Talk:Euclidean plane isometry

Support article for Wallpaper groups
I created this article as a support article for Wallpaper groups, which is about to get a major makeover. Dmharvey Talk 16:50, 4 Jun 2005 (UTC)

Please the discussion page for Wallpaper groups for more of my thoughts about what needs to be done here. Dmharvey Talk 17:03, 4 Jun 2005 (UTC)

More to do list: Dmharvey Talk 17:16, 4 Jun 2005 (UTC)
 * Would be nice to have a discussion of fixed points, orientation reversing/preserving transformations here.
 * Also would be nice to discuss how to prove this classification. Off the top of my head, I suppose given any isometry, you first compose with a translation so that the isometry has a fixed point; then translate the fixed point to the origin; now you are faced with classifying orthogonal transformations of R^2, which falls into two cases depending on whether the determinant is +1 or -1.
 * Would be nice to have a discussion of the compositions of isometries, e.g. composition of two reflections gives you a rotation, etc.

Genie here
Your wish is my command! The construction from mirrors is powerful, and provides a geometric path to many algebraic results, including composition, groups, and subgroups. My taste would place it before the coordinate approach.

However, every subsection of the mirror section cries out for illustrations. Interactive graphics would be better still. KSmrq 17:51, 11 Jun 2005 (UTC)

nice pic
nice one with the mirror composites picture. Do you think it would work better if it was split up into four separate images so we could label each one with "translation", "rotation" etc? Dmharvey Talk 9 July 2005 12:38 (UTC)


 * Thanks. I couldn't find a standard symbol that conveys the action of a glide reflection; a dotted line leaves everything to the imagination, which is unsatisfactory in this context. So I improvised. My favorite pic is the translation addition; it's pretty how the head-to-tail rule falls out. When I originally started playing with mirrors, inspired by some old geometers like Coxeter, my examples were all in 3D. It's fun to adapt the translation example, and quite an exercise to show that four mirrors in general position can be rearranged into canonical screw position. It leads to appreciation of the power of an algebraic approach, but also exercises a weak mental muscle, geometric imagination.


 * I ran into some Wikimedia limitations, both with arrangement and captions. When an image is not tagged with "frame", the associated text is treated as alternative text, not a caption. When an image is tagged with "frame", the text becomes a caption, but the sizing information is ignored. When made a readable size, the images stack too tall. To achieve the 2×2 arrangement, a single image is the easiest route. The four transforms are a logical whole, so that's not so bad. Maybe I'll find better ways, or maybe the support software will evolve. Did I mention time sink? ;-) KSmrq 2005 July 9 18:35 (UTC)


 * there have been a few discussions recently about bugs with image resizing on Village pump (technical). Not sure if that's your problem though. Dmharvey [[Image:User_dmharvey_sig.png]] Talk 00:47, 10 July 2005 (UTC)


 * No. This is a documented feature (?!). Using the "thumb" option does give a frame and resizing, but turns the transparent background opaque, so it looks terrible. (Unframed resizing does not do that; see example right.) Some other day I may tinker more. For now, notice that the list on the left gives a key: db=reflection, dp=rotation, dd=translation, dq=glide reflection. KSmrq 08:55, 2005 July 10 (UTC)

Ugly, confusing ambiguity
"Isometries as reflections" suggests that an isometry can be seen as a reflection. This is a really ugly, confusing ambiguity. I have changed it to "Isometries as combinations of reflections", which is saying what the section is about. I am puzzled by KSmrq's reversion on the grounds that he thinks it is pedantic, too long, and not needed because the section content itself says what was meant.--Patrick 08:14, 24 August 2005 (UTC)


 * Nice NPOV talk heading — not! :-)
 * In the grand spirit of Wikipedia, I suggest an article section title to meet both our concerns:
 * "Isometries as reflection group".
 * This is reasonably unambiguous, impeccably correct (see below), and links in an important connected article. It's a wee bit longer, a touch less poetic, and just a skosh pedantic. However, alternatives like "Isometries as combinations of reflections" are really ugly and only attack confusing ambiguity if we assume readers are darn dim-witted. What about the standard term "reflection group" itself? Unlike a rotation group (only rotations) and an orthogonal group (only orthogonal transformations), a reflection group is only generated by reflections, but actually contains inversions and rotations and other such odds and ends. Whoever promised mathematics would be consistent and unambiguous in its choice of names and notation?


 * One problem I see is that the reflection group article needs revision because it claims a reflection group acts on a vector space, which is too restrictive. What we want is an affine reflection group, a possibility only implicitly acknowledged in the last sentence of the article. A finite reflection group on a real affine space can contain no translations; but we obviously need the translations. I can't find my copy of Coxeter, but the MIT Press "Fundamentals of Mathematics" (ISBN 0-262-52094-X) has an extended description of geometry as reflections. (Perhaps you can find a copy of H. S. M. Coxeter's "Discrete groups generated by reflections", Ann. of Math., 35:588–621, 1934.)


 * I still prefer "Isometries as reflections"; but for the sake of not perpetuating a revert war I'll compromise. Will you? --KSmrqT 14:07, 2005 August 24 (UTC)


 * Yes, that title is better.--Patrick 20:28, 24 August 2005 (UTC)

glide reflect = reflect + translate. —Preceding unsigned comment added by Wecl0me12 (talk • contribs) 23:20, 8 January 2009 (UTC)

Reflection in a parallel line?

 * "Reflection in a parallel line corresponds to adding a vector perpendicular to it."

Maybe I've misunderstood something here, but wouldn't reflection in a parallel line result in no change? For example, taking theta to be 0 is said to give a reflection in the x-axis, and applying this transformation to (1,0) gives (cos(0),sin(0)) = (1,0). Dependent Variable (talk) 08:10, 30 November 2009 (UTC)

Missing information
There is no discussion of point reflections under reflections, which is remiss considering it's an isometry as well, being one of the two reflection isometries. Also, why do we count glide reflections separately when they're just a composition of reflection through a line and a translation? — Preceding unsigned comment added by 46.162.83.16 (talk) 08:11, 28 May 2020 (UTC)


 * Point reflection in the plane is simply a rotation by 180°, there does not seem to be a need for a separate discussion. It is convenient to have a separate name for glide reflections. After all, rotations and translations as well as glide reflections are just compositions of (at most three) reflections, so why bother to give them separate names? --Bill Cherowitzo (talk) 19:23, 28 May 2020 (UTC)


 * As Bill Cherowitzo says, reflections in a point are just a special case of rotations. The fact that we count glide reflections separately is due to the fundamental theorem on plane isometries, which states:

"Any plane isometry is the composition of at most three reflections in a line."
 * Denoting the reflection in the line $$l$$ by $$\sigma_l$$ and composition of transformations with juxtaposition from left to right, any isometry $$\phi$$ can be written as
 * $$\phi=\operatorname{id},\sigma_l,\sigma_l\sigma_k,\sigma_m\sigma_l\sigma_k$$
 * if $$l$$ is parallel to $$k$$, $$\sigma_l\sigma_k$$ is a translation, if $$l$$ intersects $$k$$, then $$\sigma_l\sigma_k$$ is a rotation (in particular, if the two lines are orthogonal, it's a reflection in a point); if the three lines are concurrent in a point or parallel to each other $$\sigma_m\sigma_l\sigma_k$$ is a reflection in a line, otherwise $$\sigma_m\sigma_l\sigma_k$$ is a glide reflection, which ends the classification. Reflections in a point might be mentioned, though, when (if?) the topic of fixed lines is addressed, in fact the fixed lines of reflections in a point are exactly all the lines passing through the fixed point of the isometry; also, they are a particular case of homothety.--Ale.rossi91 (talk) 02:48, 17 October 2020 (UTC)

About the composition of rotations
Was Euler wrong when he claimed that the composition of three rotations is a rotation? Not at all. It can easily be shown (explicit calculation!) that this is true. No need to have points in common, in contrast to the statement in this article. — Preceding unsigned comment added by MaDiehl (talk • contribs) 19:27, 6 July 2020 (UTC)


 * Could you please quote the sentence you're referring to? I could only find this one concerning the composition of rotations:

"Composition with translation produces another rotation (by the same amount, with shifted fixed point), but composition with rotation can yield either translation or rotation. It is often said that composition of two rotations produces a rotation, and Euler proved a theorem to that effect in 3D; however, this is only true for rotations sharing a fixed point."
 * The statement concerns two rotations, not three. Of course the composition of three rotations always results in a rotation because the composition of two rotations with distinct centres either yields a rotation or a translation and the composition of a rotation followed by a translation and viceversa always yields a rotation.--Ale.rossi91 (talk) 02:12, 17 October 2020 (UTC)

Very bad
written very bad — Preceding unsigned comment added by 89.139.63.123 (talk) 08:18, 12 August 2020 (UTC)
 * Ironic? Klbrain (talk) 07:19, 27 July 2021 (UTC)