|WikiProject Mathematics||(Rated Start-class, High-importance)|
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 by . Then
- 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)
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)
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. 220.127.116.11 (talk) 22:01, 22 November 2013 (UTC) (PS: The first two words of the rigid body page are "In physics...")
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)
- 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)
- The map RR defined by 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)
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)
- Turns out, I wasn't quite right with the final statement; see Intrinsic metric. - Mike Rosoft (talk) 11:45, 9 June 2010 (UTC)
- 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)
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)
- 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)
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)
- 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)
- 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)