<s>
In	O
mathematics	O
,	O
a	O
Voronoi	B-Architecture
diagram	I-Architecture
is	O
a	O
partition	O
of	O
a	O
plane	O
into	O
regions	O
close	O
to	O
each	O
of	O
a	O
given	O
set	O
of	O
objects	O
.	O
</s>
<s>
For	O
each	O
seed	O
there	O
is	O
a	O
corresponding	O
region	O
,	O
called	O
a	O
Voronoi	B-Architecture
cell	I-Architecture
,	O
consisting	O
of	O
all	O
points	O
of	O
the	O
plane	O
closer	O
to	O
that	O
seed	O
than	O
to	O
any	O
other	O
.	O
</s>
<s>
The	O
Voronoi	B-Architecture
diagram	I-Architecture
of	O
a	O
set	O
of	O
points	O
is	O
dual	O
to	O
that	O
set	O
's	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
.	O
</s>
<s>
The	O
Voronoi	B-Architecture
diagram	I-Architecture
is	O
named	O
after	O
mathematician	O
Georgy	O
Voronoy	O
,	O
and	O
is	O
also	O
called	O
a	O
Voronoi	B-Architecture
tessellation	I-Architecture
,	O
a	O
Voronoi	B-Architecture
decomposition	I-Architecture
,	O
a	O
Voronoi	B-Architecture
partition	I-Architecture
,	O
or	O
a	O
Dirichlet	B-Architecture
tessellation	I-Architecture
(	O
after	O
Peter	O
Gustav	O
Lejeune	O
Dirichlet	O
)	O
.	O
</s>
<s>
Voronoi	B-Architecture
cells	I-Architecture
are	O
also	O
known	O
as	O
Thiessen	B-Architecture
polygons	I-Architecture
.	O
</s>
<s>
Voronoi	B-Architecture
diagrams	I-Architecture
have	O
practical	O
and	O
theoretical	O
applications	O
in	O
many	O
fields	O
,	O
mainly	O
in	O
science	O
and	O
technology	O
,	O
but	O
also	O
in	O
visual	O
art	O
.	O
</s>
<s>
In	O
this	O
case	O
each	O
site	O
pk	O
is	O
simply	O
a	O
point	O
,	O
and	O
its	O
corresponding	O
Voronoi	B-Architecture
cell	I-Architecture
Rk	O
consists	O
of	O
every	O
point	O
in	O
the	O
Euclidean	O
plane	O
whose	O
distance	O
to	O
pk	O
is	O
less	O
than	O
or	O
equal	O
to	O
its	O
distance	O
to	O
any	O
other	O
pk	O
.	O
</s>
<s>
The	O
line	O
segments	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
are	O
all	O
the	O
points	O
in	O
the	O
plane	O
that	O
are	O
equidistant	O
to	O
the	O
two	O
nearest	O
sites	O
.	O
</s>
<s>
Let	O
be	O
a	O
set	O
of	O
indices	O
and	O
let	O
be	O
a	O
tuple	B-Application
(	O
indexed	O
collection	O
)	O
of	O
nonempty	O
subsets	O
(	O
the	O
sites	O
)	O
in	O
the	O
space	O
.	O
</s>
<s>
The	O
Voronoi	B-Architecture
cell	I-Architecture
,	O
or	O
Voronoi	O
region	O
,	O
,	O
associated	O
with	O
the	O
site	O
is	O
the	O
set	O
of	O
all	O
points	O
in	O
whose	O
distance	O
to	O
is	O
not	O
greater	O
than	O
their	O
distance	O
to	O
the	O
other	O
sites	O
,	O
where	O
is	O
any	O
index	O
different	O
from	O
.	O
</s>
<s>
The	O
Voronoi	B-Architecture
diagram	I-Architecture
is	O
simply	O
the	O
tuple	B-Application
of	O
cells	O
.	O
</s>
<s>
In	O
the	O
particular	O
case	O
where	O
the	O
space	O
is	O
a	O
finite-dimensional	O
Euclidean	O
space	O
,	O
each	O
site	O
is	O
a	O
point	O
,	O
there	O
are	O
finitely	O
many	O
points	O
and	O
all	O
of	O
them	O
are	O
different	O
,	O
then	O
the	O
Voronoi	B-Architecture
cells	I-Architecture
are	O
convex	O
polytopes	O
and	O
they	O
can	O
be	O
represented	O
in	O
a	O
combinatorial	O
way	O
using	O
their	O
vertices	O
,	O
sides	O
,	O
two-dimensional	O
faces	O
,	O
etc	O
.	O
</s>
<s>
Sometimes	O
the	O
induced	O
combinatorial	O
structure	O
is	O
referred	O
to	O
as	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
In	O
general	O
however	O
,	O
the	O
Voronoi	B-Architecture
cells	I-Architecture
may	O
not	O
be	O
convex	O
or	O
even	O
connected	O
.	O
</s>
<s>
Each	O
Voronoi	B-Architecture
polygon	I-Architecture
is	O
associated	O
with	O
a	O
generator	O
point	O
.	O
</s>
<s>
Then	O
,	O
as	O
expressed	O
by	O
Tran	O
et	O
al	O
,	O
"	O
all	O
locations	O
in	O
the	O
Voronoi	B-Architecture
polygon	I-Architecture
are	O
closer	O
to	O
the	O
generator	O
point	O
of	O
that	O
polygon	O
than	O
any	O
other	O
generator	O
point	O
in	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
in	O
Euclidean	O
plane	O
"	O
.	O
</s>
<s>
In	O
this	O
case	O
the	O
Voronoi	B-Architecture
cell	I-Architecture
of	O
a	O
given	O
shop	O
can	O
be	O
used	O
for	O
giving	O
a	O
rough	O
estimate	O
on	O
the	O
number	O
of	O
potential	O
customers	O
going	O
to	O
this	O
shop	O
(	O
which	O
is	O
modeled	O
by	O
a	O
point	O
in	O
our	O
city	O
)	O
.	O
</s>
<s>
The	O
corresponding	O
Voronoi	B-Architecture
diagrams	I-Architecture
look	O
different	O
for	O
different	O
distance	O
metrics	O
.	O
</s>
<s>
The	O
dual	O
graph	O
for	O
a	O
Voronoi	B-Architecture
diagram	I-Architecture
(	O
in	O
the	O
case	O
of	O
a	O
Euclidean	O
space	O
with	O
point	O
sites	O
)	O
corresponds	O
to	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
for	O
the	O
same	O
set	O
of	O
points	O
.	O
</s>
<s>
The	O
closest	B-Algorithm
pair	I-Algorithm
of	I-Algorithm
points	I-Algorithm
corresponds	O
to	O
two	O
adjacent	O
cells	O
in	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
Then	O
two	O
points	O
of	O
the	O
set	O
are	O
adjacent	O
on	O
the	O
convex	O
hull	O
if	O
and	O
only	O
if	O
their	O
Voronoi	B-Architecture
cells	I-Architecture
share	O
an	O
infinitely	O
long	O
side	O
.	O
</s>
<s>
If	O
the	O
space	O
is	O
a	O
normed	O
space	O
and	O
the	O
distance	O
to	O
each	O
site	O
is	O
attained	O
(	O
e.g.	O
,	O
when	O
a	O
site	O
is	O
a	O
compact	O
set	O
or	O
a	O
closed	O
ball	O
)	O
,	O
then	O
each	O
Voronoi	B-Architecture
cell	I-Architecture
can	O
be	O
represented	O
as	O
a	O
union	O
of	O
line	O
segments	O
emanating	O
from	O
the	O
sites	O
.	O
</s>
<s>
Voronoi	B-Architecture
cells	I-Architecture
enjoy	O
a	O
certain	O
stability	O
property	O
:	O
a	O
small	O
change	O
in	O
the	O
shapes	O
of	O
the	O
sites	O
,	O
e.g.	O
,	O
a	O
change	O
caused	O
by	O
some	O
translation	O
or	O
distortion	O
,	O
yields	O
a	O
small	O
change	O
in	O
the	O
shape	O
of	O
the	O
Voronoi	B-Architecture
cells	I-Architecture
.	O
</s>
<s>
This	O
is	O
the	O
geometric	O
stability	O
of	O
Voronoi	B-Architecture
diagrams	I-Architecture
.	O
</s>
<s>
Informal	O
use	O
of	O
Voronoi	B-Architecture
diagrams	I-Architecture
can	O
be	O
traced	O
back	O
to	O
Descartes	O
in	O
1644	O
.	O
</s>
<s>
Peter	O
Gustav	O
Lejeune	O
Dirichlet	O
used	O
two-dimensional	O
and	O
three-dimensional	O
Voronoi	B-Architecture
diagrams	I-Architecture
in	O
his	O
study	O
of	O
quadratic	O
forms	O
in	O
1850	O
.	O
</s>
<s>
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
named	O
after	O
Georgy	O
Feodosievych	O
Voronoy	O
who	O
defined	O
and	O
studied	O
the	O
general	O
n-dimensional	O
case	O
in	O
1908	O
.	O
</s>
<s>
Voronoi	B-Architecture
diagrams	I-Architecture
that	O
are	O
used	O
in	O
geophysics	O
and	O
meteorology	O
to	O
analyse	O
spatially	O
distributed	O
data	O
(	O
such	O
as	O
rainfall	O
measurements	O
)	O
are	O
called	O
Thiessen	B-Architecture
polygons	I-Architecture
after	O
American	O
meteorologist	O
Alfred	O
H	O
.	O
Thiessen	O
.	O
</s>
<s>
Other	O
equivalent	O
names	O
for	O
this	O
concept	O
(	O
or	O
particular	O
important	O
cases	O
of	O
it	O
)	O
:	O
Voronoi	O
polyhedra	O
,	O
Voronoi	B-Architecture
polygons	I-Architecture
,	O
domain(s )	O
of	O
influence	O
,	O
Voronoi	B-Architecture
decomposition	I-Architecture
,	O
Voronoi	O
tessellation(s )	O
,	O
Dirichlet	O
tessellation(s )	O
.	O
</s>
<s>
Voronoi	B-Architecture
tessellations	I-Architecture
of	O
regular	O
lattices	O
of	O
points	O
in	O
two	O
or	O
three	O
dimensions	O
give	O
rise	O
to	O
many	O
familiar	O
tessellations	O
.	O
</s>
<s>
Although	O
a	O
normal	O
Voronoi	B-Architecture
cell	I-Architecture
is	O
defined	O
as	O
the	O
set	O
of	O
points	O
closest	O
to	O
a	O
single	O
point	O
in	O
S	O
,	O
an	O
nth-order	O
Voronoi	B-Architecture
cell	I-Architecture
is	O
defined	O
as	O
the	O
set	O
of	O
points	O
having	O
a	O
particular	O
set	O
of	O
n	O
points	O
in	O
S	O
as	O
its	O
n	O
nearest	B-General_Concept
neighbors	I-General_Concept
.	O
</s>
<s>
Higher-order	O
Voronoi	B-Architecture
diagrams	I-Architecture
also	O
subdivide	O
space	O
.	O
</s>
<s>
Higher-order	O
Voronoi	B-Architecture
diagrams	I-Architecture
can	O
be	O
generated	O
recursively	O
.	O
</s>
<s>
To	O
generate	O
the	O
nth-order	O
Voronoi	B-Architecture
diagram	I-Architecture
from	O
setS	O
,	O
start	O
with	O
the	O
(	O
n−1	O
)	O
th-order	O
diagram	O
and	O
replace	O
each	O
cell	O
generated	O
by	O
X={x1,x2,...,xn−1}	O
with	O
a	O
Voronoi	B-Architecture
diagram	I-Architecture
generated	O
on	O
the	O
setS−X	O
.	O
</s>
<s>
For	O
a	O
set	O
of	O
n	O
points	O
the	O
(	O
n−1	O
)	O
th-order	O
Voronoi	B-Architecture
diagram	I-Architecture
is	O
called	O
a	O
farthest-point	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
For	O
a	O
given	O
set	O
of	O
points	O
S={p1,p2,...,pn}	O
the	O
farthest-point	O
Voronoi	B-Architecture
diagram	I-Architecture
divides	O
the	O
plane	O
into	O
cells	O
in	O
which	O
the	O
same	O
point	O
of	O
P	O
is	O
the	O
farthest	O
point	O
.	O
</s>
<s>
A	O
point	O
of	O
P	O
has	O
a	O
cell	O
in	O
the	O
farthest-point	O
Voronoi	B-Architecture
diagram	I-Architecture
if	O
and	O
only	O
if	O
it	O
is	O
a	O
vertex	O
of	O
the	O
convex	O
hull	O
of	O
P	O
.	O
Let	O
H={h1,h2,...,hk}	O
be	O
the	O
convex	O
hull	O
of	O
P	O
;	O
then	O
the	O
farthest-point	O
Voronoi	B-Architecture
diagram	I-Architecture
is	O
a	O
subdivision	O
of	O
the	O
plane	O
into	O
k	O
cells	O
,	O
one	O
for	O
each	O
point	O
in	O
H	O
,	O
with	O
the	O
property	O
that	O
a	O
point	O
q	O
lies	O
in	O
the	O
cell	O
corresponding	O
to	O
a	O
site	O
hi	O
if	O
and	O
only	O
if	O
d(q, hi )	O
>	O
d(q, pj )	O
for	O
each	O
pj∈S	O
with	O
hi	O
≠	O
pj	O
,	O
where	O
d(p, q )	O
is	O
the	O
Euclidean	O
distance	O
between	O
two	O
points	O
p	O
andq	O
.	O
</s>
<s>
The	O
boundaries	O
of	O
the	O
cells	O
in	O
the	O
farthest-point	O
Voronoi	B-Architecture
diagram	I-Architecture
have	O
the	O
structure	O
of	O
a	O
topological	O
tree	O
,	O
with	O
infinite	O
rays	O
as	O
its	O
leaves	O
.	O
</s>
<s>
Every	O
finite	O
tree	O
is	O
isomorphic	O
to	O
the	O
tree	O
formed	O
in	O
this	O
way	O
from	O
a	O
farthest-point	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
As	O
implied	O
by	O
the	O
definition	O
,	O
Voronoi	B-Architecture
cells	I-Architecture
can	O
be	O
defined	O
for	O
metrics	O
other	O
than	O
Euclidean	O
,	O
such	O
as	O
the	O
Mahalanobis	O
distance	O
or	O
Manhattan	O
distance	O
.	O
</s>
<s>
However	O
,	O
in	O
these	O
cases	O
the	O
boundaries	O
of	O
the	O
Voronoi	B-Architecture
cells	I-Architecture
may	O
be	O
more	O
complicated	O
than	O
in	O
the	O
Euclidean	O
case	O
,	O
since	O
the	O
equidistant	O
locus	O
for	O
two	O
points	O
may	O
fail	O
to	O
be	O
subspace	O
of	O
codimension	O
1	O
,	O
even	O
in	O
the	O
two-dimensional	O
case	O
.	O
</s>
<s>
A	O
weighted	B-Application
Voronoi	I-Application
diagram	I-Application
is	O
the	O
one	O
in	O
which	O
the	O
function	O
of	O
a	O
pair	O
of	O
points	O
to	O
define	O
a	O
Voronoi	B-Architecture
cell	I-Architecture
is	O
a	O
distance	O
function	O
modified	O
by	O
multiplicative	O
or	O
additive	O
weights	O
assigned	O
to	O
generator	O
points	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
case	O
of	O
Voronoi	B-Architecture
cells	I-Architecture
defined	O
using	O
a	O
distance	O
which	O
is	O
a	O
metric	O
,	O
in	O
this	O
case	O
some	O
of	O
the	O
Voronoi	B-Architecture
cells	I-Architecture
may	O
be	O
empty	O
.	O
</s>
<s>
A	O
power	B-Application
diagram	I-Application
is	O
a	O
type	O
of	O
Voronoi	B-Architecture
diagram	I-Architecture
defined	O
from	O
a	O
set	O
of	O
circles	O
using	O
the	O
power	O
distance	O
;	O
it	O
can	O
also	O
be	O
thought	O
of	O
as	O
a	O
weighted	B-Application
Voronoi	I-Application
diagram	I-Application
in	O
which	O
a	O
weight	O
defined	O
from	O
the	O
radius	O
of	O
each	O
circle	O
is	O
added	O
to	O
the	O
squared	O
Euclidean	O
distance	O
from	O
the	O
circle	O
's	O
center	O
.	O
</s>
<s>
The	O
Voronoi	B-Architecture
diagram	I-Architecture
of	O
points	O
in	O
-dimensional	O
space	O
can	O
have	O
vertices	O
,	O
requiring	O
the	O
same	O
bound	O
for	O
the	O
amount	O
of	O
memory	O
needed	O
to	O
store	O
an	O
explicit	O
description	O
of	O
it	O
.	O
</s>
<s>
Therefore	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
often	O
not	O
feasible	O
for	O
moderate	O
or	O
high	O
dimensions	O
.	O
</s>
<s>
A	O
more	O
space-efficient	O
alternative	O
is	O
to	O
use	O
approximate	O
Voronoi	B-Architecture
diagrams	I-Architecture
.	O
</s>
<s>
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
also	O
related	O
to	O
other	O
geometric	O
structures	O
such	O
as	O
the	O
medial	O
axis	O
(	O
which	O
has	O
found	O
applications	O
in	O
image	O
segmentation	O
,	O
optical	B-Application
character	I-Application
recognition	I-Application
,	O
and	O
other	O
computational	O
applications	O
)	O
,	O
straight	O
skeleton	O
,	O
and	O
zone	B-Application
diagrams	I-Application
.	O
</s>
<s>
An	O
example	O
of	O
this	O
that	O
made	O
use	O
of	O
Voronoi	B-Architecture
cells	I-Architecture
was	O
the	O
identification	O
of	O
the	O
Sabouroff	O
head	O
,	O
which	O
made	O
use	O
of	O
a	O
high-resolution	O
polygon	B-Algorithm
mesh	I-Algorithm
.	O
</s>
<s>
In	O
dialectometry	O
,	O
Voronoi	B-Architecture
cells	I-Architecture
are	O
used	O
to	O
indicate	O
a	O
supposed	O
linguistic	O
continuity	O
between	O
survey	O
points	O
.	O
</s>
<s>
In	O
biology	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
to	O
model	O
a	O
number	O
of	O
different	O
biological	O
structures	O
,	O
including	O
cells	O
and	O
bone	O
microarchitecture	O
.	O
</s>
<s>
Indeed	O
,	O
Voronoi	B-Architecture
tessellations	I-Architecture
work	O
as	O
a	O
geometrical	O
tool	O
to	O
understand	O
the	O
physical	O
constraints	O
that	O
drive	O
the	O
organization	O
of	O
biological	O
tissues	O
.	O
</s>
<s>
In	O
hydrology	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
to	O
calculate	O
the	O
rainfall	O
of	O
an	O
area	O
,	O
based	O
on	O
a	O
series	O
of	O
point	O
measurements	O
.	O
</s>
<s>
In	O
this	O
usage	O
,	O
they	O
are	O
generally	O
referred	O
to	O
as	O
Thiessen	B-Architecture
polygons	I-Architecture
.	O
</s>
<s>
In	O
ecology	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
to	O
study	O
the	O
growth	O
patterns	O
of	O
forests	O
and	O
forest	O
canopies	O
,	O
and	O
may	O
also	O
be	O
helpful	O
in	O
developing	O
predictive	O
models	O
for	O
forest	O
fires	O
.	O
</s>
<s>
In	O
computational	O
chemistry	O
,	O
ligand-binding	O
sites	O
are	O
transformed	O
into	O
Voronoi	B-Architecture
diagrams	I-Architecture
for	O
machine	O
learning	O
applications	O
(	O
e.g.	O
,	O
to	O
classify	O
binding	O
pockets	O
in	O
proteins	O
)	O
.	O
</s>
<s>
In	O
other	O
applications	O
,	O
Voronoi	B-Architecture
cells	I-Architecture
defined	O
by	O
the	O
positions	O
of	O
the	O
nuclei	O
in	O
a	O
molecule	O
are	O
used	O
to	O
compute	O
atomic	O
charges	O
.	O
</s>
<s>
In	O
astrophysics	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
to	O
generate	O
adaptative	O
smoothing	O
zones	O
on	O
images	O
,	O
adding	O
signal	O
fluxes	O
on	O
each	O
one	O
.	O
</s>
<s>
In	O
computational	O
fluid	O
dynamics	O
,	O
the	O
Voronoi	B-Architecture
tessellation	I-Architecture
of	O
a	O
set	O
of	O
points	O
can	O
be	O
used	O
to	O
define	O
the	O
computational	O
domains	O
used	O
in	O
finite	B-Algorithm
volume	I-Algorithm
methods	I-Algorithm
,	O
e.g.	O
</s>
<s>
as	O
in	O
the	O
moving-mesh	O
cosmology	O
code	O
AREPO	O
.	O
</s>
<s>
In	O
computational	B-Algorithm
physics	I-Algorithm
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
to	O
calculate	O
profiles	O
of	O
an	O
object	O
with	O
Shadowgraph	O
and	O
proton	O
radiography	O
in	O
High	O
energy	O
density	O
physics	O
.	O
</s>
<s>
In	O
medical	O
diagnosis	O
,	O
models	O
of	O
muscle	O
tissue	O
,	O
based	O
on	O
Voronoi	B-Architecture
diagrams	I-Architecture
,	O
can	O
be	O
used	O
to	O
detect	O
neuromuscular	O
diseases	O
.	O
</s>
<s>
In	O
epidemiology	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
can	O
be	O
used	O
to	O
correlate	O
sources	O
of	O
infections	O
in	O
epidemics	O
.	O
</s>
<s>
One	O
of	O
the	O
early	O
applications	O
of	O
Voronoi	B-Architecture
diagrams	I-Architecture
was	O
implemented	O
by	O
John	O
Snow	O
to	O
study	O
the	O
1854	O
Broad	O
Street	O
cholera	O
outbreak	O
in	O
Soho	O
,	O
England	O
.	O
</s>
<s>
In	O
polymer	O
physics	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
can	O
be	O
used	O
to	O
represent	O
free	O
volumes	O
of	O
polymers	O
.	O
</s>
<s>
In	O
materials	O
science	O
,	O
polycrystalline	O
microstructures	O
in	O
metallic	O
alloys	O
are	O
commonly	O
represented	O
using	O
Voronoi	B-Architecture
tessellations	I-Architecture
.	O
</s>
<s>
In	O
island	O
growth	O
,	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
is	O
used	O
to	O
estimate	O
the	O
growth	O
rate	O
of	O
individual	O
islands	O
.	O
</s>
<s>
In	O
solid-state	O
physics	O
,	O
the	O
Wigner-Seitz	O
cell	O
is	O
the	O
Voronoi	B-Architecture
tessellation	I-Architecture
of	O
a	O
solid	O
,	O
and	O
the	O
Brillouin	O
zone	O
is	O
the	O
Voronoi	B-Architecture
tessellation	I-Architecture
of	O
reciprocal	O
(	O
wavenumber	O
)	O
space	O
of	O
crystals	O
which	O
have	O
the	O
symmetry	O
of	O
a	O
space	O
group	O
.	O
</s>
<s>
In	O
aviation	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
superimposed	O
on	O
oceanic	O
plotting	O
charts	O
to	O
identify	O
the	O
nearest	O
airfield	O
for	O
in-flight	O
diversion	O
(	O
see	O
ETOPS	O
)	O
,	O
as	O
an	O
aircraft	O
progresses	O
through	O
its	O
flight	O
plan	O
.	O
</s>
<s>
In	O
architecture	O
,	O
Voronoi	B-Architecture
patterns	I-Architecture
were	O
the	O
basis	O
for	O
the	O
winning	O
entry	O
for	O
the	O
redevelopment	O
of	O
The	O
Arts	O
Centre	O
Gold	O
Coast	O
.	O
</s>
<s>
In	O
urban	O
planning	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
can	O
be	O
used	O
to	O
evaluate	O
the	O
Freight	O
Loading	O
Zone	O
system	O
.	O
</s>
<s>
In	O
mining	O
,	O
Voronoi	B-Architecture
polygons	I-Architecture
are	O
used	O
to	O
estimate	O
the	O
reserves	O
of	O
valuable	O
materials	O
,	O
minerals	O
,	O
or	O
other	O
resources	O
.	O
</s>
<s>
Exploratory	O
drillholes	O
are	O
used	O
as	O
the	O
set	O
of	O
points	O
in	O
the	O
Voronoi	B-Architecture
polygons	I-Architecture
.	O
</s>
<s>
In	O
surface	O
metrology	O
,	O
Voronoi	B-Architecture
tessellation	I-Architecture
can	O
be	O
used	O
for	O
surface	O
roughness	O
modeling	O
.	O
</s>
<s>
A	O
point	B-Data_Structure
location	I-Data_Structure
data	O
structure	O
can	O
be	O
built	O
on	O
top	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
in	O
order	O
to	O
answer	O
nearest	B-Algorithm
neighbor	I-Algorithm
queries	I-Algorithm
,	O
where	O
one	O
wants	O
to	O
find	O
the	O
object	O
that	O
is	O
closest	O
to	O
a	O
given	O
query	O
point	O
.	O
</s>
<s>
Nearest	B-Algorithm
neighbor	I-Algorithm
queries	I-Algorithm
have	O
numerous	O
applications	O
.	O
</s>
<s>
A	O
large	O
application	O
is	O
vector	B-Algorithm
quantization	I-Algorithm
,	O
commonly	O
used	O
in	O
data	B-General_Concept
compression	I-General_Concept
.	O
</s>
<s>
In	O
geometry	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
can	O
be	O
used	O
to	O
find	O
the	O
largest	B-Algorithm
empty	I-Algorithm
circle	I-Algorithm
amid	O
a	O
set	O
of	O
points	O
,	O
and	O
in	O
an	O
enclosing	O
polygon	O
;	O
e.g.	O
</s>
<s>
Voronoi	B-Architecture
diagrams	I-Architecture
together	O
with	O
farthest-point	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
for	O
efficient	O
algorithms	O
to	O
compute	O
the	O
roundness	O
of	O
a	O
set	O
of	O
points	O
.	O
</s>
<s>
In	O
networking	B-Architecture
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
can	O
be	O
used	O
in	O
derivations	O
of	O
the	O
capacity	O
of	O
a	O
wireless	B-General_Concept
network	I-General_Concept
.	O
</s>
<s>
In	O
computer	O
graphics	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
to	O
calculate	O
3D	O
shattering	O
/	O
fracturing	O
geometry	O
patterns	O
.	O
</s>
<s>
In	O
autonomous	O
robot	B-General_Concept
navigation	I-General_Concept
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
to	O
find	O
clear	O
routes	O
.	O
</s>
<s>
In	O
machine	O
learning	O
,	O
Voronoi	B-Architecture
diagrams	I-Architecture
are	O
used	O
to	O
do	O
1-NN	B-General_Concept
classifications	O
.	O
</s>
<s>
In	O
global	O
scene	O
reconstruction	O
,	O
including	O
with	O
random	O
sensor	O
sites	O
and	O
unsteady	O
wake	O
flow	O
,	O
geophysical	O
data	O
,	O
and	O
3D	O
turbulence	O
data	O
,	O
Voronoi	B-Architecture
tesselations	I-Architecture
are	O
used	O
with	O
deep	B-Algorithm
learning	I-Algorithm
.	O
</s>
<s>
In	O
user	B-Application
interface	I-Application
development	O
,	O
Voronoi	B-Architecture
patterns	I-Architecture
can	O
be	O
used	O
to	O
compute	O
the	O
best	O
hover	O
state	O
for	O
a	O
given	O
point	O
.	O
</s>
<s>
The	O
map	O
of	O
school	O
zones	O
is	O
therefore	O
a	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
Ukrainian	O
Pastry	O
chef	O
Dinara	O
Kasko	O
uses	O
the	O
mathematical	O
principles	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
to	O
create	O
silicone	O
molds	O
made	O
with	O
a	O
3D	O
printer	O
to	O
shape	O
her	O
original	O
cakes	O
.	O
</s>
<s>
Several	O
efficient	O
algorithms	O
are	O
known	O
for	O
constructing	O
Voronoi	B-Architecture
diagrams	I-Architecture
,	O
either	O
directly	O
(	O
as	O
the	O
diagram	O
itself	O
)	O
or	O
indirectly	O
by	O
starting	O
with	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
and	O
then	O
obtaining	O
its	O
dual	O
.	O
</s>
<s>
Direct	O
algorithms	O
include	O
Fortune	O
's	O
algorithm	O
,	O
an	O
O( n	O
log(n )	O
)	O
algorithm	O
for	O
generating	O
a	O
Voronoi	B-Architecture
diagram	I-Architecture
from	O
a	O
set	O
of	O
points	O
in	O
a	O
plane	O
.	O
</s>
<s>
Bowyer	B-Algorithm
–	I-Algorithm
Watson	I-Algorithm
algorithm	I-Algorithm
,	O
an	O
O( n	O
log(n )	O
)	O
to	O
O(n2 )	O
algorithm	O
for	O
generating	O
a	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
in	O
any	O
number	O
of	O
dimensions	O
,	O
can	O
be	O
used	O
in	O
an	O
indirect	O
algorithm	O
for	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
The	O
Jump	B-Protocol
Flooding	I-Protocol
Algorithm	I-Protocol
can	O
generate	O
approximate	O
Voronoi	B-Architecture
diagrams	I-Architecture
in	O
constant	O
time	O
and	O
is	O
suited	O
for	O
use	O
on	O
commodity	O
graphics	O
hardware	O
.	O
</s>
<s>
Lloyd	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
its	O
generalization	O
via	O
the	O
Linde	B-Algorithm
–	I-Algorithm
Buzo	I-Algorithm
–	I-Algorithm
Gray	I-Algorithm
algorithm	I-Algorithm
(	O
aka	O
k-means	B-Algorithm
clustering	I-Algorithm
)	O
,	O
use	O
the	O
construction	O
of	O
Voronoi	B-Architecture
diagrams	I-Architecture
as	O
a	O
subroutine	O
.	O
</s>
<s>
These	O
methods	O
alternate	O
between	O
steps	O
in	O
which	O
one	O
constructs	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
for	O
a	O
set	O
of	O
seed	O
points	O
,	O
and	O
steps	O
in	O
which	O
the	O
seed	O
points	O
are	O
moved	O
to	O
new	O
locations	O
that	O
are	O
more	O
central	O
within	O
their	O
cells	O
.	O
</s>
<s>
These	O
methods	O
can	O
be	O
used	O
in	O
spaces	O
of	O
arbitrary	O
dimension	O
to	O
iteratively	O
converge	O
towards	O
a	O
specialized	O
form	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
,	O
called	O
a	O
Centroidal	B-Application
Voronoi	I-Application
tessellation	I-Application
,	O
where	O
the	O
sites	O
have	O
been	O
moved	O
to	O
points	O
that	O
are	O
also	O
the	O
geometric	O
centers	O
of	O
their	O
cells	O
.	O
</s>
<s>
Voronoi	O
meshes	B-Algorithm
can	O
also	O
be	O
generated	O
in	O
3D	O
.	O
</s>
<s>
Mathematica	O
13.1	O
added	O
support	O
for	O
rendering	O
3D	O
Voronoi	O
mesh	B-Algorithm
decompositions	O
of	O
space	O
.	O
</s>
