<s>
In	O
mathematics	O
and	O
computational	O
geometry	O
,	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
(	O
also	O
known	O
as	O
a	O
Delone	B-Algorithm
triangulation	I-Algorithm
)	O
for	O
a	O
given	O
set	O
of	O
discrete	O
points	O
in	O
a	O
general	O
position	O
is	O
a	O
triangulation	B-Algorithm
such	O
that	O
no	O
point	O
in	O
is	O
inside	O
the	O
circumcircle	O
of	O
any	O
triangle	O
in	O
.	O
</s>
<s>
Delaunay	B-Algorithm
triangulations	I-Algorithm
maximize	O
the	O
minimum	O
of	O
all	O
the	O
angles	O
of	O
the	O
triangles	O
in	O
the	O
triangulation	B-Algorithm
;	O
they	O
tend	O
to	O
avoid	O
sliver	O
triangles	O
.	O
</s>
<s>
The	O
triangulation	B-Algorithm
is	O
named	O
after	O
Boris	O
Delaunay	O
for	O
his	O
work	O
on	O
this	O
topic	O
from	O
1934	O
.	O
</s>
<s>
For	O
a	O
set	O
of	O
points	O
on	O
the	O
same	O
line	O
there	O
is	O
no	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
(	O
the	O
notion	O
of	O
triangulation	B-Algorithm
is	O
degenerate	O
for	O
this	O
case	O
)	O
.	O
</s>
<s>
For	O
four	O
or	O
more	O
points	O
on	O
the	O
same	O
circle	O
(	O
e.g.	O
,	O
the	O
vertices	O
of	O
a	O
rectangle	O
)	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
not	O
unique	O
:	O
each	O
of	O
the	O
two	O
possible	O
triangulations	B-Algorithm
that	O
split	O
the	O
quadrangle	O
into	O
two	O
triangles	O
satisfies	O
the	O
"	O
Delaunay	O
condition	O
"	O
,	O
i.e.	O
,	O
the	O
requirement	O
that	O
the	O
circumcircles	O
of	O
all	O
triangles	O
have	O
empty	O
interiors	O
.	O
</s>
<s>
By	O
considering	O
circumscribed	O
spheres	O
,	O
the	O
notion	O
of	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
extends	O
to	O
three	O
and	O
higher	O
dimensions	O
.	O
</s>
<s>
However	O
,	O
in	O
these	O
cases	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
not	O
guaranteed	O
to	O
exist	O
or	O
be	O
unique	O
.	O
</s>
<s>
The	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
a	O
discrete	O
point	O
set	O
in	O
general	O
position	O
corresponds	O
to	O
the	O
dual	O
graph	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
for	O
.	O
</s>
<s>
The	O
circumcenters	O
of	O
Delaunay	O
triangles	O
are	O
the	O
vertices	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
In	O
the	O
2D	O
case	O
,	O
the	O
Voronoi	O
vertices	O
are	O
connected	O
via	O
edges	O
,	O
that	O
can	O
be	O
derived	O
from	O
adjacency-relationships	O
of	O
the	O
Delaunay	O
triangles	O
:	O
If	O
two	O
triangles	O
share	O
an	O
edge	O
in	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
,	O
their	O
circumcenters	O
are	O
to	O
be	O
connected	O
with	O
an	O
edge	O
in	O
the	O
Voronoi	B-Architecture
tesselation	I-Architecture
.	O
</s>
<s>
Three	O
or	O
more	O
collinear	O
points	O
,	O
where	O
the	O
circumcircles	O
are	O
of	O
infinite	O
radii	B-Protocol
.	O
</s>
<s>
Four	O
or	O
more	O
points	O
on	O
a	O
perfect	O
circle	O
,	O
where	O
the	O
triangulation	B-Algorithm
is	O
ambiguous	O
and	O
all	O
circumcenters	O
are	O
trivially	O
identical	O
.	O
</s>
<s>
Edges	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
going	O
to	O
infinity	O
are	O
not	O
defined	O
by	O
this	O
relation	O
in	O
case	O
of	O
a	O
finite	O
set	O
.	O
</s>
<s>
If	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
calculated	O
using	O
the	O
Bowyer	B-Algorithm
–	I-Algorithm
Watson	I-Algorithm
algorithm	I-Algorithm
then	O
the	O
circumcenters	O
of	O
triangles	O
having	O
a	O
common	O
vertex	B-Algorithm
with	O
the	O
"	O
super	O
"	O
triangle	O
should	O
be	O
ignored	O
.	O
</s>
<s>
For	O
a	O
set	O
of	O
points	O
in	O
the	O
(	O
-dimensional	O
)	O
Euclidean	O
space	O
,	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
a	O
triangulation	B-Algorithm
such	O
that	O
no	O
point	O
in	O
is	O
inside	O
the	O
circum-hypersphere	O
of	O
any	O
-simplex	O
in	O
.	O
</s>
<s>
It	O
is	O
known	O
that	O
there	O
exists	O
a	O
unique	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
for	O
if	O
is	O
a	O
set	O
of	O
points	O
in	O
general	O
position	O
;	O
that	O
is	O
,	O
the	O
affine	O
hull	O
of	O
is	O
-dimensional	O
and	O
no	O
set	O
of	O
points	O
in	O
lie	O
on	O
the	O
boundary	O
of	O
a	O
ball	O
whose	O
interior	O
does	O
not	O
intersect	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
a	O
set	O
of	O
points	O
in	O
-dimensional	O
Euclidean	O
space	O
can	O
be	O
converted	O
to	O
the	O
problem	O
of	O
finding	O
the	O
convex	O
hull	O
of	O
a	O
set	O
of	O
points	O
in	O
(	O
)	O
-dimensional	O
space	O
.	O
</s>
<s>
As	O
the	O
convex	O
hull	O
is	O
unique	O
,	O
so	O
is	O
the	O
triangulation	B-Algorithm
,	O
assuming	O
all	O
facets	O
of	O
the	O
convex	O
hull	O
are	O
simplices	O
.	O
</s>
<s>
The	O
union	O
of	O
all	O
simplices	O
in	O
the	O
triangulation	B-Algorithm
is	O
the	O
convex	O
hull	O
of	O
the	O
points	O
.	O
</s>
<s>
The	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
contains	O
simplices	O
.	O
</s>
<s>
In	O
the	O
plane	O
(	O
)	O
,	O
if	O
there	O
are	O
vertices	O
on	O
the	O
convex	O
hull	O
,	O
then	O
any	O
triangulation	B-Algorithm
of	O
the	O
points	O
has	O
at	O
most	O
triangles	O
,	O
plus	O
one	O
exterior	O
face	O
(	O
see	O
Euler	O
characteristic	O
)	O
.	O
</s>
<s>
If	O
points	O
are	O
distributed	O
according	O
to	O
a	O
Poisson	O
process	O
in	O
the	O
plane	O
with	O
constant	O
intensity	O
,	O
then	O
each	O
vertex	B-Algorithm
has	O
on	O
average	O
six	O
surrounding	O
triangles	O
.	O
</s>
<s>
In	O
the	O
plane	O
,	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
maximizes	O
the	O
minimum	O
angle	O
.	O
</s>
<s>
Compared	O
to	O
any	O
other	O
triangulation	B-Algorithm
of	O
the	O
points	O
,	O
the	O
smallest	O
angle	O
in	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
at	O
least	O
as	O
large	O
as	O
the	O
smallest	O
angle	O
in	O
any	O
other	O
.	O
</s>
<s>
However	O
,	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
does	O
not	O
necessarily	O
minimize	O
the	O
maximum	O
angle	O
.	O
</s>
<s>
The	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
also	O
does	O
not	O
necessarily	O
minimize	O
the	O
length	O
of	O
the	O
edges	O
.	O
</s>
<s>
If	O
a	O
circle	O
passing	O
through	O
two	O
of	O
the	O
input	O
points	O
does	O
n't	O
contain	O
any	O
other	O
input	O
points	O
in	O
its	O
interior	O
,	O
then	O
the	O
segment	O
connecting	O
the	O
two	O
points	O
is	O
an	O
edge	O
of	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
the	O
given	O
points	O
.	O
</s>
<s>
Each	O
triangle	O
of	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
a	O
set	O
of	O
points	O
in	O
-dimensional	O
spaces	O
corresponds	O
to	O
a	O
facet	O
of	O
convex	O
hull	O
of	O
the	O
projection	O
of	O
the	O
points	O
onto	O
a	O
(	O
)	O
-dimensional	O
paraboloid	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
The	O
closest	O
neighbor	O
to	O
any	O
point	O
is	O
on	O
an	O
edge	O
in	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
since	O
the	O
nearest	O
neighbor	O
graph	O
is	O
a	O
subgraph	O
of	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
.	O
</s>
<s>
The	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
a	O
geometric	B-Algorithm
spanner	I-Algorithm
:	O
In	O
the	O
plane	O
(	O
)	O
,	O
the	O
shortest	O
path	O
between	O
two	O
vertices	O
,	O
along	O
Delaunay	O
edges	O
,	O
is	O
known	O
to	O
be	O
no	O
longer	O
than	O
1.998	O
times	O
the	O
Euclidean	O
distance	O
between	O
them	O
.	O
</s>
<s>
Many	O
algorithms	O
for	O
computing	O
Delaunay	B-Algorithm
triangulations	I-Algorithm
rely	O
on	O
fast	O
operations	O
for	O
detecting	O
when	O
a	O
point	O
is	O
within	O
a	O
triangle	O
's	O
circumcircle	O
and	O
an	O
efficient	O
data	O
structure	O
for	O
storing	O
triangles	O
and	O
edges	O
.	O
</s>
<s>
This	O
leads	O
to	O
a	O
straightforward	O
algorithm	O
:	O
construct	O
any	O
triangulation	B-Algorithm
of	O
the	O
points	O
,	O
and	O
then	O
flip	O
edges	O
until	O
no	O
triangle	O
is	O
non-Delaunay	O
.	O
</s>
<s>
The	O
most	O
straightforward	O
way	O
of	O
efficiently	O
computing	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
to	O
repeatedly	O
add	O
one	O
vertex	B-Algorithm
at	O
a	O
time	O
,	O
retriangulating	O
the	O
affected	O
parts	O
of	O
the	O
graph	O
.	O
</s>
<s>
When	O
a	O
vertex	B-Algorithm
is	O
added	O
,	O
we	O
split	O
in	O
three	O
the	O
triangle	O
that	O
contains	O
,	O
then	O
we	O
apply	O
the	O
flip	O
algorithm	O
.	O
</s>
<s>
While	O
the	O
technique	O
extends	O
to	O
higher	O
dimension	O
(	O
as	O
proved	O
by	O
Edelsbrunner	O
and	O
Shah	O
)	O
,	O
the	O
runtime	O
can	O
be	O
exponential	O
in	O
the	O
dimension	O
even	O
if	O
the	O
final	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
small	O
.	O
</s>
<s>
The	O
Bowyer	B-Algorithm
–	I-Algorithm
Watson	I-Algorithm
algorithm	I-Algorithm
provides	O
another	O
approach	O
for	O
incremental	O
construction	O
.	O
</s>
<s>
It	O
gives	O
an	O
alternative	O
to	O
edge	O
flipping	O
for	O
computing	O
the	O
Delaunay	O
triangles	O
containing	O
a	O
newly	O
inserted	O
vertex	B-Algorithm
.	O
</s>
<s>
proposed	O
another	O
version	O
of	O
incremental	O
algorithm	O
based	O
on	O
rip-and-tent	O
,	O
which	O
is	O
practical	O
and	O
highly	O
parallelized	O
with	O
polylogarithmic	O
span	B-Operating_System
.	O
</s>
<s>
A	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
algorithm	I-Algorithm
for	O
triangulations	B-Algorithm
in	O
two	O
dimensions	O
was	O
developed	O
by	O
Lee	O
and	O
Schachter	O
and	O
improved	O
by	O
Guibas	O
and	O
Stolfi	O
and	O
later	O
by	O
Dwyer	O
.	O
</s>
<s>
The	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
computed	O
for	O
each	O
set	O
,	O
and	O
then	O
the	O
two	O
sets	O
are	O
merged	O
along	O
the	O
splitting	O
line	O
.	O
</s>
<s>
A	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
paradigm	I-Algorithm
to	O
performing	O
a	O
triangulation	B-Algorithm
in	O
dimensions	O
is	O
presented	O
in	O
"	O
DeWall	O
:	O
A	O
fast	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
Delaunay	B-Algorithm
triangulation	I-Algorithm
algorithm	O
in	O
Ed	O
"	O
by	O
P	O
.	O
Cignoni	O
,	O
C	O
.	O
Montani	O
,	O
R	O
.	O
Scopigno	O
.	O
</s>
<s>
The	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
algorithm	I-Algorithm
has	O
been	O
shown	O
to	O
be	O
the	O
fastest	O
DT	O
generation	O
technique	O
sequentially	O
.	O
</s>
<s>
Sweephull	O
is	O
a	O
hybrid	O
technique	O
for	O
2D	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
that	O
uses	O
a	O
radially	O
propagating	O
sweep-hull	O
,	O
and	O
a	O
flipping	O
algorithm	O
.	O
</s>
<s>
The	O
sweep-hull	O
is	O
created	O
sequentially	O
by	O
iterating	O
a	O
radially-sorted	O
set	O
of	O
2D	O
points	O
,	O
and	O
connecting	O
triangles	O
to	O
the	O
visible	O
part	O
of	O
the	O
convex	O
hull	O
,	O
which	O
gives	O
a	O
non-overlapping	O
triangulation	B-Algorithm
.	O
</s>
<s>
The	O
Euclidean	O
minimum	O
spanning	O
tree	O
of	O
a	O
set	O
of	O
points	O
is	O
a	O
subset	O
of	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
the	O
same	O
points	O
,	O
and	O
this	O
can	O
be	O
exploited	O
to	O
compute	O
it	O
efficiently	O
.	O
</s>
<s>
For	O
modelling	O
terrain	O
or	O
other	O
objects	O
given	O
a	O
point	B-Algorithm
cloud	I-Algorithm
,	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
gives	O
a	O
nice	O
set	O
of	O
triangles	O
to	O
use	O
as	O
polygons	O
in	O
the	O
model	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
avoids	O
narrow	O
triangles	O
(	O
as	O
they	O
have	O
large	O
circumcircles	O
compared	O
to	O
their	O
area	O
)	O
.	O
</s>
<s>
See	O
triangulated	B-Algorithm
irregular	I-Algorithm
network	I-Algorithm
.	O
</s>
<s>
Delaunay	B-Algorithm
triangulations	I-Algorithm
can	O
be	O
used	O
to	O
determine	O
the	O
density	O
or	O
intensity	O
of	O
points	O
samplings	O
by	O
means	O
of	O
the	O
Delaunay	B-Algorithm
tessellation	I-Algorithm
field	I-Algorithm
estimator	I-Algorithm
(	O
DTFE	B-Algorithm
)	O
.	O
</s>
<s>
Delaunay	B-Algorithm
triangulations	I-Algorithm
are	O
often	O
used	O
to	O
generate	B-Architecture
meshes	I-Architecture
for	O
space-discretised	O
solvers	O
such	O
as	O
the	O
finite	B-Application
element	I-Application
method	I-Application
and	O
the	O
finite	B-Algorithm
volume	I-Algorithm
method	I-Algorithm
of	O
physics	O
simulation	O
,	O
because	O
of	O
the	O
angle	O
guarantee	O
and	O
because	O
fast	O
triangulation	B-Algorithm
algorithms	O
have	O
been	O
developed	O
.	O
</s>
<s>
The	O
increasing	O
popularity	O
of	O
finite	B-Application
element	I-Application
method	I-Application
and	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
techniques	O
increases	O
the	O
incentive	O
to	O
improve	O
automatic	O
meshing	O
algorithms	O
.	O
</s>
<s>
The	O
stretched	B-Algorithm
grid	I-Algorithm
method	I-Algorithm
allows	O
the	O
generation	O
of	O
pseudo-regular	O
meshes	O
that	O
meet	O
the	O
Delaunay	O
criteria	O
easily	O
and	O
quickly	O
in	O
a	O
one-step	O
solution	O
.	O
</s>
<s>
Constrained	B-Algorithm
Delaunay	I-Algorithm
triangulation	I-Algorithm
has	O
found	O
applications	O
in	O
path	O
planning	O
in	O
automated	O
driving	O
and	O
topographic	O
surveying	O
.	O
</s>
