In Coxeter's "Introduction to Geometry", I was reading the section about Glide Reflections (page 44, section 3.3), which states
Every opposite isometry with no invariant point is a glide reflection.
I see no issues with the proof, however I seem to have a "counter example" of a reflection followed by a rotation not centred on the line of reflection, specifically:
Consider a line l and a point O not on l, then let S be the isometry of reflection through l and then a rotation of any non-zero angle about O (call it 30 degrees to be concrete)
As this is a product of an opposite and direct isometry, it is an opposite isometry.
There can't be a fixed point:
- Any point on l is fixed by the reflection and as the center of rotation O is not on l, it unfixes all of these points, so none of the points on l are fixed by S
- For any point not on l, a rotation that mapped its reflection through l back to itself must have be centred at l, but O is not centred at l. Hence none of the points not on l are fixed by S either.
So S is an opposite isometry with no fixed points, but it is evidently not a glide reflection as a rotation is not a translation, so we have a "contradiction".
I would like to know where exactly my reasoning fails and why.
Actually S will be a glide reflection, although its axis of reflection will not be parallel to l. For an example see https://www.geogebra.org/geometry/gqx3a7kx . The quadrilateral $CDEF$ is reflected in the line $AB$ to give $C'D'E'F'$. This is then rotated 30 deg. clockwise about point $O$ to give $C''D''E''F''$. I have then reflected this last one in the second straight line giving a translate of our original polygon.