<s>
A	O
triangulation	O
of	O
a	O
set	O
of	O
points	O
in	O
the	O
Euclidean	O
space	O
is	O
a	O
simplicial	O
complex	O
that	O
covers	O
the	O
convex	B-Algorithm
hull	I-Algorithm
of	O
,	O
and	O
whose	O
vertices	O
belong	O
to	O
.	O
</s>
<s>
A	O
particularly	O
interesting	O
kind	O
of	O
triangulations	O
are	O
the	O
Delaunay	B-Algorithm
triangulations	I-Algorithm
.	O
</s>
<s>
They	O
are	O
the	O
geometric	O
duals	O
of	O
Voronoi	B-Architecture
diagrams	I-Architecture
.	O
</s>
<s>
The	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
a	O
set	O
of	O
points	O
in	O
the	O
plane	O
contains	O
the	O
Gabriel	O
graph	O
,	O
the	O
nearest	O
neighbor	O
graph	O
and	O
the	O
minimal	O
spanning	O
tree	O
of	O
.	O
</s>
<s>
Some	O
triangulations	O
of	O
a	O
set	O
of	O
points	O
can	O
be	O
obtained	O
by	O
lifting	O
the	O
points	O
of	O
into	O
(	O
which	O
amounts	O
to	O
add	O
a	O
coordinate	O
to	O
each	O
point	O
of	O
)	O
,	O
by	O
computing	O
the	O
convex	B-Algorithm
hull	I-Algorithm
of	O
the	O
lifted	O
set	O
of	O
points	O
,	O
and	O
by	O
projecting	O
the	O
lower	O
faces	O
of	O
this	O
convex	B-Algorithm
hull	I-Algorithm
back	O
on	O
.	O
</s>
<s>
When	O
the	O
points	O
are	O
lifted	O
to	O
the	O
paraboloid	O
of	O
equation	O
,	O
this	O
construction	O
results	O
in	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
.	O
</s>
<s>
Note	O
that	O
,	O
in	O
order	O
for	O
this	O
construction	O
to	O
provide	O
a	O
triangulation	O
,	O
the	O
lower	O
convex	B-Algorithm
hull	I-Algorithm
of	O
the	O
lifted	O
set	O
of	O
points	O
needs	O
to	O
be	O
simplicial	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
Delaunay	B-Algorithm
triangulations	I-Algorithm
,	O
this	O
amounts	O
to	O
require	O
that	O
no	O
points	O
of	O
lie	O
in	O
the	O
same	O
sphere	O
.	O
</s>
<s>
Every	O
triangulation	O
of	O
any	O
set	O
of	O
points	O
in	O
the	O
plane	O
has	O
triangles	O
and	O
edges	O
where	O
is	O
the	O
number	O
of	O
points	O
of	O
in	O
the	O
boundary	O
of	O
the	O
convex	B-Algorithm
hull	I-Algorithm
of	O
.	O
</s>
<s>
Triangle	O
Splitting	O
Algorithm	O
:	O
Find	O
the	O
convex	B-Algorithm
hull	I-Algorithm
of	O
the	O
point	O
set	O
and	O
triangulate	O
this	O
hull	O
as	O
a	O
polygon	O
.	O
</s>
