|WikiProject Mathematics||(Rated Start-class, Low-importance)|
I signal an incoherence in terminology: the reflection article says a reflection has only one eigenvalue -1 (and all its examples adhere to this) so a "point reflection" is not a reflection (actually, in the plane I would consider it a rotation rather than a reflection). I think the proper term is "point symmetry" (which I just redirected here; it used to point to symmetry group for no apparent reason), and would suggest a corresponding page move. But I'm not particularly acquainted with English geometry literature, so I'll stand corrected if this is common terminology. However the reflection through the origin article does call the use of "reflection" an abuse of language. Marc van Leeuwen (talk) 15:40, 4 April 2010 (UTC)
- Point reflections do not fall under the framework for a reflection described in the reflection (mathematics) article, but it is nonetheless the common terminology for this transformation (see Google books for examples). "Point symmetry" refers to a slightly different concept, in the same way that reflection symmetry is different from reflection. Jim (talk) 16:02, 4 April 2010 (UTC)
The only meaningful distinctions I can see that could be made would be:
- affine vs. vector (reflection through any point vs. reflection through the origin);
- low dimensions (2D, 3D) for novices vs. arbitrary dimension (n-dimensions) for initiates.
For such a simple topic I think these topics can all effectively be covered in a single page, though the current page could use some work.
Point reflection as special case of uniform scaling or homothety
If I correctly understand the text of this article:
- when the point P coincides with the origin, point reflection is just a special case of uniform scaling: uniform scaling with scale factor equal to -1 (which is an example of linear transformation).
- when P does not coincide with the origin, point reflection is just a special case of homothetic transformation: homothety with homothetic center coinciding with P, and scale factor = -1 (an example of non-linear affine transformation).
An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric.
Is there a compact set that is symmetric about a point P that is not its "center"?