<s>
In	O
computational	O
geometry	O
,	O
a	O
power	B-Application
diagram	I-Application
,	O
also	O
called	O
a	O
Laguerre	O
–	O
Voronoi	B-Architecture
diagram	I-Architecture
,	O
Dirichlet	B-Architecture
cell	I-Architecture
complex	O
,	O
radical	O
Voronoi	B-Architecture
tesselation	I-Architecture
or	O
a	O
sectional	O
Dirichlet	B-Architecture
tesselation	I-Architecture
,	O
is	O
a	O
partition	O
of	O
the	O
Euclidean	O
plane	O
into	O
polygonal	O
cells	O
defined	O
from	O
a	O
set	O
of	O
circles	O
.	O
</s>
<s>
The	O
power	B-Application
diagram	I-Application
is	O
a	O
form	O
of	O
generalized	O
Voronoi	B-Architecture
diagram	I-Architecture
,	O
and	O
coincides	O
with	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
of	O
the	O
circle	O
centers	O
in	O
the	O
case	O
that	O
all	O
the	O
circles	O
have	O
equal	O
radii	O
.	O
</s>
<s>
The	O
power	B-Application
diagram	I-Application
of	O
a	O
set	O
of	O
n	O
circles	O
Ci	O
is	O
a	O
partition	O
of	O
the	O
plane	O
into	O
n	O
regions	O
Ri	O
(	O
called	O
cells	O
)	O
,	O
such	O
that	O
a	O
point	O
P	O
belongs	O
to	O
Ri	O
whenever	O
circle	O
Ci	O
is	O
the	O
circle	O
minimizing	O
the	O
power	O
of	O
P	O
.	O
</s>
<s>
In	O
the	O
case	O
n	O
=	O
2	O
,	O
the	O
power	B-Application
diagram	I-Application
consists	O
of	O
two	O
halfplanes	O
,	O
separated	O
by	O
a	O
line	O
called	O
the	O
radical	O
axis	O
or	O
chordale	O
of	O
the	O
two	O
circles	O
.	O
</s>
<s>
More	O
generally	O
,	O
in	O
any	O
power	B-Application
diagram	I-Application
,	O
each	O
cell	O
Ri	O
is	O
a	O
convex	O
polygon	B-General_Concept
,	O
the	O
intersection	O
of	O
the	O
halfspaces	O
bounded	O
by	O
the	O
radical	O
axes	O
of	O
circle	O
Ci	O
with	O
each	O
other	O
circle	O
.	O
</s>
<s>
The	O
power	B-Application
diagram	I-Application
may	O
be	O
seen	O
as	O
a	O
weighted	O
form	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
of	O
a	O
set	O
of	O
point	O
sites	O
,	O
a	O
partition	O
of	O
the	O
plane	O
into	O
cells	O
within	O
which	O
one	O
of	O
the	O
sites	O
is	O
closer	O
than	O
all	O
the	O
other	O
sites	O
.	O
</s>
<s>
Other	O
forms	O
of	O
weighted	B-Application
Voronoi	I-Application
diagram	I-Application
include	O
the	O
additively	O
weighted	B-Application
Voronoi	I-Application
diagram	I-Application
,	O
in	O
which	O
each	O
site	O
has	O
a	O
weight	O
that	O
is	O
added	O
to	O
its	O
distance	O
before	O
comparing	O
it	O
to	O
the	O
distances	O
to	O
the	O
other	O
sites	O
,	O
and	O
the	O
multiplicatively	O
weighted	B-Application
Voronoi	I-Application
diagram	I-Application
,	O
in	O
which	O
the	O
weight	O
of	O
a	O
site	O
is	O
multiplied	O
by	O
its	O
distance	O
before	O
comparing	O
it	O
to	O
the	O
distances	O
to	O
the	O
other	O
sites	O
.	O
</s>
<s>
In	O
contrast	O
,	O
in	O
the	O
power	B-Application
diagram	I-Application
,	O
we	O
may	O
view	O
each	O
circle	O
center	O
as	O
a	O
site	O
,	O
and	O
each	O
circle	O
's	O
squared	O
radius	O
as	O
a	O
weight	O
that	O
is	O
subtracted	O
from	O
the	O
squared	O
Euclidean	O
distance	O
before	O
comparing	O
it	O
to	O
other	O
squared	O
distances	O
.	O
</s>
<s>
In	O
the	O
case	O
that	O
all	O
the	O
circle	O
radii	O
are	O
equal	O
,	O
this	O
subtraction	O
makes	O
no	O
difference	O
to	O
the	O
comparison	O
,	O
and	O
the	O
power	B-Application
diagram	I-Application
coincides	O
with	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
A	O
planar	O
power	B-Application
diagram	I-Application
may	O
also	O
be	O
interpreted	O
as	O
a	O
planar	O
cross-section	O
of	O
an	O
unweighted	O
three-dimensional	O
Voronoi	B-Architecture
diagram	I-Architecture
.	O
</s>
<s>
Like	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
,	O
the	O
power	B-Application
diagram	I-Application
may	O
be	O
generalized	O
to	O
Euclidean	O
spaces	O
of	O
any	O
dimension	O
.	O
</s>
<s>
The	O
power	B-Application
diagram	I-Application
of	O
n	O
spheres	O
in	O
d	O
dimensions	O
is	O
combinatorially	O
equivalent	O
to	O
the	O
intersection	O
of	O
a	O
set	O
of	O
n	O
upward-facing	O
halfspaces	O
in	O
d+1	O
dimensions	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Two-dimensional	O
power	B-Application
diagrams	I-Application
may	O
be	O
constructed	O
by	O
an	O
algorithm	O
that	O
runs	O
in	O
time	O
O(nlogn )	O
.	O
</s>
<s>
More	O
generally	O
,	O
because	O
of	O
the	O
equivalence	O
with	O
higher-dimensional	O
halfspace	O
intersections	O
,	O
d-dimensional	O
power	B-Application
diagrams	I-Application
(	O
for	O
d>2	O
)	O
may	O
be	O
constructed	O
by	O
an	O
algorithm	O
that	O
runs	O
in	O
time	O
.	O
</s>
<s>
The	O
power	B-Application
diagram	I-Application
may	O
be	O
used	O
as	O
part	O
of	O
an	O
efficient	O
algorithm	O
for	O
computing	O
the	O
volume	O
of	O
a	O
union	O
of	O
spheres	O
.	O
</s>
<s>
Intersecting	O
each	O
sphere	O
with	O
its	O
power	B-Application
diagram	I-Application
cell	O
gives	O
its	O
contribution	O
to	O
the	O
total	O
union	O
,	O
from	O
which	O
the	O
volume	O
may	O
be	O
computed	O
in	O
time	O
proportional	O
to	O
the	O
complexity	O
of	O
the	O
power	B-Application
diagram	I-Application
.	O
</s>
<s>
Other	O
applications	O
of	O
power	B-Application
diagrams	I-Application
include	O
data	B-General_Concept
structures	I-General_Concept
for	O
testing	O
whether	O
a	O
point	O
belongs	O
to	O
a	O
union	O
of	O
disks	O
,	O
algorithms	O
for	O
constructing	O
the	O
boundary	O
of	O
a	O
union	O
of	O
disks	O
,	O
and	O
algorithms	O
for	O
finding	O
the	O
closest	O
two	O
balls	O
in	O
a	O
set	O
of	O
balls	O
.	O
</s>
<s>
defined	O
power	B-Application
diagrams	I-Application
and	O
used	O
them	O
to	O
show	O
that	O
the	O
boundary	O
of	O
a	O
union	O
of	O
n	O
circular	O
disks	O
can	O
always	O
be	O
illuminated	O
from	O
at	O
most	O
2n	O
point	O
light	O
sources	O
.	O
</s>
<s>
Power	B-Application
diagrams	I-Application
have	O
appeared	O
in	O
the	O
literature	O
under	O
other	O
names	O
including	O
the	O
"	O
Laguerre	O
–	O
Voronoi	B-Architecture
diagram	I-Architecture
"	O
,	O
"	O
Dirichlet	B-Architecture
cell	I-Architecture
complex	O
"	O
,	O
"	O
radical	O
Voronoi	B-Architecture
tesselation	I-Architecture
"	O
and	O
"	O
sectional	O
Dirichlet	B-Architecture
tesselation	I-Architecture
"	O
.	O
</s>
