If you rotate a 2-dim. figure around two different center points with the same angle, you need an extra translation to move the images into the same position.
For simplicity, we assume that one center is the origin, and the other is $c=(c_x, c_y, 1)^T$ (homogeneous coordinates in projective space). Let’s determine the equations to convert the rotation $R$ around $c$ into a rotation around the origin $O=(0,0, 1)^T$, followed by a translation $t=(t_x, t_y, 1)^T$.