<s>
In	O
geometry	O
,	O
a	O
point	B-Algorithm
reflection	I-Algorithm
(	O
point	B-Algorithm
inversion	I-Algorithm
,	O
central	B-Algorithm
inversion	I-Algorithm
,	O
or	O
inversion	B-Algorithm
through	I-Algorithm
a	I-Algorithm
point	I-Algorithm
)	O
is	O
a	O
type	O
of	O
isometry	O
of	O
Euclidean	O
space	O
.	O
</s>
<s>
An	O
object	O
that	O
is	O
invariant	O
under	O
a	O
point	B-Algorithm
reflection	I-Algorithm
is	O
said	O
to	O
possess	O
point	B-Algorithm
symmetry	I-Algorithm
;	O
if	O
it	O
is	O
invariant	O
under	O
point	B-Algorithm
reflection	I-Algorithm
through	O
its	O
center	O
,	O
it	O
is	O
said	O
to	O
possess	O
central	B-Algorithm
symmetry	I-Algorithm
or	O
to	O
be	O
centrally	B-Algorithm
symmetric	I-Algorithm
.	O
</s>
<s>
Point	B-Algorithm
reflection	I-Algorithm
can	O
be	O
classified	O
as	O
an	O
affine	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
Namely	O
,	O
it	O
is	O
an	O
isometric	O
involutive	B-Algorithm
affine	B-Algorithm
transformation	I-Algorithm
,	O
which	O
has	O
exactly	O
one	O
fixed	O
point	O
,	O
which	O
is	O
the	O
point	O
of	O
inversion	O
.	O
</s>
<s>
It	O
is	O
equivalent	O
to	O
a	O
homothetic	B-Algorithm
transformation	I-Algorithm
with	O
scale	O
factor	O
equal	O
to	O
−1	O
.	O
</s>
<s>
The	O
term	O
reflection	B-Algorithm
is	O
loose	O
,	O
and	O
considered	O
by	O
some	O
an	O
abuse	O
of	O
language	O
,	O
with	O
inversion	O
preferred	O
;	O
however	O
,	O
point	B-Algorithm
reflection	I-Algorithm
is	O
widely	O
used	O
.	O
</s>
<s>
Such	O
maps	O
are	O
involutions	B-Algorithm
,	O
meaning	O
that	O
they	O
have	O
order	O
2	O
–	O
they	O
are	O
their	O
own	O
inverse	O
:	O
applying	O
them	O
twice	O
yields	O
the	O
identity	O
map	O
–	O
which	O
is	O
also	O
true	O
of	O
other	O
maps	O
called	O
reflections	O
.	O
</s>
<s>
More	O
narrowly	O
,	O
a	O
reflection	B-Algorithm
refers	O
to	O
a	O
reflection	B-Algorithm
in	O
a	O
hyperplane	O
(	O
dimensional	O
affine	O
subspace	O
–	O
a	O
point	O
on	O
the	O
line	O
,	O
a	O
line	O
in	O
the	O
plane	O
,	O
a	O
plane	O
in	O
3-space	O
)	O
,	O
with	O
the	O
hyperplane	O
being	O
fixed	O
,	O
but	O
more	O
broadly	O
reflection	B-Algorithm
is	O
applied	O
to	O
any	O
involution	B-Algorithm
of	O
Euclidean	O
space	O
,	O
and	O
the	O
fixed	O
set	O
(	O
an	O
affine	O
space	O
of	O
dimension	O
k	O
,	O
where	O
)	O
is	O
called	O
the	O
mirror	O
.	O
</s>
<s>
In	O
terms	O
of	O
linear	O
algebra	O
,	O
assuming	O
the	O
origin	O
is	O
fixed	O
,	O
involutions	B-Algorithm
are	O
exactly	O
the	O
diagonalizable	B-Algorithm
maps	O
with	O
all	O
eigenvalues	O
either	O
1	O
or	O
−1	O
.	O
</s>
<s>
Reflection	B-Algorithm
in	O
a	O
hyperplane	O
has	O
a	O
single	O
1	O
eigenvalue	O
(	O
and	O
multiplicity	O
on	O
the	O
1	O
eigenvalue	O
)	O
,	O
while	O
point	B-Algorithm
reflection	I-Algorithm
has	O
only	O
the	O
−1	O
eigenvalue	O
(	O
with	O
multiplicity	O
n	O
)	O
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
a	O
point	B-Algorithm
reflection	I-Algorithm
is	O
the	O
same	O
as	O
a	O
rotation	O
of	O
180	O
degrees	O
.	O
</s>
<s>
In	O
three	O
dimensions	O
,	O
a	O
point	B-Algorithm
reflection	I-Algorithm
can	O
be	O
described	O
as	O
a	O
180-degree	O
rotation	O
composed	O
with	O
reflection	B-Algorithm
across	O
a	O
plane	O
perpendicular	O
to	O
the	O
axis	O
of	O
rotation	O
.	O
</s>
<s>
In	O
dimension	O
n	O
,	O
point	B-Algorithm
reflections	I-Algorithm
are	O
orientation-preserving	O
if	O
n	O
is	O
even	O
,	O
and	O
orientation-reversing	O
if	O
n	O
is	O
odd	O
.	O
</s>
<s>
In	O
the	O
case	O
where	O
p	O
is	O
the	O
origin	O
,	O
point	B-Algorithm
reflection	I-Algorithm
is	O
simply	O
the	O
negation	O
of	O
the	O
vector	O
a	O
.	O
</s>
<s>
This	O
mapping	O
is	O
an	O
isometric	O
involutive	B-Algorithm
affine	B-Algorithm
transformation	I-Algorithm
which	O
has	O
exactly	O
one	O
fixed	O
point	O
,	O
which	O
is	O
P	O
.	O
</s>
<s>
When	O
the	O
inversion	O
point	O
P	O
coincides	O
with	O
the	O
origin	O
,	O
point	B-Algorithm
reflection	I-Algorithm
is	O
equivalent	O
to	O
a	O
special	O
case	O
of	O
uniform	B-Algorithm
scaling	I-Algorithm
:	O
uniform	B-Algorithm
scaling	I-Algorithm
with	O
scale	O
factor	O
equal	O
to	O
−1	O
.	O
</s>
<s>
This	O
is	O
an	O
example	O
of	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
When	O
P	O
does	O
not	O
coincide	O
with	O
the	O
origin	O
,	O
point	B-Algorithm
reflection	I-Algorithm
is	O
equivalent	O
to	O
a	O
special	O
case	O
of	O
homothetic	B-Algorithm
transformation	I-Algorithm
:	O
homothety	B-Algorithm
with	O
homothetic	O
center	O
coinciding	O
with	O
P	O
,	O
and	O
scale	O
factor	O
−1	O
.	O
</s>
<s>
(	O
This	O
is	O
an	O
example	O
of	O
non-linear	O
affine	B-Algorithm
transformation	I-Algorithm
.	O
)	O
</s>
<s>
The	O
composition	O
of	O
two	O
point	B-Algorithm
reflections	I-Algorithm
is	O
a	O
translation	B-Algorithm
.	O
</s>
<s>
Specifically	O
,	O
point	B-Algorithm
reflection	I-Algorithm
at	O
p	O
followed	O
by	O
point	B-Algorithm
reflection	I-Algorithm
at	O
q	O
is	O
translation	B-Algorithm
by	O
the	O
vector	O
2( q−	O
p	O
)	O
.	O
</s>
<s>
The	O
set	O
consisting	O
of	O
all	O
point	B-Algorithm
reflections	I-Algorithm
and	O
translations	O
is	O
Lie	O
subgroup	O
of	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
It	O
is	O
precisely	O
the	O
subgroup	O
of	O
the	O
Euclidean	B-Algorithm
group	I-Algorithm
that	O
fixes	O
the	O
line	O
at	O
infinity	O
pointwise	O
.	O
</s>
<s>
In	O
the	O
case	O
n	O
=	O
1	O
,	O
the	O
point	B-Algorithm
reflection	I-Algorithm
group	I-Algorithm
is	O
the	O
full	O
isometry	B-Algorithm
group	I-Algorithm
of	O
the	O
line	O
.	O
</s>
<s>
Point	B-Algorithm
reflection	I-Algorithm
across	O
the	O
center	O
of	O
a	O
sphere	O
yields	O
the	O
antipodal	O
map	O
.	O
</s>
<s>
A	O
symmetric	O
space	O
is	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
with	O
an	O
isometric	O
reflection	B-Algorithm
across	O
each	O
point	O
.	O
</s>
<s>
Given	O
the	O
point	O
and	O
its	O
reflection	B-Algorithm
with	O
respect	O
to	O
the	O
point	O
,	O
the	O
latter	O
is	O
the	O
midpoint	O
of	O
the	O
segment	O
;	O
</s>
<s>
In	O
even-dimensional	O
Euclidean	O
space	O
,	O
say	O
2N-dimensional	O
space	O
,	O
the	O
inversion	B-Algorithm
in	I-Algorithm
a	I-Algorithm
point	I-Algorithm
P	O
is	O
equivalent	O
to	O
N	O
rotations	O
over	O
angles	O
in	O
each	O
plane	O
of	O
an	O
arbitrary	O
set	O
of	O
N	O
mutually	O
orthogonal	O
planes	O
intersecting	O
at	O
P	O
.	O
These	O
rotations	O
are	O
mutually	O
commutative	O
.	O
</s>
<s>
Therefore	O
,	O
inversion	B-Algorithm
in	I-Algorithm
a	I-Algorithm
point	I-Algorithm
in	O
even-dimensional	O
space	O
is	O
an	O
orientation-preserving	O
isometry	O
or	O
direct	B-Algorithm
isometry	I-Algorithm
.	O
</s>
<s>
In	O
odd-dimensional	O
Euclidean	O
space	O
,	O
say	O
(	O
2N+1	O
)	O
-dimensional	O
space	O
,	O
it	O
is	O
equivalent	O
to	O
N	O
rotations	O
over	O
in	O
each	O
plane	O
of	O
an	O
arbitrary	O
set	O
of	O
N	O
mutually	O
orthogonal	O
planes	O
intersecting	O
at	O
P	O
,	O
combined	O
with	O
the	O
reflection	B-Algorithm
in	O
the	O
2N-dimensional	O
subspace	O
spanned	O
by	O
these	O
rotation	O
planes	O
.	O
</s>
<s>
Therefore	O
,	O
it	O
reverses	O
rather	O
than	O
preserves	O
orientation	O
,	O
it	O
is	O
an	O
indirect	B-Algorithm
isometry	I-Algorithm
.	O
</s>
<s>
Geometrically	O
in	O
3D	O
it	O
amounts	O
to	O
rotation	O
about	O
an	O
axis	O
through	O
P	O
by	O
an	O
angle	O
of	O
180°	O
,	O
combined	O
with	O
reflection	B-Algorithm
in	O
the	O
plane	O
through	O
P	O
which	O
is	O
perpendicular	O
to	O
the	O
axis	O
;	O
the	O
result	O
does	O
not	O
depend	O
on	O
the	O
orientation	O
(	O
in	O
the	O
other	O
sense	O
)	O
of	O
the	O
axis	O
.	O
</s>
<s>
The	O
following	O
point	B-Algorithm
groups	I-Algorithm
in	I-Algorithm
three	I-Algorithm
dimensions	I-Algorithm
contain	O
inversion	O
:	O
</s>
<s>
Closely	O
related	O
to	O
inverse	O
in	O
a	O
point	O
is	O
reflection	B-Algorithm
in	O
respect	O
to	O
a	O
plane	O
,	O
which	O
can	O
be	O
thought	O
of	O
as	O
a	O
"	O
inversion	O
in	O
a	O
plane	O
"	O
.	O
</s>
<s>
Molecules	O
contain	O
an	O
inversion	B-Algorithm
center	I-Algorithm
when	O
a	O
point	O
exists	O
through	O
which	O
all	O
atoms	O
can	O
reflect	O
while	O
retaining	O
symmetry	O
.	O
</s>
<s>
In	O
crystallography	O
,	O
the	O
presence	O
of	O
inversion	B-Algorithm
centers	I-Algorithm
distinguishes	O
between	O
centrosymmetric	O
and	O
noncentrosymmetric	O
compounds	O
.	O
</s>
<s>
Polyhedra	O
containing	O
inversion	B-Algorithm
centers	I-Algorithm
are	O
known	O
as	O
centrosymmetric	O
,	O
while	O
those	O
without	O
are	O
noncentrosymmetric	O
.	O
</s>
<s>
Six-coordinate	O
octahedra	O
are	O
an	O
example	O
of	O
centrosymmetric	O
polyhedra	O
,	O
as	O
the	O
central	O
atom	O
acts	O
as	O
an	O
inversion	B-Algorithm
center	I-Algorithm
through	O
which	O
the	O
six	O
bonded	O
atoms	O
retain	O
symmetry	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
that	O
bonding	O
geometries	O
with	O
odd	O
coordination	O
numbers	O
must	O
be	O
noncentrosymmetric	O
,	O
because	O
these	O
polyhedra	O
will	O
not	O
contain	O
inversion	B-Algorithm
centers	I-Algorithm
.	O
</s>
<s>
Disorder	O
can	O
influence	O
the	O
centrosymmetry	O
of	O
certain	O
polyhedra	O
as	O
well	O
,	O
depending	O
on	O
whether	O
or	O
not	O
the	O
occupancy	O
is	O
split	O
over	O
an	O
already-present	O
inversion	B-Algorithm
center	I-Algorithm
.	O
</s>
<s>
The	O
presence	O
of	O
noncentrosymmetric	O
polyhedra	O
does	O
not	O
guarantee	O
that	O
the	O
point	O
group	O
will	O
be	O
the	O
same	O
—	O
two	O
noncentrosymmetric	O
shapes	O
can	O
be	O
oriented	O
in	O
space	O
in	O
a	O
manner	O
which	O
contains	O
an	O
inversion	B-Algorithm
center	I-Algorithm
between	O
the	O
two	O
.	O
</s>
<s>
Two	O
tetrahedra	O
facing	O
each	O
other	O
can	O
have	O
an	O
inversion	B-Algorithm
center	I-Algorithm
in	O
the	O
middle	O
,	O
because	O
the	O
orientation	O
allows	O
for	O
each	O
atom	O
to	O
have	O
a	O
reflected	O
pair	O
.	O
</s>
<s>
The	O
lack	O
of	O
symmetry	O
via	O
inversion	B-Algorithm
centers	I-Algorithm
can	O
allow	O
for	O
areas	O
of	O
the	O
crystal	O
to	O
interact	O
differently	O
with	O
incoming	O
light	O
.	O
</s>
<s>
The	O
applications	O
for	O
nonlinear	O
materials	O
are	O
still	O
being	O
researched	O
,	O
but	O
these	O
properties	O
stem	O
from	O
the	O
presence	O
of	O
(	O
or	O
lack	O
thereof	O
)	O
an	O
inversion	B-Algorithm
center	I-Algorithm
.	O
</s>
<s>
The	O
operation	O
commutes	O
with	O
every	O
other	O
linear	B-Architecture
transformation	I-Architecture
,	O
but	O
not	O
with	O
translation	B-Algorithm
:	O
it	O
is	O
in	O
the	O
center	O
of	O
the	O
general	O
linear	O
group	O
.	O
</s>
<s>
"	O
Inversion	O
"	O
without	O
indicating	O
"	O
in	O
a	O
point	O
"	O
,	O
"	O
in	O
a	O
line	O
"	O
or	O
"	O
in	O
a	O
plane	O
"	O
,	O
means	O
this	O
inversion	O
;	O
in	O
physics	O
3-dimensional	O
reflection	B-Algorithm
through	O
the	O
origin	O
is	O
also	O
called	O
a	O
parity	O
transformation	O
.	O
</s>
<s>
In	O
mathematics	O
,	O
reflection	B-Algorithm
through	O
the	O
origin	O
refers	O
to	O
the	O
point	B-Algorithm
reflection	I-Algorithm
of	O
Euclidean	O
space	O
Rn	O
across	O
the	O
origin	O
of	O
the	O
Cartesian	O
coordinate	O
system	O
.	O
</s>
<s>
Reflection	B-Algorithm
through	O
the	O
origin	O
is	O
an	O
orthogonal	O
transformation	O
corresponding	O
to	O
scalar	O
multiplication	O
by	O
,	O
and	O
can	O
also	O
be	O
written	O
as	O
,	O
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
It	O
is	O
a	O
product	O
of	O
n	O
orthogonal	O
reflections	O
(	O
reflection	B-Algorithm
through	O
the	O
axes	O
of	O
any	O
orthogonal	B-Algorithm
basis	I-Algorithm
)	O
;	O
note	O
that	O
orthogonal	O
reflections	O
commute	O
.	O
</s>
<s>
It	O
is	O
the	O
longest	O
element	O
of	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
of	O
signed	O
permutations	O
.	O
</s>
<s>
Analogously	O
,	O
it	O
is	O
a	O
longest	O
element	O
of	O
the	O
orthogonal	O
group	O
,	O
with	O
respect	O
to	O
the	O
generating	O
set	O
of	O
reflections	O
:	O
elements	O
of	O
the	O
orthogonal	O
group	O
all	O
have	O
length	O
at	O
most	O
n	O
with	O
respect	O
to	O
the	O
generating	O
set	O
of	O
reflections	O
,	O
and	O
reflection	B-Algorithm
through	O
the	O
origin	O
has	O
length	O
n	O
,	O
though	O
it	O
is	O
not	O
unique	O
in	O
this	O
:	O
other	O
maximal	O
combinations	O
of	O
rotations	O
(	O
and	O
possibly	O
reflections	O
)	O
also	O
have	O
maximal	O
length	O
.	O
</s>
<s>
In	O
SO(2r )	O
,	O
reflection	B-Algorithm
through	O
the	O
origin	O
is	O
the	O
farthest	O
point	O
from	O
the	O
identity	O
element	O
with	O
respect	O
to	O
the	O
usual	O
metric	O
.	O
</s>
<s>
In	O
O( 2r	O
+	O
1	O
)	O
,	O
reflection	B-Algorithm
through	O
the	O
origin	O
is	O
not	O
in	O
SO( 2r+1	O
)	O
(	O
it	O
is	O
in	O
the	O
non-identity	O
component	O
)	O
,	O
and	O
there	O
is	O
no	O
natural	O
sense	O
in	O
which	O
it	O
is	O
a	O
"	O
farther	O
point	O
"	O
than	O
any	O
other	O
point	O
in	O
the	O
non-identity	O
component	O
,	O
but	O
it	O
does	O
provide	O
a	O
base	O
point	O
in	O
the	O
other	O
component	O
.	O
</s>
<s>
Reflection	B-Algorithm
through	O
the	O
identity	O
extends	O
to	O
an	O
automorphism	O
of	O
a	O
Clifford	O
algebra	O
,	O
called	O
the	O
main	O
involution	B-Algorithm
or	O
grade	O
involution	B-Algorithm
.	O
</s>
<s>
Reflection	B-Algorithm
through	O
the	O
identity	O
lifts	O
to	O
a	O
pseudoscalar	O
.	O
</s>
