<s>
In	O
Euclidean	O
geometry	O
,	O
a	O
translation	O
is	O
a	O
geometric	B-Algorithm
transformation	I-Algorithm
that	O
moves	O
every	O
point	O
of	O
a	O
figure	O
,	O
shape	O
or	O
space	O
by	O
the	O
same	O
distance	O
in	O
a	O
given	O
direction	O
.	O
</s>
<s>
If	O
is	O
a	O
translation	O
,	O
then	O
the	O
image	O
of	O
a	O
subset	O
under	O
the	O
function	O
is	O
the	O
translate	B-Algorithm
of	O
by	O
.	O
</s>
<s>
The	O
translate	B-Algorithm
of	O
by	O
is	O
often	O
written	O
.	O
</s>
<s>
In	O
geometry	O
,	O
a	O
vertical	B-Algorithm
translation	I-Algorithm
(	O
also	O
known	O
as	O
vertical	B-Algorithm
shift	I-Algorithm
)	O
is	O
a	O
translation	O
of	O
a	O
geometric	O
object	O
in	O
a	O
direction	O
parallel	O
to	O
the	O
vertical	O
axis	O
of	O
the	O
Cartesian	O
coordinate	O
system	O
.	O
</s>
<s>
Often	O
,	O
vertical	B-Algorithm
translations	I-Algorithm
are	O
considered	O
for	O
the	O
graph	B-Application
of	I-Application
a	I-Application
function	I-Application
.	O
</s>
<s>
If	O
f	O
is	O
any	O
function	O
ofx	O
,	O
then	O
the	O
graph	O
of	O
the	O
function	O
f(x )	O
+c	O
(	O
whose	O
values	O
are	O
given	O
by	O
adding	O
a	O
constant	O
c	O
to	O
the	O
values	O
of	O
f	O
)	O
may	O
be	O
obtained	O
by	O
a	O
vertical	B-Algorithm
translation	I-Algorithm
of	O
the	O
graph	O
of	O
f(x )	O
by	O
distance	O
c	O
.	O
For	O
this	O
reason	O
the	O
function	O
f(x )	O
+c	O
is	O
sometimes	O
called	O
a	O
vertical	O
translate	B-Algorithm
of	O
f(x )	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
antiderivatives	B-Algorithm
of	O
a	O
function	O
all	O
differ	O
from	O
each	O
other	O
by	O
a	O
constant	O
of	O
integration	O
and	O
are	O
therefore	O
vertical	O
translates	B-Algorithm
of	O
each	O
other	O
.	O
</s>
<s>
In	O
function	B-Application
graphing	I-Application
,	O
a	O
horizontal	B-Algorithm
translation	I-Algorithm
is	O
a	O
transformation	B-Algorithm
which	O
results	O
in	O
a	O
graph	O
that	O
is	O
equivalent	O
to	O
shifting	O
the	O
base	O
graph	O
left	O
or	O
right	O
in	O
the	O
direction	O
of	O
the	O
x-axis	O
.	O
</s>
<s>
If	O
function	O
transformation	B-Algorithm
was	O
talked	O
about	O
in	O
terms	O
of	O
geometric	B-Algorithm
transformations	I-Algorithm
it	O
may	O
be	O
clearer	O
why	O
functions	O
translate	B-Algorithm
horizontally	O
the	O
way	O
they	O
do	O
.	O
</s>
<s>
Taking	O
the	O
parabola	O
y	O
=	O
x2	O
,	O
a	O
horizontal	B-Algorithm
translation	I-Algorithm
5	O
units	O
to	O
the	O
right	O
would	O
be	O
represented	O
by	O
T(x,y )	O
=	O
(	O
x	O
+	O
5	O
,	O
y	O
)	O
.	O
</s>
<s>
Now	O
we	O
must	O
connect	O
this	O
transformation	B-Algorithm
notation	O
to	O
an	O
algebraic	O
notation	O
.	O
</s>
<s>
When	O
considering	O
spacetime	B-Protocol
,	O
a	O
change	O
of	O
time	O
coordinate	O
is	O
considered	O
to	O
be	O
a	O
translation	O
.	O
</s>
<s>
The	O
translation	B-Algorithm
operator	I-Algorithm
turns	O
a	O
function	O
of	O
the	O
original	O
position	O
,	O
,	O
into	O
a	O
function	O
of	O
the	O
final	O
position	O
,	O
.	O
</s>
<s>
The	O
translation	B-Algorithm
operator	I-Algorithm
can	O
act	O
on	O
many	O
kinds	O
of	O
functions	O
,	O
such	O
as	O
when	O
the	O
translation	B-Algorithm
operator	I-Algorithm
acts	O
on	O
a	O
wavefunction	O
,	O
which	O
is	O
studied	O
in	O
the	O
field	O
of	O
quantum	O
mechanics	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
translations	O
forms	O
the	O
translation	B-Algorithm
group	I-Algorithm
,	O
which	O
is	O
isomorphic	O
to	O
the	O
space	O
itself	O
,	O
and	O
a	O
normal	O
subgroup	O
of	O
Euclidean	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Because	O
translation	O
is	O
commutative	O
,	O
the	O
translation	B-Algorithm
group	I-Algorithm
is	O
abelian	O
.	O
</s>
<s>
There	O
are	O
an	O
infinite	O
number	O
of	O
possible	O
translations	O
,	O
so	O
the	O
translation	B-Algorithm
group	I-Algorithm
is	O
an	O
infinite	O
group	O
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
relativity	O
,	O
due	O
to	O
the	O
treatment	O
of	O
space	O
and	O
time	O
as	O
a	O
single	O
spacetime	B-Protocol
,	O
translations	O
can	O
also	O
refer	O
to	O
changes	O
in	O
the	O
time	O
coordinate	O
.	O
</s>
<s>
One	O
kind	O
of	O
subgroup	O
of	O
the	O
three-dimensional	O
translation	B-Algorithm
group	I-Algorithm
are	O
the	O
lattice	O
groups	O
,	O
which	O
are	O
infinite	O
groups	O
,	O
but	O
unlike	O
the	O
translation	B-Algorithm
groups	I-Algorithm
,	O
are	O
finitely	O
generated	O
.	O
</s>
<s>
A	O
translation	O
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
with	O
no	O
fixed	O
points	O
.	O
</s>
<s>
Nevertheless	O
,	O
there	O
is	O
a	O
common	O
workaround	B-Application
using	O
homogeneous	O
coordinates	O
to	O
represent	O
a	O
translation	O
of	O
a	O
vector	O
space	O
with	O
matrix	O
multiplication	O
:	O
Write	O
the	O
3-dimensional	O
vector	O
using	O
4	O
homogeneous	O
coordinates	O
as	O
.	O
</s>
<s>
To	O
translate	B-Algorithm
an	O
object	O
by	O
a	O
vector	O
,	O
each	O
homogeneous	O
vector	O
(	O
written	O
in	O
homogeneous	O
coordinates	O
)	O
can	O
be	O
multiplied	O
by	O
this	O
translation	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
The	O
inverse	O
of	O
a	O
translation	B-Algorithm
matrix	I-Algorithm
can	O
be	O
obtained	O
by	O
reversing	O
the	O
direction	O
of	O
the	O
vector	O
:	O
</s>
<s>
While	O
geometric	O
translation	O
is	O
often	O
viewed	O
as	O
an	O
active	O
process	O
that	O
changes	O
the	O
position	O
of	O
a	O
geometric	O
object	O
,	O
a	O
similar	O
result	O
can	O
be	O
achieved	O
by	O
a	O
passive	O
transformation	B-Algorithm
that	O
moves	O
the	O
coordinate	O
system	O
itself	O
but	O
leaves	O
the	O
object	O
fixed	O
.	O
</s>
<s>
The	O
passive	O
version	O
of	O
an	O
active	O
geometric	O
translation	O
is	O
known	O
as	O
a	O
translation	B-Algorithm
of	I-Algorithm
axes	I-Algorithm
.	O
</s>
<s>
A	O
common	O
example	O
is	O
a	O
periodic	O
function	O
,	O
which	O
is	O
an	O
eigenfunction	B-Algorithm
of	O
a	O
translation	B-Algorithm
operator	I-Algorithm
.	O
</s>
