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Given $$\cos(\omega)a_x-\sin(\omega)a_y+b_x=c_x$$ and $$\sin(\omega)a_x+\cos(\omega)a_y+b_y=c_y$$ we have that $$a_x^2 + a_y^2 - b_x^2 + 2b_xc_x - b_y^2 + 2b_yc_y - c_x^2 - c_y^2=0$$ Notice that $\omega$ does not appear in this equation.

If we have two more point correspondences of the form that $(a_x,a_y)$ maps to $(c_x,c_y)$ with a rotation depending on $\omega$ and a translation given by $(b_x,b_y)$, this equation is useful to calculate $(b_x,b_y)$ independently of $\omega$.

Is there a compact way to prove this equation? A CAS library produced it for me.

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I'm not sure what you mean by "two more point correspondences", but if I understand correctly, you're just trying to prove the third equation given the first two. Take the $b$'s to the right-hand side, square the equations, add them, use $\cos^2\omega+\sin^2\omega=1$ and take everything to the left-hand side.

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