<s>
In	O
Euclidean	O
geometry	O
,	O
two-dimensional	O
rotations	B-General_Concept
and	O
reflections	O
are	O
two	O
kinds	O
of	O
Euclidean	O
plane	O
isometries	O
which	O
are	O
related	O
to	O
one	O
another	O
.	O
</s>
<s>
A	O
pair	O
of	O
rotations	B-General_Concept
about	O
the	O
same	O
point	O
will	O
be	O
equivalent	O
to	O
another	O
rotation	O
about	O
point	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
composition	O
of	O
a	O
reflection	B-Algorithm
and	O
a	O
rotation	O
,	O
or	O
of	O
a	O
rotation	O
and	O
a	O
reflection	B-Algorithm
(	O
composition	O
is	O
not	O
commutative	O
)	O
,	O
will	O
be	O
equivalent	O
to	O
a	O
reflection	B-Algorithm
.	O
</s>
<s>
Let	O
a	O
reflection	B-Algorithm
about	O
a	O
line	O
through	O
the	O
origin	O
which	O
makes	O
an	O
angle	O
with	O
the	O
-axis	O
be	O
denoted	O
as	O
.	O
</s>
<s>
Let	O
these	O
rotations	B-General_Concept
and	O
reflections	O
operate	O
on	O
all	O
points	O
on	O
the	O
plane	O
,	O
and	O
let	O
these	O
points	O
be	O
represented	O
by	O
position	O
vectors	O
.	O
</s>
<s>
Then	O
a	O
rotation	O
can	O
be	O
represented	O
as	O
a	O
matrix	B-Architecture
,	O
</s>
<s>
and	O
likewise	O
for	O
a	O
reflection	B-Algorithm
,	O
</s>
<s>
With	O
these	O
definitions	O
of	O
coordinate	B-General_Concept
rotation	I-General_Concept
and	O
reflection	B-Algorithm
,	O
the	O
following	O
four	O
identities	O
hold	O
:	O
</s>
<s>
These	O
equations	O
can	O
be	O
proved	O
through	O
straightforward	O
matrix	B-Architecture
multiplication	O
and	O
application	O
of	O
trigonometric	O
identities	O
,	O
specifically	O
the	O
sum	O
and	O
difference	O
identities	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
reflections	O
in	O
lines	O
through	O
the	O
origin	O
and	O
rotations	B-General_Concept
about	O
the	O
origin	O
,	O
together	O
with	O
the	O
operation	O
of	O
composition	O
of	O
reflections	O
and	O
rotations	B-General_Concept
,	O
forms	O
a	O
group	O
.	O
</s>
<s>
Every	O
reflection	B-Algorithm
is	O
its	O
own	O
inverse	O
.	O
</s>
<s>
Composition	O
has	O
closure	O
and	O
is	O
associative	O
,	O
since	O
matrix	B-Architecture
multiplication	O
is	O
associative	O
.	O
</s>
<s>
Notice	O
that	O
both	O
and	O
have	O
been	O
represented	O
with	O
orthogonal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Rotation	O
matrices	O
have	O
a	O
determinant	O
of+1	O
,	O
and	O
reflection	B-Algorithm
matrices	O
have	O
a	O
determinant	O
of−1	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
orthogonal	O
two-dimensional	O
matrices	O
together	O
with	O
matrix	B-Architecture
multiplication	O
form	O
the	O
orthogonal	O
group	O
:	O
.	O
</s>
<s>
The	O
following	O
table	O
gives	O
examples	O
of	O
rotation	O
and	O
reflection	B-Algorithm
matrix	B-Architecture
:	O
</s>
