>
It	O
is	O
trivial	O
to	O
triangulate	O
any	O
convex	O
polygon	B-General_Concept
in	O
linear	O
time	O
into	O
a	O
fan	B-Algorithm
triangulation	I-Algorithm
,	O
by	O
adding	O
diagonals	O
from	O
one	O
vertex	O
to	O
all	O
other	O
non-nearest	O
neighbor	O
vertices	O
.	O
</s>
<s>
A	O
monotone	B-Algorithm
polygon	I-Algorithm
can	O
be	O
triangulated	O
in	O
linear	O
time	O
with	O
either	O
the	O
algorithm	O
of	O
A	O
.	O
Fournier	O
and	O
D.Y.	O
</s>
<s>
One	O
way	O
to	O
triangulate	O
a	O
simple	O
polygon	B-General_Concept
is	O
based	O
on	O
the	O
two	O
ears	O
theorem	O
,	O
as	O
the	O
fact	O
that	O
any	O
simple	O
polygon	B-General_Concept
with	O
at	O
least	O
4	O
vertices	O
without	O
holes	O
has	O
at	O
least	O
two	O
'	O
ears	O
 '	O
,	O
which	O
are	O
triangles	O
with	O
two	O
sides	O
being	O
the	O
edges	O
of	O
the	O
polygon	B-General_Concept
and	O
the	O
third	O
one	O
completely	O
inside	O
it	O
.	O
</s>
<s>
The	O
algorithm	O
then	O
consists	O
of	O
finding	O
such	O
an	O
ear	O
,	O
removing	O
it	O
from	O
the	O
polygon	B-General_Concept
(	O
which	O
results	O
in	O
a	O
new	O
polygon	B-General_Concept
that	O
still	O
meets	O
the	O
conditions	O
)	O
and	O
repeating	O
until	O
there	O
is	O
only	O
one	O
triangle	O
left	O
.	O
</s>
<s>
This	O
algorithm	O
is	O
easy	O
to	O
implement	O
,	O
but	O
slower	O
than	O
some	O
other	O
algorithms	O
,	O
and	O
it	O
only	O
works	O
on	O
polygons	B-General_Concept
without	O
holes	O
.	O
</s>
<s>
This	O
method	O
is	O
known	O
as	O
ear	B-Algorithm
clipping	I-Algorithm
and	O
sometimes	O
ear	O
trimming	O
.	O
</s>
<s>
A	O
simple	O
polygon	B-General_Concept
is	O
monotone	O
with	O
respect	O
to	O
a	O
line	O
,	O
if	O
any	O
line	O
orthogonal	O
to	O
intersects	O
the	O
polygon	B-General_Concept
at	O
most	O
twice	O
.	O
</s>
<s>
A	O
monotone	B-Algorithm
polygon	I-Algorithm
can	O
be	O
split	O
into	O
two	O
monotone	O
chains	O
.	O
</s>
<s>
A	O
polygon	B-General_Concept
that	O
is	O
monotone	O
with	O
respect	O
to	O
the	O
y-axis	O
is	O
called	O
y-monotone	O
.	O
</s>
<s>
A	O
monotone	B-Algorithm
polygon	I-Algorithm
with	O
vertices	O
can	O
be	O
triangulated	O
in	O
time	O
.	O
</s>
<s>
Assuming	O
a	O
given	O
polygon	B-General_Concept
is	O
y-monotone	O
,	O
the	O
greedy	B-Algorithm
algorithm	I-Algorithm
begins	O
by	O
walking	O
on	O
one	O
chain	O
of	O
the	O
polygon	B-General_Concept
from	O
top	O
to	O
bottom	O
while	O
adding	O
diagonals	O
whenever	O
it	O
is	O
possible	O
.	O
</s>
<s>
It	O
is	O
easy	O
to	O
see	O
that	O
the	O
algorithm	O
can	O
be	O
applied	O
to	O
any	O
monotone	B-Algorithm
polygon	I-Algorithm
.	O
</s>
<s>
If	O
a	O
polygon	B-General_Concept
is	O
not	O
monotone	O
,	O
it	O
can	O
be	O
partitioned	O
into	O
monotone	O
subpolygons	O
in	O
time	O
using	O
a	O
sweep-line	B-Algorithm
approach	I-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
does	O
not	O
require	O
the	O
polygon	B-General_Concept
to	O
be	O
simple	O
,	O
thus	O
it	O
can	O
be	O
applied	O
to	O
polygons	B-General_Concept
with	O
holes	O
.	O
</s>
<s>
A	O
useful	O
graph	O
that	O
is	O
often	O
associated	O
with	O
a	O
triangulation	B-Algorithm
of	O
a	O
polygon	B-General_Concept
is	O
the	O
dual	O
graph	O
.	O
</s>
<s>
Given	O
a	O
triangulation	B-Algorithm
of	O
,	O
one	O
defines	O
the	O
graph	O
as	O
the	O
graph	O
whose	O
vertex	O
set	O
are	O
the	O
triangles	O
of	O
,	O
two	O
vertices	O
(	O
triangles	O
)	O
being	O
adjacent	O
if	O
and	O
only	O
if	O
they	O
share	O
a	O
diagonal	O
.	O
</s>
<s>
Until	O
1988	O
,	O
whether	O
a	O
simple	O
polygon	B-General_Concept
can	O
be	O
triangulated	O
faster	O
than	O
time	O
was	O
an	O
open	O
problem	O
in	O
computational	O
geometry	O
.	O
</s>
<s>
Then	O
,	O
discovered	O
an	O
-time	O
algorithm	O
for	O
triangulation	B-Algorithm
,	O
later	O
simplified	O
by	O
.	O
</s>
<s>
Bernard	O
Chazelle	O
showed	O
in	O
1991	O
that	O
any	O
simple	O
polygon	B-General_Concept
can	O
be	O
triangulated	O
in	O
linear	O
time	O
,	O
though	O
the	O
proposed	O
algorithm	O
is	O
very	O
complex	O
.	O
</s>
<s>
Seidel	O
's	O
decomposition	O
algorithm	O
and	O
Chazelle	O
's	O
triangulation	B-Algorithm
method	O
are	O
discussed	O
in	O
detail	O
in	O
.	O
</s>
<s>
The	O
time	O
complexity	O
of	O
triangulation	B-Algorithm
of	O
an	O
-vertex	O
polygon	B-General_Concept
with	O
holes	O
has	O
an	O
lower	O
bound	O
,	O
in	O
algebraic	O
computation	O
tree	O
models	O
of	O
computation	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
compute	O
the	O
number	O
of	O
distinct	O
triangulations	B-Algorithm
of	O
a	O
simple	O
polygon	B-General_Concept
in	O
polynomial	O
time	O
using	O
dynamic	B-Algorithm
programming	I-Algorithm
,	O
and	O
(	O
based	O
on	O
this	O
counting	O
algorithm	O
)	O
to	O
generate	O
uniformly	O
random	O
triangulations	B-Algorithm
in	O
polynomial	O
time	O
.	O
</s>
<s>
However	O
,	O
counting	O
the	O
triangulations	B-Algorithm
of	O
a	O
polygon	B-General_Concept
with	O
holes	O
is	O
#P	O
-complete	O
,	O
making	O
it	O
unlikely	O
that	O
it	O
can	O
be	O
done	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
Both	O
triangulation	B-Algorithm
problems	O
are	O
a	O
special	O
case	O
of	O
triangulation	B-Algorithm
(	O
geometry	O
)	O
and	O
a	O
special	O
case	O
of	O
polygon	B-General_Concept
partition	O
.	O
</s>
<s>
Minimum-weight	O
triangulation	B-Algorithm
is	O
a	O
triangulation	B-Algorithm
in	O
which	O
the	O
goal	O
is	O
to	O
minimize	O
the	O
total	O
edge	O
length	O
.	O
</s>
<s>
A	O
point-set	B-Algorithm
triangulation	I-Algorithm
is	O
a	O
polygon	B-Algorithm
triangulation	I-Algorithm
of	O
the	O
convex	O
hull	O
of	O
a	O
set	O
of	O
points	O
.	O
</s>
<s>
A	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
another	O
way	O
to	O
create	O
a	O
triangulation	B-Algorithm
based	O
on	O
a	O
set	O
of	O
points	O
.	O
</s>
<s>
The	O
associahedron	O
is	O
a	O
polytope	O
whose	O
vertices	O
correspond	O
to	O
the	O
triangulations	B-Algorithm
of	O
a	O
convex	O
polygon	B-General_Concept
.	O
</s>
<s>
Polygon	B-General_Concept
triangle	O
covering	O
,	O
in	O
which	O
the	O
triangles	O
may	O
overlap	O
.	O
</s>
<s>
Tiling	O
by	O
polygons	B-General_Concept
,	O
where	O
the	O
goal	O
is	O
to	O
cover	O
the	O
entire	O
plane	O
with	O
polygons	B-General_Concept
of	O
pre-specified	O
shapes	O
.	O
</s>
