Talk:Point reflection

Reflection?

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)[reply]

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)[reply]

Merge

I’ve just merged inversion in a point and reflection through the origin (the latter of which I wrote, not knowing of this page) to this page, as they cover the same topic.

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.

—Nils von Barth (nbarth) (talk) 09:28, 14 April 2010 (UTC)[reply]

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).

I think this should be mentioned in the article. I'll propose an edit. Feel free to improve. Paolo.dL (talk) 17:12, 23 February 2012 (UTC)[reply]

Novel distinction?

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"?

Apparently the word "center" does not have a mathematical definition, only an everyday language one. 76.118.180.76 (talk) 15:22, 17 December 2015 (UTC)[reply]

Terminology

Mainly to @Jacobolus, a question is what should the focus of this article be. If it is (as the first sentence implies) just about geometry then the whole section on crystals/molecules does not belong, and also many of the current links and redirects are probably incorrect. If, instead, it is more general then the "in geometry" should be changed to be more general, and the lead.

I strongly favor being inclusive. There is a fair bit of math etc in symmetry, see for instance Massimo's paper here. I was intending to add some of this (maybe drag Massimo in) and other such as spin, commutation in QM etc, but we should discuss focus here first. Obviously I am strongly in favor of an inclusive approach. Ldm1954 (talk) 19:35, 20 October 2024 (UTC)[reply]

@Ldm1954 I would consider crystallography to be more or less a branch of geometry, or at least substantially overlapping with and built on part of geometry. We don't necessarily need to lead with "In geometry, ...", but a point reflection is most certainly a geometric transformation (I wish that article was better as an overview). –jacobolus (t) 20:07, 20 October 2024 (UTC)[reply]

Aspects of crystallography certainly overlap with geometry, but I think few card-carrying crystallographers would agree that it is a branch of geometry (myself included). There are definitely many components of modern crystallography that go way beyond geometry.

I don't think we are that far apart -- have to go to an event now.

N.B., the page crystallography is also not great. Ldm1954 (talk) 20:35, 20 October 2024 (UTC)[reply]

Fair enough. I'm by no means a crystallographer, so I'll defer to you on that one.

When I notice and have feel motivated sometimes, I've been trying to clean up the lead sections of geometric transformation articles such as Glide reflection and Translation (geometry) to be more grammatically coherent and clearer to non-experts. These could still use work (in the lead sections, and more generally throughout), as could Reflection (mathematics), Improper rotation, Scaling (geometry), Homothety, Affine transformation, Homography, Motion (geometry), Isometry, etc. We don't even have an article about screw motions: Screw displacement redirects to Chasles' theorem (kinematics) while Screw motion redirects to Screw theory; in my opinion we should make a Screw motion article and merge Screw axis into it (maybe best accomplished by moving Screw axis), and also describe screw symmetry, redirecting Screw symmetry (currently a red link!) and Helical symmetry (currently a redirect to Symmetry (geometry) § Helical symmetry). Circle inversion should perhaps its own article instead of a redirect to Inversive geometry, even though there is substantial overlap between these topics. Möbius transformation is unfortunately described on Wikipedia as a fractional linear transformation of Complex numbers rather than as a circle-preserving geometric transformation generated from circle inversions which can be conveniently modeled by the isomorphic group of fractional linear transformations of complex numbers when the base space is the Euclidean plane (with one point at infinity)). And so on.

It would be nice if Geometric transformation and Transformation geometry were both significantly improved from their current state. If you or anyone else is interested, it would be nice to come up with a clearer plan to collect good sources and make some kind of clear outline of relevant topics and figure out both inter- and intra-article organization. –jacobolus (t) 21:53, 20 October 2024 (UTC)[reply]