<s>
In	O
geometry	O
,	O
a	O
triangulation	B-Algorithm
is	O
a	O
subdivision	O
of	O
a	O
planar	O
object	O
into	O
triangles	O
,	O
and	O
by	O
extension	O
the	O
subdivision	O
of	O
a	O
higher-dimension	O
geometric	O
object	O
into	O
simplices	O
.	O
</s>
<s>
Triangulations	B-Algorithm
of	O
a	O
three-dimensional	O
volume	O
would	O
involve	O
subdividing	O
it	O
into	O
tetrahedra	O
packed	O
together	O
.	O
</s>
<s>
In	O
most	O
instances	O
,	O
the	O
triangles	O
of	O
a	O
triangulation	B-Algorithm
are	O
required	O
to	O
meet	O
edge-to-edge	O
and	O
vertex-to-vertex	O
.	O
</s>
<s>
Different	O
types	O
of	O
triangulations	B-Algorithm
may	O
be	O
defined	O
,	O
depending	O
both	O
on	O
what	O
geometric	O
object	O
is	O
to	O
be	O
subdivided	O
and	O
on	O
how	O
the	O
subdivision	O
is	O
determined	O
.	O
</s>
<s>
A	O
triangulation	B-Algorithm
of	O
is	O
a	O
subdivision	O
of	O
into	O
-dimensional	O
simplices	O
such	O
that	O
any	O
two	O
simplices	O
in	O
intersect	O
in	O
a	O
common	O
face	O
(	O
a	O
simplex	O
of	O
any	O
lower	O
dimension	O
)	O
or	O
not	O
at	O
all	O
,	O
and	O
any	O
bounded	B-Algorithm
set	I-Algorithm
in	O
intersects	O
only	O
finitely	O
many	O
simplices	O
in	O
.	O
</s>
<s>
A	O
point-set	B-Algorithm
triangulation	I-Algorithm
,	O
i.e.	O
,	O
a	O
triangulation	B-Algorithm
of	O
a	O
discrete	O
set	O
of	O
points	O
,	O
is	O
a	O
subdivision	O
of	O
the	O
convex	O
hull	O
of	O
the	O
points	O
into	O
simplices	O
such	O
that	O
any	O
two	O
simplices	O
intersect	O
in	O
a	O
common	O
face	O
of	O
any	O
dimension	O
or	O
not	O
at	O
all	O
and	O
such	O
that	O
the	O
set	O
of	O
vertices	O
of	O
the	O
simplices	O
are	O
contained	O
in	O
.	O
</s>
<s>
Frequently	O
used	O
and	O
studied	O
point	B-Algorithm
set	I-Algorithm
triangulations	I-Algorithm
include	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
(	O
for	O
points	O
in	O
general	O
position	O
,	O
the	O
set	O
of	O
simplices	O
that	O
are	O
circumscribed	O
by	O
an	O
open	O
ball	O
that	O
contains	O
no	O
input	O
points	O
)	O
and	O
the	O
minimum-weight	O
triangulation	B-Algorithm
(	O
the	O
point	B-Algorithm
set	I-Algorithm
triangulation	I-Algorithm
minimizing	O
the	O
sum	O
of	O
the	O
edge	O
lengths	O
)	O
.	O
</s>
<s>
In	O
cartography	O
,	O
a	O
triangulated	B-Algorithm
irregular	I-Algorithm
network	I-Algorithm
is	O
a	O
point	B-Algorithm
set	I-Algorithm
triangulation	I-Algorithm
of	O
a	O
set	O
of	O
two-dimensional	O
points	O
together	O
with	O
elevations	O
for	O
each	O
point	O
.	O
</s>
<s>
Lifting	O
each	O
point	O
from	O
the	O
plane	O
to	O
its	O
elevated	O
height	O
lifts	O
the	O
triangles	O
of	O
the	O
triangulation	B-Algorithm
into	O
three-dimensional	O
surfaces	O
,	O
which	O
form	O
an	O
approximation	O
of	O
a	O
three-dimensional	O
landform	O
.	O
</s>
<s>
A	O
polygon	B-Algorithm
triangulation	I-Algorithm
is	O
a	O
subdivision	O
of	O
a	O
given	O
polygon	B-General_Concept
into	O
triangles	O
meeting	O
edge-to-edge	O
,	O
again	O
with	O
the	O
property	O
that	O
the	O
set	O
of	O
triangle	O
vertices	O
coincides	O
with	O
the	O
set	O
of	O
vertices	O
of	O
the	O
polygon	B-General_Concept
.	O
</s>
<s>
Polygon	B-Algorithm
triangulations	I-Algorithm
may	O
be	O
found	O
in	O
linear	O
time	O
and	O
form	O
the	O
basis	O
of	O
several	O
important	O
geometric	O
algorithms	O
,	O
including	O
a	O
simple	O
approximate	O
solution	O
to	O
the	O
art	O
gallery	O
problem	O
.	O
</s>
<s>
The	O
constrained	B-Algorithm
Delaunay	I-Algorithm
triangulation	I-Algorithm
is	O
an	O
adaptation	O
of	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
from	O
point	O
sets	O
to	O
polygons	B-General_Concept
or	O
,	O
more	O
generally	O
,	O
to	O
planar	O
straight-line	O
graphs	O
.	O
</s>
<s>
A	O
triangulation	B-Algorithm
of	I-Algorithm
a	I-Algorithm
surface	I-Algorithm
consists	O
of	O
a	O
net	O
of	O
triangles	O
with	O
points	O
on	O
a	O
given	O
surface	O
covering	O
the	O
surface	O
partly	O
or	O
totally	O
.	O
</s>
<s>
In	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
triangulations	B-Algorithm
are	O
often	O
used	O
as	O
the	O
mesh	B-Algorithm
(	O
in	O
this	O
case	O
,	O
a	O
triangle	B-Algorithm
mesh	I-Algorithm
)	O
underlying	O
a	O
computation	O
.	O
</s>
<s>
In	O
order	O
to	O
be	O
suitable	O
as	O
finite	B-Application
element	I-Application
meshes	B-Algorithm
,	O
a	O
triangulation	B-Algorithm
must	O
have	O
well-shaped	O
triangles	O
,	O
according	O
to	O
criteria	O
that	O
depend	O
on	O
the	O
details	O
of	O
the	O
finite	B-Application
element	I-Application
simulation	O
(	O
see	O
mesh	B-Algorithm
quality	O
)	O
;	O
for	O
instance	O
,	O
some	O
methods	O
require	O
that	O
all	O
triangles	O
be	O
right	O
or	O
acute	O
,	O
forming	O
nonobtuse	B-Algorithm
meshes	I-Algorithm
.	O
</s>
<s>
Many	O
meshing	O
techniques	O
are	O
known	O
,	O
including	O
Delaunay	B-Algorithm
refinement	I-Algorithm
algorithms	O
such	O
as	O
Chew	B-Algorithm
's	I-Algorithm
second	I-Algorithm
algorithm	I-Algorithm
and	O
Ruppert	O
's	O
algorithm	O
.	O
</s>
<s>
In	O
more	O
general	O
topological	O
spaces	O
,	O
triangulations	B-Algorithm
of	O
a	O
space	O
generally	O
refer	O
to	O
simplicial	O
complexes	O
that	O
are	O
homeomorphic	O
to	O
the	O
space	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
triangulation	B-Algorithm
may	O
also	O
be	O
generalized	O
somewhat	O
to	O
subdivisions	O
into	O
shapes	O
related	O
to	O
triangles	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
pseudotriangulation	O
of	O
a	O
point	O
set	O
is	O
a	O
partition	O
of	O
the	O
convex	O
hull	O
of	O
the	O
points	O
into	O
pseudotriangles	O
--	O
polygons	B-General_Concept
that	O
,	O
like	O
triangles	O
,	O
have	O
exactly	O
three	O
convex	O
vertices	O
.	O
</s>
<s>
As	O
in	O
point	B-Algorithm
set	I-Algorithm
triangulations	I-Algorithm
,	O
pseudotriangulations	O
are	O
required	O
to	O
have	O
their	O
vertices	O
at	O
the	O
given	O
input	O
points	O
.	O
</s>
