<s>
In	O
computational	O
geometry	O
,	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
is	O
a	O
point	B-Algorithm
set	I-Algorithm
triangulation	I-Algorithm
in	O
which	O
the	O
nearest	B-Algorithm
neighbor	I-Algorithm
of	O
any	O
point	O
p	O
within	O
the	O
triangulation	O
is	O
one	O
of	O
the	O
vertices	O
of	O
the	O
triangle	O
containing	O
p	O
.	O
</s>
<s>
Alternatively	O
,	O
it	O
is	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
in	O
which	O
each	O
internal	O
edge	O
crosses	O
its	O
dual	O
Voronoi	B-Architecture
diagram	I-Architecture
edge	O
.	O
</s>
<s>
Pitteway	B-Algorithm
triangulations	I-Algorithm
are	O
named	O
after	O
Michael	O
Pitteway	O
,	O
who	O
studied	O
them	O
in	O
1973	O
.	O
</s>
<s>
Not	O
every	O
point	O
set	O
supports	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
.	O
</s>
<s>
When	O
such	O
a	O
triangulation	O
exists	O
it	O
is	O
a	O
special	O
case	O
of	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
,	O
and	O
consists	O
of	O
the	O
union	O
of	O
the	O
Gabriel	O
graph	O
and	O
convex	O
hull	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
was	O
introduced	O
by	O
.	O
</s>
<s>
The	O
name	O
"	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
"	O
was	O
given	O
by	O
.	O
</s>
<s>
points	O
out	O
that	O
not	O
every	O
point	O
set	O
supports	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
any	O
triangulation	O
of	O
a	O
regular	O
pentagon	O
includes	O
a	O
central	O
isosceles	O
triangle	O
such	O
that	O
a	O
point	O
p	O
near	O
the	O
midpoint	O
of	O
one	O
of	O
the	O
triangle	O
sides	O
has	O
its	O
nearest	B-Algorithm
neighbor	I-Algorithm
outside	O
the	O
triangle	O
.	O
</s>
<s>
When	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
exists	O
,	O
the	O
midpoint	O
of	O
each	O
edge	O
interior	O
to	O
the	O
triangulation	O
must	O
have	O
the	O
two	O
edge	O
endpoints	O
as	O
its	O
nearest	O
neighbors	O
,	O
for	O
any	O
other	O
neighbor	O
would	O
violate	O
the	O
Pitteway	O
property	O
for	O
nearby	O
points	O
in	O
one	O
of	O
the	O
two	O
adjacent	O
triangles	O
.	O
</s>
<s>
Thus	O
,	O
a	O
circle	O
having	O
that	O
edge	O
as	O
diameter	O
must	O
be	O
empty	O
of	O
vertices	O
,	O
so	O
the	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
consists	O
of	O
the	O
Gabriel	O
graph	O
together	O
with	O
the	O
convex	O
hull	O
of	O
the	O
point	O
set	O
.	O
</s>
<s>
Conversely	O
,	O
when	O
the	O
Gabriel	O
graph	O
and	O
convex	O
hull	O
together	O
form	O
a	O
triangulation	O
,	O
it	O
is	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
.	O
</s>
<s>
Since	O
all	O
Gabriel	O
graph	O
and	O
convex	O
hull	O
edges	O
are	O
part	O
of	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
,	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
,	O
when	O
it	O
exists	O
,	O
is	O
unique	O
for	O
points	O
in	O
general	O
position	O
and	O
coincides	O
with	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
.	O
</s>
<s>
However	O
point	O
sets	O
with	O
no	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
will	O
still	O
have	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
.	O
</s>
<s>
In	O
the	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
,	O
each	O
edge	O
pq	O
either	O
belongs	O
to	O
the	O
convex	O
hull	O
or	O
crosses	O
the	O
edge	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
that	O
separates	O
the	O
cells	O
containing	O
p	O
and	O
q	O
.	O
</s>
<s>
In	O
some	O
references	O
this	O
property	O
is	O
used	O
to	O
define	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
,	O
as	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
in	O
which	O
all	O
internal	O
Delaunay	O
edges	O
cross	O
their	O
dual	O
Voronoi	O
edges	O
.	O
</s>
<s>
However	O
,	O
a	O
Pitteway	B-Algorithm
triangulation	I-Algorithm
may	O
include	O
convex	O
hull	O
edges	O
that	O
do	O
not	O
cross	O
their	O
duals	O
.	O
</s>
