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An affine transformation is a linear transformation followed by a translation. They are morphism between affine spaces.

A rigid transformation consists of a rotation and a translation. I was wondering what are the spaces between which the rigid transformations can become morphisms?

what are the spaces between which the rotation transformations can become morphisms?

Same question for similarity transformation and projective transformation.

Thanks and regards!

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translations do not preserve the inner product. I am not sure exactly what you're asking. Are you asking whether rigid transformations preserve some structure in addition to the inner product (and whatever comes from it)? – Qiaochu Yuan Sep 21 '10 at 14:25
Sorry. Made a mistake. Now making changes to the original post. – Tim Sep 21 '10 at 15:12
up vote 1 down vote accepted

If by "rotation" you mean "orientation-preserving isometry of $\mathbb{R}^n$ fixing the origin," then

  • the spaces between which rigid transformations are morphisms are the "affine oriented (finite-dimensional) inner product spaces" over $\mathbb{R}$, by which I mean torsors over a finite-dimensional oriented inner product space over $\mathbb{R}$,
  • the spaces between which rotations are morphisms are the oriented finite-dimensional inner product spaces over $\mathbb{R}$, and
  • the spaces between which projective transformations are morphisms are the projective spaces over $\mathbb{R}$.

I cannot off the top of my head think of a good name for the spaces between which similarities are morphisms. I think you are looking for "conformal affine spaces," e.g. torsors over $\mathbb{R}^n$ equipped with a notion of oriented angle (but not the full inner product).

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The truth of yur statement of course depends on what you mean by structure :) Is everything preserved by motions definable in terms of/determined by the metric and the orientation? – Mariano Suárez-Alvarez Sep 21 '10 at 14:36

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