Talk:Isometry

Early thread

Is f:x->|x| an path isometry from R->R? f(1)-f(-1)=0 but 1-(-1)=2.

Right, but that's not the length of the curve. I assume you're talking about defining

\gamma :[-1,1]\to \mathbb {R}

by

\gamma (t)=t

. Then

{\mbox{Length}}(\gamma )=\int _{-1}^{1}\left|\,{d\gamma  \over dt}\,\right|dt=\int _{-1}^{1}|1|dt=2

But

{\mbox{Length}}(f\circ \gamma )=\int _{-1}^{0}\left|\,{d(f\circ \gamma ) \over dt}\,\right|dt+\int _{0}^{1}\left|\,{d(f\circ \gamma ) \over dt}\,\right|dt=\int _{-1}^{0}|-1|dt+\int _{0}^{1}|1|dt=2

So that's not a counterexample.

IMHO, the second paragraph should be moved to much later in the article. It's definitely way too involved to be part of the initial summary.--Paul 17:07, September 11, 2005 (UTC)

Recent deletion

The example I just deleted does not belong either in this article or even in isometry group, it rather belongs in Euclidean group and there only. It is way too specific to provide any useful illustration anywhere else. A good example in the right article is invaluable, a good example in the wrong article is useless ranting distracting form the point of the article. Oleg Alexandrov (talk) 04:05, 9 October 2005 (UTC)[reply]

Rigid Motion or Translation + Rotation?

I see the term "rigid motion" used on this page repeatedly, but when I click the link for the definition it takes me to the page for "rigid body". I glean that "rigid motion" as used here is just another word for translation and/or rotation. I propose using those terms instead. One reason I propose this is that, although used by physicists, this is clearly a general mathematics concept. There needn't actually be any kind of literal movement involved, and referring to "motion" without reference to precise meaning suggests to me we are restricting the concept of isometry to contexts involving positions and times. Alternatively, a nice mathematically general definition for "rigid motion" could be provided, so as to make it clear we are not making such a restriction. Thanks. 24.233.151.201 (talk) 22:01, 22 November 2013 (UTC) (PS: The first two words of the rigid body page are "In physics...")[reply]

Equivalence Class or Quotient Set?

I clicked through quotient set because I was not familiar with the term. I was expecting something similar to quotient group, but it redirects equivalence class. I have a hard time believing quotient set is the wikipedia standard. Surely equivalence class is more common?? (This may be a US-centric view). --Jpawloski 14:37, 16 February 2006 (UTC)[reply]

Okay, well, never mind. Next time I'll read the page first. I will continue, however, to post without thinking. --Jpawloski 14:38, 16 February 2006 (UTC)[reply]

Huh?

The map R $\to$ R defined by $x\mapsto |x|$ is a path isometry but not a global isometry.

Am I missing something? The map x->abs(x) is not an isometry (under Euclidean metric); it doesn't preserve the distance between 1 and -1. (It isn't even injective.) - Mike Rosoft (talk) 11:12, 9 June 2010 (UTC)[reply]

Removed the example completely. (It's been there for quite a bit of time; see this revision.) As for "path isometry", note that length of a curve isn't defined in a general metric space. - Mike Rosoft (talk) 11:23, 9 June 2010 (UTC)[reply]

Turns out, I wasn't quite right with the final statement; see Intrinsic metric. - Mike Rosoft (talk) 11:45, 9 June 2010 (UTC)[reply]
- Yeah, should the example really be removed? Assuming it's correct, which it seems to be offhand, it seems useful. I hadn't seen the concept of "path isometry" before. Sniffnoy (talk) 16:38, 9 June 2010 (UTC)[reply]
Well, the key question is: how, precisely, is path isometry defined? But x->abs(x) obviously isn't an isometry. (Can a function be a path isometry without being an isometry?) - Mike Rosoft (talk) 18:04, 9 June 2010 (UTC)[reply]
- Evidently so - that's precisely what it was serving as an example of! Path isometry is pretty clearly defined in the article; it's a map that preserves lengths of paths. Sniffnoy (talk) 04:11, 10 June 2010 (UTC)[reply]

I have tried to clarify things with consistency between lead and definitions section, and restored the example in modified form.--Patrick (talk) 08:07, 10 June 2010 (UTC)[reply]

Further question: is path isometry defined on a general metric space, or just on R (or R^n)? Or, more to the point: how (if at all) do you define the length of a curve on a general metric space? - Mike Rosoft (talk) 15:53, 10 June 2010 (UTC)[reply]

Yes, length of a curve is defined in general metric spaces. See Arc length#Definition or Curve#Lengths of curves. Sniffnoy (talk) 21:10, 10 June 2010 (UTC)[reply]

Okay, but the fact remains that earlier in the article, "isometry" is said to be automatically injective. The example is a valid path isometry, however it needs to be stated that not all path isometries are injective. The sequencing of the article makes it seem as though both global and path isometries are subsets of isometries, but the latter clearly isn't true; instead, we have that every isometry (global or not) is a path isometry, but not the converse.Jtabbsvt (talk) 11:47, 10 July 2013 (UTC)[reply]

I'm not seeing what the problem is here? The article does state -- twice -- that "path isometry" is a weaker notion than "isometry". If you still think this is misleading and it should explicitly state that path isometries need not be injective, well, go ahead and add that. I don't think it needs a whole resequencing. Perhaps I'll go and add that right after doing this. Sniffnoy (talk) 22:05, 10 July 2013 (UTC)[reply]

- Understood, but the real problem is that in the beginning of the definitions, we're told that there are two varieties: global and path. Aren't there three: global, "plain ol'," and path? Or at the very least, two, with one being "plain ol'," of which global is a subset, and the other being path? Yes, path is weaker than global, but the missing distinction is that it is also weaker than "plain ol'." When I get to a laptop and not my phone, I'll edit. The rest of the problems spring from that. It wasn't until I came to the talk page that I fully understood what was meant, and that seems counterproductive. No offense to the author(s), who perhaps simply have a better intuition already in their minds. Jtabbsvt (talk) 02:15, 11 July 2013 (UTC)[reply]

- - Oh, OK. I guess that makes sense then. Sniffnoy (talk) 06:39, 12 July 2013 (UTC)[reply]

Isometric scaling paragraph in lead

The addition of a paragraph in the lead to an unrelated topic (Allometry, aka isometric scaling) in response to an IP's insertion of a mention thereof seems strange and out-of-place to me. At most, a hatnote could be devoted to this. —Quondum 19:37, 2 February 2014 (UTC)[reply]

You're right, it felt that way to me as well ... as I put it in. The hatnote was getting too crowded and I didn't want to just remove the statement, even though it was incorrect. I replaced it with a correct statement, but it really doesn't belong. Let's just put this down to too much late night editing. Bill Cherowitzo (talk) 06:08, 3 February 2014 (UTC)[reply]

That still leaves the question of what to do with it. My inclination is towards an even more crowded hatnote, if "isometry" is used sufficiently in that context that interested readers may end up here; otherwise to simply remove it. A further (and not unreasonable) alternative is to move all the hatnotes to a disambiguation page, leaving a hatnote pointing to that page. —Quondum 17:24, 3 February 2014 (UTC)[reply]

Done but it was a little more complicated (more than one disambiguation page) and may require some adjustment. Bill Cherowitzo (talk) 18:24, 3 February 2014 (UTC)[reply]

Wow, yes, I see that the disambiguation pages are a little confusing, but at least this page is essentially sorted now. —Quondum 03:57, 4 February 2014 (UTC)[reply]

What is the relationship between Isometric and "angle and distance preserving"?

I've seen a lot in physics about "angle and distance preserving" transformations (when talking about metric spaces). It would be nice to see something explaining the relationship between concepts such as isometry and that.216.96.76.37 (talk) 13:59, 28 May 2015 (UTC)[reply]

An isometry is precisely a distance-preserving function. On the other hand, it is not immediately clear how to define angles in a general metric space. (Angles between what entities?) JoergenB (talk) 23:48, 23 August 2015 (UTC)[reply]

The metric defines a dot-product between vectors in the tangent manifold. Whenever you got a dot product, you can write the angle as the arc-cosine of the dot product of two unit length vectors. And you know what unit length is, because the dot product tells you that. This article should be expanded to explain such things. 67.198.37.16 (talk) 02:34, 10 November 2020 (UTC)[reply]

Citations

Does this article really need sources? I think that most of it's pretty open source/common knowledge. — Preceding unsigned comment added by Schuddeboomw (talk • contribs) 14:30, 10 February 2016 (UTC)[reply]

Yes, absolutely, it does. As the comment above immediately shows, what is "common knowledge" to you is totally opaque to others. Anyway, this article is sorely deficient in even the most basic details as to what isometries are. It needs to be expanded. 67.198.37.16 (talk) 02:36, 10 November 2020 (UTC)[reply]

Is C^n always assumed to have the Euclidean norm?

In the last example (maps on C^n), shouldn't C^n be "C^n with the Euclidean norm"? Or does Wikipedia say somewhere that C^n is always assumed to have that norm? Vaughan Pratt (talk) 16:43, 2 December 2021 (UTC)[reply]

C^n is not always assumed to have any norm, but, when metric properties are considered, the norm defined by the dot product is generally implicitly implied, if not otherwise specified. On the other hand, I am not sure that the term "Euclidean norm" is common for complex vector spaces. Nevertheless, I have clarified the example. D.Lazard (talk) 19:10, 2 December 2021 (UTC)[reply]