Rigid motions in the Poincaré disk

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The diagram below shows an example of a rigid motion in the Poincaré disk generated by reflection through the green arc from \(X\) to \(Y\). We can see the effect of this reflection on the blue hyperbolic line connecting \(A\) and \(B\), which reflects to the blue line connecting \(A'\) and \(B'\), and the orange hyperbolic circle containing \(Q\) and centred at \(P\), which reflects to a hyperbolic circle centred at \(P'\) and containing \(Q'\).

You can move the points \(X\) and \(Y\) to change the green arc (and hence change the rigid motion). You can also move the points \(A\) and \(B\) to change the blue hyperbolic line, and move the points \(P\) and \(Q\) to change the orange hyperbolic circle.