<s>
In	O
mathematics	O
,	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
(	O
also	O
called	O
Euclidean	B-Algorithm
transformation	I-Algorithm
or	O
Euclidean	B-Algorithm
isometry	I-Algorithm
)	O
is	O
a	O
geometric	B-Algorithm
transformation	I-Algorithm
of	O
a	O
Euclidean	O
space	O
that	O
preserves	O
the	O
Euclidean	O
distance	O
between	O
every	O
pair	O
of	O
points	O
.	O
</s>
<s>
The	O
rigid	B-Algorithm
transformations	I-Algorithm
include	O
rotations	B-General_Concept
,	O
translations	B-Algorithm
,	O
reflections	B-Algorithm
,	O
or	O
any	O
sequence	O
of	O
these	O
.	O
</s>
<s>
Reflections	B-Algorithm
are	O
sometimes	O
excluded	O
from	O
the	O
definition	O
of	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
by	O
requiring	O
that	O
the	O
transformation	O
also	O
preserve	O
the	O
handedness	O
of	O
objects	O
in	O
the	O
Euclidean	O
space	O
.	O
</s>
<s>
To	O
avoid	O
ambiguity	O
,	O
a	O
transformation	O
that	O
preserves	O
handedness	O
is	O
known	O
as	O
a	O
proper	O
rigid	B-Algorithm
transformation	I-Algorithm
,	O
or	O
rototranslation	B-Algorithm
.	O
</s>
<s>
Any	O
proper	O
rigid	B-Algorithm
transformation	I-Algorithm
can	O
be	O
decomposed	O
into	O
a	O
rotation	B-General_Concept
followed	O
by	O
a	O
translation	B-Algorithm
,	O
while	O
any	O
improper	O
rigid	B-Algorithm
transformation	I-Algorithm
can	O
be	O
decomposed	O
into	O
an	O
improper	B-Algorithm
rotation	I-Algorithm
followed	O
by	O
a	O
translation	B-Algorithm
,	O
or	O
into	O
a	O
sequence	O
of	O
reflections	B-Algorithm
.	O
</s>
<s>
Any	O
object	O
will	O
keep	O
the	O
same	O
shape	O
and	O
size	O
after	O
a	O
proper	O
rigid	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
All	O
rigid	B-Algorithm
transformations	I-Algorithm
are	O
examples	O
of	O
affine	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
The	O
set	O
of	O
all	O
(	O
proper	O
and	O
improper	O
)	O
rigid	B-Algorithm
transformations	I-Algorithm
is	O
a	O
mathematical	O
group	O
called	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
,	O
denoted	O
for	O
-dimensional	O
Euclidean	O
spaces	O
.	O
</s>
<s>
The	O
set	O
of	O
proper	O
rigid	B-Algorithm
transformations	I-Algorithm
is	O
called	O
special	B-Algorithm
Euclidean	I-Algorithm
group	I-Algorithm
,	O
denoted	O
.	O
</s>
<s>
In	O
kinematics	O
,	O
proper	O
rigid	B-Algorithm
transformations	I-Algorithm
in	O
a	O
3-dimensional	O
Euclidean	O
space	O
,	O
denoted	O
SE(3 )	O
,	O
are	O
used	O
to	O
represent	O
the	O
linear	O
and	O
angular	O
displacement	O
of	O
rigid	O
bodies	O
.	O
</s>
<s>
According	O
to	O
Chasles	O
 '	O
theorem	O
,	O
every	O
rigid	B-Algorithm
transformation	I-Algorithm
can	O
be	O
expressed	O
as	O
a	O
screw	O
displacement	O
.	O
</s>
<s>
where	O
(	O
i.e.	O
,	O
is	O
an	O
orthogonal	O
transformation	O
)	O
,	O
and	O
is	O
a	O
vector	O
giving	O
the	O
translation	B-Algorithm
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
proper	O
rigid	B-Algorithm
transformation	I-Algorithm
has	O
,	O
in	O
addition	O
,	O
</s>
<s>
which	O
means	O
that	O
R	O
does	O
not	O
produce	O
a	O
reflection	O
,	O
and	O
hence	O
it	O
represents	O
a	O
rotation	B-General_Concept
(	O
an	O
orientation-preserving	O
orthogonal	O
transformation	O
)	O
.	O
</s>
<s>
Indeed	O
,	O
when	O
an	O
orthogonal	O
transformation	B-Algorithm
matrix	I-Algorithm
produces	O
a	O
reflection	O
,	O
its	O
determinant	O
is	O
−1	O
.	O
</s>
<s>
Using	O
this	O
distance	O
formula	O
,	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
has	O
the	O
property	O
,	O
</s>
<s>
It	O
is	O
easy	O
to	O
show	O
that	O
this	O
is	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
by	O
showing	O
that	O
the	O
distance	O
between	O
translated	O
vectors	O
equal	O
the	O
distance	O
between	O
the	O
original	O
vectors	O
:	O
</s>
<s>
A	O
linear	O
transformation	O
is	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
if	O
it	O
satisfies	O
the	O
condition	O
,	O
</s>
<s>
Matrices	O
that	O
satisfy	O
this	O
condition	O
are	O
called	O
orthogonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Orthogonal	B-Algorithm
matrices	I-Algorithm
with	O
determinant	O
−1	O
are	O
reflections	B-Algorithm
,	O
and	O
those	O
with	O
determinant	O
+1	O
are	O
rotations	B-General_Concept
.	O
</s>
<s>
Notice	O
that	O
the	O
set	O
of	O
orthogonal	B-Algorithm
matrices	I-Algorithm
can	O
be	O
viewed	O
as	O
consisting	O
of	O
two	O
manifolds	O
in	O
separated	O
by	O
the	O
set	O
of	O
singular	O
matrices	O
.	O
</s>
<s>
The	O
set	O
of	O
rotation	B-General_Concept
matrices	O
is	O
called	O
the	O
special	O
orthogonal	O
group	O
,	O
and	O
denoted	O
.	O
</s>
