<s>
In	O
geometry	O
,	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
two	O
sets	O
of	O
position	O
vectors	O
A	O
and	O
B	O
in	O
Euclidean	O
space	O
is	O
formed	O
by	O
adding	O
each	O
vector	O
in	O
A	O
to	O
each	O
vector	O
in	O
B	O
:	O
</s>
<s>
The	O
Minkowski	B-Algorithm
difference	I-Algorithm
(	O
also	O
Minkowski	O
subtraction	O
,	O
Minkowski	O
decomposition	O
,	O
or	O
geometric	O
difference	O
)	O
is	O
the	O
corresponding	O
inverse	O
,	O
where	O
produces	O
a	O
set	O
that	O
could	O
be	O
summed	O
with	O
B	O
to	O
recover	O
A	O
.	O
</s>
<s>
This	O
is	O
defined	O
as	O
the	O
complement	O
of	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
the	O
complement	O
of	O
A	O
with	O
the	O
reflection	O
of	O
B	O
about	O
the	O
origin	O
.	O
</s>
<s>
This	O
definition	O
allows	O
a	O
symmetrical	O
relationship	O
between	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
and	O
difference	O
.	O
</s>
<s>
In	O
2D	O
image	B-Algorithm
processing	I-Algorithm
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
and	O
difference	O
are	O
known	O
as	O
dilation	B-Algorithm
and	O
erosion	B-Algorithm
.	O
</s>
<s>
An	O
alternative	O
definition	O
of	O
the	O
Minkowski	B-Algorithm
difference	I-Algorithm
is	O
sometimes	O
used	O
for	O
computing	O
intersection	O
of	O
convex	O
shapes	O
.	O
</s>
<s>
Instead	O
it	O
replaces	O
the	O
vector	O
addition	O
of	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
with	O
a	O
vector	O
subtraction	O
.	O
</s>
<s>
For	O
Minkowski	B-Algorithm
addition	I-Algorithm
,	O
the	O
,	O
containing	O
only	O
the	O
zero	O
vector	O
,	O
0	O
,	O
is	O
an	O
identity	O
element	O
:	O
for	O
every	O
subset	O
S	O
of	O
a	O
vector	O
space	O
,	O
</s>
<s>
The	O
empty	O
set	O
is	O
important	O
in	O
Minkowski	B-Algorithm
addition	I-Algorithm
,	O
because	O
the	O
empty	O
set	O
annihilates	O
every	O
other	O
subset	O
:	O
for	O
every	O
subset	O
S	O
of	O
a	O
vector	O
space	O
,	O
its	O
sum	O
with	O
the	O
empty	O
set	O
is	O
empty	O
:	O
</s>
<s>
For	O
another	O
example	O
,	O
consider	O
the	O
Minkowski	B-Algorithm
sums	I-Algorithm
of	O
open	O
or	O
closed	O
balls	O
in	O
the	O
field	O
which	O
is	O
either	O
the	O
real	O
numbers	O
or	O
complex	O
numbers	O
If	O
is	O
the	O
closed	O
ball	O
of	O
radius	O
centered	O
at	O
in	O
then	O
for	O
any	O
and	O
also	O
will	O
hold	O
for	O
any	O
scalar	O
such	O
that	O
the	O
product	O
is	O
defined	O
(	O
which	O
happens	O
when	O
or	O
)	O
.	O
</s>
<s>
The	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
a	O
closed	O
ball	O
and	O
an	O
open	O
ball	O
is	O
an	O
open	O
ball	O
.	O
</s>
<s>
More	O
generally	O
,	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
an	O
open	O
subset	O
with	O
other	O
set	O
will	O
be	O
an	O
open	O
subset	O
.	O
</s>
<s>
If	O
is	O
the	O
graph	B-Application
of	O
and	O
if	O
and	O
is	O
the	O
-axis	O
in	O
then	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
these	O
two	O
closed	O
subsets	O
of	O
the	O
plane	O
is	O
the	O
open	O
set	O
consisting	O
of	O
everything	O
other	O
than	O
the	O
-axis	O
.	O
</s>
<s>
This	O
shows	O
that	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
two	O
closed	O
sets	O
is	O
not	O
necessarily	O
a	O
closed	O
set	O
.	O
</s>
<s>
However	O
,	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
two	O
closed	O
subsets	O
will	O
be	O
a	O
closed	O
subset	O
if	O
at	O
least	O
one	O
of	O
these	O
sets	O
is	O
also	O
a	O
compact	O
subset	O
.	O
</s>
<s>
Minkowski	B-Algorithm
addition	I-Algorithm
behaves	O
well	O
with	O
respect	O
to	O
the	O
operation	O
of	O
taking	O
convex	O
hulls	O
,	O
as	O
shown	O
by	O
the	O
following	O
proposition	O
:	O
</s>
<s>
For	O
all	O
non-empty	O
subsets	O
and	O
of	O
a	O
real	O
vector	O
space	O
,	O
the	O
convex	O
hull	O
of	O
their	O
Minkowski	B-Algorithm
sum	I-Algorithm
is	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
their	O
convex	O
hulls	O
:	O
</s>
<s>
Minkowski	B-Algorithm
sums	I-Algorithm
act	O
linearly	O
on	O
the	O
perimeter	O
of	O
two-dimensional	O
convex	O
bodies	O
:	O
the	O
perimeter	O
of	O
the	O
sum	O
equals	O
the	O
sum	O
of	O
perimeters	O
.	O
</s>
<s>
Additionally	O
,	O
if	O
is	O
(	O
the	O
interior	O
of	O
)	O
a	O
curve	O
of	O
constant	O
width	O
,	O
then	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
and	O
of	O
its	O
rotation	O
is	O
a	O
disk	O
.	O
</s>
<s>
Minkowski	B-Algorithm
addition	I-Algorithm
plays	O
a	O
central	O
role	O
in	O
mathematical	B-Algorithm
morphology	I-Algorithm
.	O
</s>
<s>
It	O
arises	O
in	O
the	O
brush-and-stroke	O
paradigm	O
of	O
2D	O
computer	O
graphics	O
(	O
with	O
various	O
uses	O
,	O
notably	O
by	O
Donald	O
E	O
.	O
Knuth	O
in	O
Metafont	B-Operating_System
)	O
,	O
and	O
as	O
the	O
solid	O
sweep	O
operation	O
of	O
3D	O
computer	O
graphics	O
.	O
</s>
<s>
Minkowski	B-Algorithm
sums	I-Algorithm
are	O
used	O
in	O
motion	O
planning	O
of	O
an	O
object	O
among	O
obstacles	O
.	O
</s>
<s>
In	O
the	O
simple	O
model	O
of	O
translational	O
motion	O
of	O
an	O
object	O
in	O
the	O
plane	O
,	O
where	O
the	O
position	O
of	O
an	O
object	O
may	O
be	O
uniquely	O
specified	O
by	O
the	O
position	O
of	O
a	O
fixed	O
point	O
of	O
this	O
object	O
,	O
the	O
configuration	O
space	O
are	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
the	O
set	O
of	O
obstacles	O
and	O
the	O
movable	O
object	O
placed	O
at	O
the	O
origin	O
and	O
rotated	O
180	O
degrees	O
.	O
</s>
<s>
In	O
numerical	O
control	O
machining	O
,	O
the	O
programming	O
of	O
the	O
NC	O
tool	O
exploits	O
the	O
fact	O
that	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
the	O
cutting	O
piece	O
with	O
its	O
trajectory	O
gives	O
the	O
shape	O
of	O
the	O
cut	O
in	O
the	O
material	O
.	O
</s>
<s>
In	O
OpenSCAD	B-Language
Minkowski	B-Algorithm
sums	I-Algorithm
are	O
used	O
to	O
outline	O
a	O
shape	O
with	O
another	O
shape	O
creating	O
a	O
composite	O
of	O
both	O
shapes	O
.	O
</s>
<s>
Minkowski	B-Algorithm
sums	I-Algorithm
are	O
also	O
frequently	O
used	O
in	O
aggregation	O
theory	O
when	O
individual	O
objects	O
to	O
be	O
aggregated	O
are	O
characterized	O
via	O
sets	O
.	O
</s>
<s>
Minkowski	B-Algorithm
sums	I-Algorithm
,	O
specifically	O
Minkowski	B-Algorithm
differences	I-Algorithm
,	O
are	O
often	O
used	O
alongside	O
GJK	O
algorithms	O
to	O
compute	O
collision	O
detection	O
for	O
convex	O
hulls	O
in	O
physics	O
engines	O
.	O
</s>
<s>
For	O
two	O
convex	O
polygons	O
and	O
in	O
the	O
plane	O
with	O
and	O
vertices	O
,	O
their	O
Minkowski	B-Algorithm
sum	I-Algorithm
is	O
a	O
convex	O
polygon	O
with	O
at	O
most	O
+	O
vertices	O
and	O
may	O
be	O
computed	O
in	O
time	O
O( 	O
+	O
)	O
by	O
a	O
very	O
simple	O
procedure	O
,	O
which	O
may	O
be	O
informally	O
described	O
as	O
follows	O
.	O
</s>
<s>
Let	O
us	O
merge	B-Algorithm
the	I-Algorithm
ordered	I-Algorithm
sequences	I-Algorithm
of	O
the	O
directed	O
edges	O
from	O
and	O
into	O
a	O
single	O
ordered	O
sequence	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
the	O
resulting	O
polygonal	O
chain	O
will	O
in	O
fact	O
be	O
a	O
convex	O
polygon	O
which	O
is	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
and	O
.	O
</s>
<s>
If	O
one	O
polygon	O
is	O
convex	O
and	O
another	O
one	O
is	O
not	O
,	O
the	O
complexity	O
of	O
their	O
Minkowski	B-Algorithm
sum	I-Algorithm
is	O
O(nm )	O
.	O
</s>
<s>
If	O
both	O
of	O
them	O
are	O
nonconvex	O
,	O
their	O
Minkowski	B-Algorithm
sum	I-Algorithm
complexity	O
is	O
O((mn )	O
2	O
)	O
.	O
</s>
<s>
There	O
is	O
also	O
a	O
notion	O
of	O
the	O
essential	O
Minkowski	B-Algorithm
sum	I-Algorithm
+e	O
of	O
two	O
subsets	O
of	O
Euclidean	O
space	O
.	O
</s>
<s>
For	O
K	O
and	O
L	O
compact	O
convex	O
subsets	O
in	O
,	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
can	O
be	O
described	O
by	O
the	O
support	O
function	O
of	O
the	O
convex	O
sets	O
:	O
</s>
