# 2D rotation matrix plus another 2D rotation matrix is a similarity matrix (scaled 2D rotation)

## weblog/

Consider we have two rotation matrices:
``````R1 = / cos(θ1) -sin(θ1) \
\ sin(θ1)  cos(θ1) /

R2 = / cos(θ2) -sin(θ2) \
\ sin(θ2)  cos(θ2) /
``````
Then we can show that:
``````R1 + R2 = s * R3
``````
where s is a scalar and R3 is another 2D rotation. First we just add the components to get:
``````R1 + R2 = / cos(θ1)+cos(θ2)   -(sin(θ1)+sin(θ2)) \
\ sin(θ1)+sin(θ2)    cos(θ1)+cos(θ2)   /
``````
Notice we already have the right pattern:
``````R1 + R2 =  / A -B \
\ B  A /
``````
we just need to show that we can pull out a common scaling term and angle from A and B. We can use trigonometric identities to turn the sums of cosines and the sums of sines into products, namely:
``````cos(θ1)+cos(θ2) = 2*cos((θ1+θ2)/2)*cos((θ1-θ2)/2)
sin(θ1)+sin(θ2) = 2*sin((θ1+θ2)/2)*cos((θ1-θ2)/2)
``````
Now we declare that:
``````s = 2*cos((θ1-θ2)/2)
θ3 = (θ1+θ2)/2
``````
Then it follows that:
``````R1 + R2
=
/ s*cos((θ1+θ2)/2)  -s*sin((θ1+θ2)/2) \
\ s*sin((θ1+θ2)/2)   s*cos((θ1+θ2)/2) /
=
s * / cos(θ3)  -sin(θ3) \
\ sin(θ3)   cos(θ3) /
=
s * R3
``````