<s>
A	O
reflection	O
is	O
an	O
involution	B-Algorithm
:	O
when	O
applied	O
twice	O
in	O
succession	O
,	O
every	O
point	O
returns	O
to	O
its	O
original	O
location	O
,	O
and	O
every	O
geometrical	O
object	O
is	O
restored	O
to	O
its	O
original	O
state	O
.	O
</s>
<s>
The	O
term	O
reflection	O
is	O
sometimes	O
used	O
for	O
a	O
larger	O
class	O
of	O
mappings	O
from	O
a	O
Euclidean	O
space	O
to	O
itself	O
,	O
namely	O
the	O
non-identity	O
isometries	O
that	O
are	O
involutions	B-Algorithm
.	O
</s>
<s>
would	O
look	O
like	O
a	O
d	O
.	O
This	O
operation	O
is	O
also	O
known	O
as	O
a	O
central	B-Algorithm
inversion	I-Algorithm
,	O
and	O
exhibits	O
Euclidean	O
space	O
as	O
a	O
symmetric	O
space	O
.	O
</s>
<s>
Some	O
mathematicians	O
use	O
"	O
flip	B-Algorithm
"	O
as	O
a	O
synonym	O
for	O
"	O
reflection	O
"	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
for	O
a	O
reflection	O
is	O
orthogonal	B-Algorithm
with	O
determinant	O
−1	O
and	O
eigenvalues	O
−1	O
,	O
1	O
,	O
1	O
,	O
...	O
,	O
1	O
.	O
</s>
<s>
The	O
product	O
of	O
two	O
such	O
matrices	O
is	O
a	O
special	B-Algorithm
orthogonal	I-Algorithm
matrix	I-Algorithm
that	O
represents	O
a	O
rotation	B-General_Concept
.	O
</s>
<s>
Every	O
rotation	B-General_Concept
is	O
the	O
result	O
of	O
reflecting	O
in	O
an	O
even	O
number	O
of	O
reflections	O
in	O
hyperplanes	O
through	O
the	O
origin	O
,	O
and	O
every	O
improper	B-Algorithm
rotation	I-Algorithm
is	O
the	O
result	O
of	O
reflecting	O
in	O
an	O
odd	O
number	O
.	O
</s>
<s>
Thus	O
reflections	O
generate	O
the	O
orthogonal	B-Algorithm
group	O
,	O
and	O
this	O
result	O
is	O
known	O
as	O
the	O
Cartan	O
–	O
Dieudonné	O
theorem	O
.	O
</s>
<s>
Similarly	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
,	O
which	O
consists	O
of	O
all	O
isometries	O
of	O
Euclidean	O
space	O
,	O
is	O
generated	O
by	O
reflections	O
in	O
affine	O
hyperplanes	O
.	O
</s>
<s>
In	O
general	O
,	O
a	O
group	O
generated	O
by	O
reflections	O
in	O
affine	O
hyperplanes	O
is	O
known	O
as	O
a	O
reflection	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
finite	O
groups	O
generated	O
in	O
this	O
way	O
are	O
examples	O
of	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
saying	O
that	O
a	O
reflection	O
of	O
across	O
is	O
equal	O
to	O
2	O
times	O
the	O
projection	B-Algorithm
of	O
on	O
,	O
minus	O
the	O
vector	O
.	O
</s>
<s>
Note	O
that	O
the	O
second	O
term	O
in	O
the	O
above	O
equation	O
is	O
just	O
twice	O
the	O
vector	B-Algorithm
projection	I-Algorithm
of	O
onto	O
.	O
</s>
<s>
Since	O
these	O
reflections	O
are	O
isometries	O
of	O
Euclidean	O
space	O
fixing	O
the	O
origin	O
they	O
may	O
be	O
represented	O
by	O
orthogonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
where	O
denotes	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
and	O
is	O
the	O
transpose	O
of	O
a	O
.	O
</s>
