<s>
In	O
mathematics	O
,	O
a	O
Euclidean	B-Algorithm
group	I-Algorithm
is	O
the	O
group	O
of	O
(	O
Euclidean	O
)	O
isometries	O
of	O
a	O
Euclidean	O
space	O
;	O
that	O
is	O
,	O
the	O
transformations	O
of	O
that	O
space	O
that	O
preserve	O
the	O
Euclidean	O
distance	O
between	O
any	O
two	O
points	O
(	O
also	O
called	O
Euclidean	B-Algorithm
transformations	I-Algorithm
)	O
.	O
</s>
<s>
The	O
Euclidean	B-Algorithm
group	I-Algorithm
E(n )	O
comprises	O
all	O
translations	B-Algorithm
,	O
rotations	B-General_Concept
,	O
and	O
reflections	B-Algorithm
of	O
;	O
and	O
arbitrary	O
finite	O
combinations	O
of	O
them	O
.	O
</s>
<s>
The	O
Euclidean	B-Algorithm
group	I-Algorithm
can	O
be	O
seen	O
as	O
the	O
symmetry	O
group	O
of	O
the	O
space	O
itself	O
,	O
and	O
contains	O
the	O
group	O
of	O
symmetries	O
of	O
any	O
figure	O
(	O
subset	O
)	O
of	O
that	O
space	O
.	O
</s>
<s>
A	O
Euclidean	B-Algorithm
isometry	I-Algorithm
can	O
be	O
direct	O
or	O
indirect	O
,	O
depending	O
on	O
whether	O
it	O
preserves	O
the	O
handedness	O
of	O
figures	O
.	O
</s>
<s>
The	O
direct	O
Euclidean	B-Algorithm
isometries	I-Algorithm
form	O
a	O
subgroup	O
,	O
the	O
special	B-Algorithm
Euclidean	I-Algorithm
group	I-Algorithm
,	O
often	O
denoted	O
SE(n )	O
,	O
whose	O
elements	O
are	O
called	O
rigid	B-Algorithm
motions	I-Algorithm
or	O
Euclidean	B-Algorithm
motions	I-Algorithm
.	O
</s>
<s>
They	O
comprise	O
arbitrary	O
combinations	O
of	O
translations	B-Algorithm
and	O
rotations	B-General_Concept
,	O
but	O
not	O
reflections	B-Algorithm
.	O
</s>
<s>
Of	O
these	O
,	O
n	O
can	O
be	O
attributed	O
to	O
available	O
translational	B-Algorithm
symmetry	O
,	O
and	O
the	O
remaining	O
to	O
rotational	O
symmetry	O
.	O
</s>
<s>
The	O
direct	O
isometries	O
(	O
i.e.	O
,	O
isometries	O
preserving	O
the	O
handedness	O
of	O
chiral	O
subsets	O
)	O
comprise	O
a	O
subgroup	O
of	O
E(n )	O
,	O
called	O
the	O
special	B-Algorithm
Euclidean	I-Algorithm
group	I-Algorithm
and	O
usually	O
denoted	O
by	O
E+( n	O
)	O
or	O
SE(n )	O
.	O
</s>
<s>
They	O
include	O
the	O
translations	B-Algorithm
and	O
rotations	B-General_Concept
,	O
and	O
combinations	O
thereof	O
;	O
including	O
the	O
identity	O
transformation	O
,	O
but	O
excluding	O
any	O
reflections	B-Algorithm
.	O
</s>
<s>
For	O
any	O
fixed	O
indirect	O
isometry	O
R	O
,	O
such	O
as	O
a	O
reflection	B-Algorithm
about	O
some	O
hyperplane	O
,	O
every	O
other	O
indirect	O
isometry	O
can	O
be	O
obtained	O
by	O
the	O
composition	B-Application
of	O
R	O
with	O
some	O
direct	O
isometry	O
.	O
</s>
<s>
The	O
natural	O
topology	B-Architecture
of	O
Euclidean	O
space	O
implies	O
a	O
topology	B-Architecture
for	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
E(n )	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
the	O
special	B-Algorithm
Euclidean	I-Algorithm
group	I-Algorithm
SE(n )	O
=	O
E+( n	O
)	O
is	O
connected	O
in	O
this	O
topology	B-Architecture
.	O
</s>
<s>
For	O
that	O
reason	O
,	O
the	O
direct	O
Euclidean	B-Algorithm
isometries	I-Algorithm
are	O
also	O
called	O
"	O
rigid	B-Algorithm
motions	I-Algorithm
"	O
.	O
</s>
<s>
The	O
Euclidean	B-Algorithm
groups	I-Algorithm
are	O
not	O
only	O
topological	O
groups	O
,	O
they	O
are	O
Lie	O
groups	O
,	O
so	O
that	O
calculus	O
notions	O
can	O
be	O
adapted	O
immediately	O
to	O
this	O
setting	O
.	O
</s>
<s>
The	O
Euclidean	B-Algorithm
group	I-Algorithm
E(n )	O
is	O
a	O
subgroup	O
of	O
the	O
affine	O
group	O
for	O
n	O
dimensions	O
,	O
and	O
in	O
such	O
a	O
way	O
as	O
to	O
respect	O
the	O
semidirect	O
product	O
structure	O
of	O
both	O
groups	O
.	O
</s>
<s>
by	O
a	O
single	O
square	B-Algorithm
matrix	I-Algorithm
of	O
size	O
,	O
as	O
explained	O
for	O
the	O
affine	O
group	O
.	O
</s>
<s>
In	O
the	O
terms	O
of	O
Felix	O
Klein	O
's	O
Erlangen	O
programme	O
,	O
we	O
read	O
off	O
from	O
this	O
that	O
Euclidean	O
geometry	O
,	O
the	O
geometry	O
of	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
of	O
symmetries	O
,	O
is	O
,	O
therefore	O
,	O
a	O
specialisation	O
of	O
affine	O
geometry	O
.	O
</s>
<s>
The	O
Euclidean	B-Algorithm
group	I-Algorithm
is	O
a	O
subgroup	O
of	O
the	O
group	O
of	O
affine	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
It	O
has	O
as	O
subgroups	O
the	O
translational	B-Algorithm
group	O
T(n )	O
,	O
and	O
the	O
orthogonal	O
group	O
O(n )	O
.	O
</s>
<s>
T(n )	O
is	O
a	O
normal	O
subgroup	O
of	O
E(n )	O
:	O
for	O
every	O
translation	O
t	O
and	O
every	O
isometry	O
u	O
,	O
the	O
composition	B-Application
is	O
again	O
a	O
translation	O
.	O
</s>
<s>
They	O
are	O
represented	O
as	O
a	O
translation	O
followed	O
by	O
a	O
rotation	O
,	O
rather	O
than	O
a	O
translation	O
followed	O
by	O
some	O
kind	O
of	O
reflection	B-Algorithm
(	O
in	O
dimensions	O
2	O
and	O
3	O
,	O
these	O
are	O
the	O
familiar	O
reflections	B-Algorithm
in	O
a	O
mirror	O
line	O
or	O
plane	O
,	O
which	O
may	O
be	O
taken	O
to	O
include	O
the	O
origin	O
,	O
or	O
in	O
3D	O
,	O
a	O
rotoreflection	B-Algorithm
)	O
.	O
</s>
<s>
Countably	O
infinite	O
groups	O
without	O
arbitrarily	O
small	O
translations	B-Algorithm
,	O
rotations	B-General_Concept
,	O
or	O
combinations	O
i.e.	O
,	O
for	O
every	O
point	O
the	O
set	O
of	O
images	O
under	O
the	O
isometries	O
is	O
topologically	O
discrete	O
(	O
e.g.	O
,	O
for	O
a	O
group	O
generated	O
by	O
m	O
translations	B-Algorithm
in	O
independent	O
directions	O
,	O
and	O
possibly	O
a	O
finite	O
point	O
group	O
)	O
.	O
</s>
<s>
Countably	O
infinite	O
groups	O
with	O
arbitrarily	O
small	O
translations	B-Algorithm
,	O
rotations	B-General_Concept
,	O
or	O
combinations	O
In	O
this	O
case	O
there	O
are	O
points	O
for	O
which	O
the	O
set	O
of	O
images	O
under	O
the	O
isometries	O
is	O
not	O
closed	O
.	O
</s>
<s>
Non-countable	O
groups	O
,	O
where	O
there	O
are	O
points	O
for	O
which	O
the	O
set	O
of	O
images	O
under	O
the	O
isometries	O
is	O
not	O
closed	O
(	O
e.g.	O
,	O
in	O
2D	O
all	O
translations	B-Algorithm
in	O
one	O
direction	O
,	O
and	O
all	O
translations	B-Algorithm
by	O
rational	O
distances	O
in	O
another	O
direction	O
)	O
.	O
</s>
<s>
all	O
isometries	O
which	O
are	O
a	O
combination	O
of	O
a	O
rotation	O
about	O
some	O
axis	O
and	O
a	O
proportional	O
translation	O
along	O
the	O
axis	O
;	O
in	O
general	O
this	O
is	O
combined	O
with	O
k-fold	O
rotational	O
isometries	O
about	O
the	O
same	O
axis	O
(	O
)	O
;	O
the	O
set	O
of	O
images	O
of	O
a	O
point	O
under	O
the	O
isometries	O
is	O
a	O
k-fold	O
helix	B-Application
;	O
in	O
addition	O
there	O
may	O
be	O
a	O
2-fold	O
rotation	O
about	O
a	O
perpendicularly	O
intersecting	O
axis	O
,	O
and	O
hence	O
a	O
k-fold	O
helix	B-Application
of	O
such	O
axes	O
.	O
</s>
<s>
for	O
any	O
point	O
group	O
:	O
the	O
group	O
of	O
all	O
isometries	O
which	O
are	O
a	O
combination	O
of	O
an	O
isometry	O
in	O
the	O
point	O
group	O
and	O
a	O
translation	O
;	O
for	O
example	O
,	O
in	O
the	O
case	O
of	O
the	O
group	O
generated	O
by	O
inversion	O
in	O
the	O
origin	O
:	O
the	O
group	O
of	O
all	O
translations	B-Algorithm
and	O
inversion	O
in	O
all	O
points	O
;	O
this	O
is	O
the	O
generalized	O
dihedral	B-Algorithm
group	I-Algorithm
of	O
R3	O
,	O
Dih(R3 )	O
.	O
</s>
<s>
See	O
also	O
3D	O
isometries	O
that	O
leave	O
the	O
origin	O
fixed	O
,	O
space	O
group	O
,	O
involution	B-Algorithm
.	O
</s>
<s>
For	O
some	O
isometry	O
pairs	O
composition	B-Application
does	O
not	O
depend	O
on	O
order	O
:	O
</s>
<s>
The	O
translations	B-Algorithm
by	O
a	O
given	O
distance	O
in	O
any	O
direction	O
form	O
a	O
conjugacy	O
class	O
;	O
the	O
translation	B-Algorithm
group	I-Algorithm
is	O
the	O
union	O
of	O
those	O
for	O
all	O
distances	O
.	O
</s>
<s>
In	O
1D	O
,	O
all	O
reflections	B-Algorithm
are	O
in	O
the	O
same	O
class	O
.	O
</s>
<s>
In	O
2D	O
,	O
rotations	B-General_Concept
by	O
the	O
same	O
angle	O
in	O
either	O
direction	O
are	O
in	O
the	O
same	O
class	O
.	O
</s>
<s>
Glide	B-Algorithm
reflections	I-Algorithm
with	O
translation	O
by	O
the	O
same	O
distance	O
are	O
in	O
the	O
same	O
class	O
.	O
</s>
<s>
Rotations	B-General_Concept
by	O
the	O
same	O
angle	O
are	O
in	O
the	O
same	O
class	O
.	O
</s>
<s>
Rotations	B-General_Concept
about	O
an	O
axis	O
combined	O
with	O
translation	O
along	O
that	O
axis	O
are	O
in	O
the	O
same	O
class	O
if	O
the	O
angle	O
is	O
the	O
same	O
and	O
the	O
translation	O
distance	O
is	O
the	O
same	O
.	O
</s>
<s>
Reflections	B-Algorithm
in	O
a	O
plane	O
combined	O
with	O
translation	O
in	O
that	O
plane	O
by	O
the	O
same	O
distance	O
are	O
in	O
the	O
same	O
class	O
.	O
</s>
<s>
Rotations	B-General_Concept
about	O
an	O
axis	O
by	O
the	O
same	O
angle	O
not	O
equal	O
to	O
180°	O
,	O
combined	O
with	O
reflection	B-Algorithm
in	O
a	O
plane	O
perpendicular	O
to	O
that	O
axis	O
,	O
are	O
in	O
the	O
same	O
class	O
.	O
</s>
