An Interactive Figure illustrating Conjugation

You can drag the points in this figure and click (once) on the Hide and Show buttons. Try dragging points on the circles to change the angles of rotation.

1. In the figure is shown a point P that is rotated by angle s around B to get P'.
2. Then this whole figure is rotated by angle s around A, sending B to C, P to Q and P' to Q'.
3. The relationship between Q and Q' can be expressed two ways. On the one hand, Q' is obtained by rotating Q by angle s and center C. On the other, one can travel from Q back to P by A_(-s) then to P' by B_t and then to Q' by A_s.
4. This shows that C_t = A_s B_t A_(-s). The point J traces out this path. If you drag J to be close to Q, you can see that the image J''' gets close to Q', J' gets close to P and Q'' gets close to P'. From the triangle CJJ''' you can see that that J is rotated by angle s with center C to get to J''' directly.

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