mirror image of mirror rotating around out-of-plane point

Hi group,

more a geometry question than a pure optics question:
Suppose you have a mirror rotating around a point on its surface, the
mirror image of a point is obtained trivially: It rotates with 2 times
the angle of the mirror and simply traces a circle around the axis of
rotation.

Things are slightly more complicated in practice: my system uses a
scanning mirror [1] that rotates around a point about 4 mm below the
surface. I am trying to figure out what type of curve a mirrored point
traces when I rotate the mirror, but I am still a bit puzzled. Before
I start doing the dirty math of rotation, mirror and translation
operations, does anyone have an intuitive way to solve this problem? I
have a gut-feeling that it might be a non-circular path...

Note that the mirror is part of some interferometric device, so I am
not only sensitive to tilt but also to piston. I need to know the
exact point. The rotation angles involved are pretty small, so I guess
that in the end I can approximate any solution with a new circle. The
mirror is used with 45 degrees of incidence.

Thanks,
Bas

[1] http://www.physikinstrumente.com/en/products/prdetail.php?sortnr=300710
.

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