<s>
Thus	O
the	O
conjugacy	O
class	O
within	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
E(n )	O
of	O
a	O
translation	O
is	O
the	O
set	O
of	O
all	O
translations	O
by	O
the	O
same	O
distance	O
.	O
</s>
<s>
The	O
smallest	O
subgroup	O
of	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
containing	O
all	O
translations	O
by	O
a	O
given	O
distance	O
is	O
the	O
set	O
of	O
all	O
translations	O
.	O
</s>
<s>
Thus	O
there	O
is	O
a	O
partition	O
of	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
with	O
in	O
each	O
subset	O
one	O
isometries	O
that	O
keeps	O
the	O
origins	O
fixed	O
,	O
and	O
its	O
combination	O
with	O
all	O
translations	O
.	O
</s>
<s>
Each	O
isometry	O
is	O
given	O
by	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
A	O
in	O
O(n )	O
and	O
a	O
vector	O
b	O
:	O
</s>
<s>
The	O
conjugate	O
of	O
the	O
inversion	B-Algorithm
in	I-Algorithm
a	I-Algorithm
point	I-Algorithm
by	O
a	O
translation	O
is	O
the	O
inversion	O
in	O
the	O
translated	O
point	O
,	O
etc	O
.	O
</s>
<s>
Thus	O
the	O
conjugacy	O
class	O
within	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
E(n )	O
of	O
inversion	B-Algorithm
in	I-Algorithm
a	I-Algorithm
point	I-Algorithm
is	O
the	O
set	O
of	O
inversions	O
in	O
all	O
points	O
.	O
</s>
<s>
Since	O
a	O
combination	O
of	O
two	O
inversions	O
is	O
a	O
translation	O
,	O
the	O
conjugate	O
closure	O
of	O
a	O
singleton	O
containing	O
inversion	B-Algorithm
in	I-Algorithm
a	I-Algorithm
point	I-Algorithm
is	O
the	O
set	O
of	O
all	O
translations	O
and	O
the	O
inversions	O
in	O
all	O
points	O
.	O
</s>
<s>
This	O
is	O
the	O
generalized	O
dihedral	B-Algorithm
group	I-Algorithm
dih	O
(	O
Rn	O
)	O
.	O
</s>
<s>
The	O
conjugacy	O
class	O
within	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
E(3 )	O
of	O
a	O
rotation	O
about	O
an	O
axis	O
is	O
a	O
rotation	O
by	O
the	O
same	O
angle	O
about	O
any	O
axis	O
.	O
</s>
<s>
Two	O
isometry	O
groups	O
are	O
said	O
to	O
be	O
equal	O
up	O
to	O
conjugacy	O
with	O
respect	O
to	O
affine	B-Algorithm
transformations	I-Algorithm
if	O
there	O
is	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
such	O
that	O
all	O
elements	O
of	O
one	O
group	O
are	O
obtained	O
by	O
taking	O
the	O
conjugates	O
by	O
that	O
affine	B-Algorithm
transformation	I-Algorithm
of	O
all	O
elements	O
of	O
the	O
other	O
group	O
.	O
</s>
<s>
Note	O
however	O
that	O
the	O
conjugate	O
with	O
respect	O
to	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
of	O
an	O
isometry	O
is	O
in	O
general	O
not	O
an	O
isometry	O
,	O
although	O
volume	O
(	O
in	O
2D	O
:	O
area	O
)	O
and	O
orientation	O
are	O
preserved	O
.	O
</s>
