show that there is one point whose co-ordinates do not alter due to a rigid motion

Question

I have chosen a point $A(x,y)$.
And let the origin be shifted to $(a,b)$
Now $A(x+a,y+b)$ If the rotation is $\theta$,

$$x= X\cos\theta - Y\sin\theta$$ $$y= X\sin\theta + Y\cos\theta$$

Now I wonder what to do.

Answer 1

We need to show that exist $x$ fixed point such that

$$x=Rx+x_0 \implies(I-R)x=x_0$$

and thus that

• for $x_0=0 \implies x=0$ (no translation)

or

• for $x_0\neq 0 \quad (I-R)$ is invertible and $x=(I-R)^{-1}x_0$