   "Points at infinity:" Hyperbolic space?
  I'm looking for a nice way to represent and to talk about "points at infinity" in a particular sense.  I.e., I want to compactify R^n in a certain way, and I'd like to know what this compactification is called and if there's a convenient way to work with it.  I think it might be hyperbolic space, H^n, but there are a few things I haven't been able to figure out if that's the case.Not projective spaceIt's probably easiest to describe what I want by saying what I don't want:  Whereas real projective space identifies "opposite infinities" with one another, I would like them to be separate.Two constructions A constructionDisk methodHere's a construction for what I want:  Identify R^n with the open unit ball B^n in R^n via the diffeomorphism phi:B^n-->R^n defined,    1phi(p) = ------------- p .1 - ||p||^2Moreover, define the distance between two points byd(p1,p2) = ||phi(p1) - phi(p2)|| .What I'd like, now, is the closed ball with this metric.  (This requires allowing the metric to take values in the extended reals, but I think that's ok.)Homogeneous coordinatesAnother construction would use homogeneous coordinates.  However, whereas real projective space RP^n can be constructed in this way using the equivalence relation(x1,w1) ~ (x2,w2) <==> There exists a real number 'y' s.t.   (x1,w1) = y (x2, w2) ,what I want would require(x1,w1) ~ (x2,w2) <==> There exists a positive real number 'y' s.t.   (x1,w1) = y (x2, w2) .Hyperbolic space?All this is extremely similar to either the Klein model or the Poincare disk model for hyperbolic space, so I think this might be what I actually want.  My problem, if this is the case, is that in order for this to be useful I need an isometry from R^n with the Euclidean metric, to the Poisson/Klein disk with associated metric.  I.e., in order to compactify R^n, I need to know what the right way to map it to the disk is.I don't think that the answer is a conformal map, because these preserve angles, which I don't care about.  What I want is something that preserves distances -- i.e., an isometry.Questions- Is what I want actually hyperbolic space?  If so, what's the isometry from R^n to the open Poincare or Klein disk? (Or how can I figure this out?)- If what I want is not hyperbolic space, what is it?Thanks in advance everybody for any replies! :-)[Edited by - Emergent on May 27, 2010 12:18:10 PM]
  I am not sure what you want, because I think you described two different things. In your first construction, you get something of the same dimensionality than the space you start with, so if you start with R^1, you want to attach -inf and +inf to the straight line. What you get is topologically a closed segment. Now in your projective-like construction, you end up with something that is one dimension smaller than the initial R^n. So if you start with R^2 you end up with a circle.So the 1-dimensional space that you are describing is topologically a segment in the first definition, but topologically a circle in the second definition.If you want to define something that is isometric to R^n, it won't be a hyperbolic space, because R^n has zero curvature and the hyperbolic space has negative curvature.;Quote:Original post by alvaroif you start with R^1, you want to attach -inf and +inf to the straight line.Exactly.  Which would be the extended reals.  But I'd like something that works in more dimensions.Quote:Original post by alvaroNow in your projective-like construction, you end up with something that is one dimension smaller than the initial R^n.I guess I wasn't clear enough here.  When I wrote a typical point as (x,w), I meant that x was in R^n and that w in R was the "extra coordinate."  This put the dimensionality at n+1, and then the equivalence relation brought it back down to n.Quote:Original post by alvaroSo if you start with R^2 you end up with a circle.Unless I've made a mistake, it's not a circle since I've added the restriction that the multiplier needs to be positive.Quote:Original post by alvaroIf you want to define something that is isometric to R^n, it won't be a hyperbolic space, because R^n has zero curvature and the hyperbolic space has negative curvature.Oooh, good call...   ;Quote:Original post by EmergentQuote:Original post by alvaroSo if you start with R^2 you end up with a circle.Unless I've made a mistake, it's not a circle since I've added the restriction that the multiplier needs to be positive.Still a circle. Think about it.;Quote:Original post by alvaroStill a circle. Think about it.*sigh* Yeeeaaaah.... (thanks). *strikes from original post*I think the disk method is ok though...   ;  I don't see what you gain going back and forth between the extended R^n representation and the closed ball representation.You can just define your space as R^n together with one point at infinity for each "semi-direction" (defined as a class of vectors that are proportional to each other with a positive proportionality constant), and a "metric" that is the usual metric between points of R^n and +inf if one of the two points involved is at infinity (of course dist(P,P)=0 even if P is at infinity).I've seen a similar construction in a paper by Marco Brunella, except he replaced the origin with a sphere, using some form of blow-up. Your construction is equivalent to adding a single point at infinity (getting a hyper-sphere) and then replacing it with a sphere the same way that paper does. The metric properties weren't very important in the context of the paper, but the possibility of extending analytic structures was.What are you trying to do with this space?;Quote:Original post by alvaroYou can just define your space as R^n together with one point at infinity for each "semi-direction" (defined as a class of vectors that are proportional to each other with a positive proportionality constant), and a "metric" that is the usual metric between points of R^n and +inf if one of the two points involved is at infinity (of course dist(P,P)=0 even if P is at infinity).Yeah, that sounds equivalent.  I suppose another way to express what you wrote would be to define the space as --- EDITED ---(R^n x {1}) U (S^{n-1} x {2})which would contain two disjoint subsets, one on the (...,1) plane (the points), and another on the (...,2) plane (the semi-directions).--- EDITED ---The only difference would be that your "semi-directions" seem to include 0 whereas S^{n-1} doesn't (but this is an element of the set I don't need anyway).Why "semi"-direction, though?  Is "direction" reserved for lines rather than rays?Quote:What are you trying to do with this space?It has to do with an optimization problem which may or may not be bounded.  If it's not bounded, it still makes sense to ask what the "best direction" is -- and it'd be nice to be able to treat the bounded and unbounded cases in the same way simultaneously.[Edited by - Emergent on May 28, 2010 9:01:36 AM];Quote:Original post by Emergentwhich would contain both n-tuples (the points) and (n-1)-tuples (the semi-directions).  The only difference would be that your "semi-directions" seem to include 0 whereas S^{n-1} doesn't (but this is an element of the set I don't need anyway).Oh, yes. I forgot to remove 0, as is usually done in the construction of projective space.Quote:Why "semi"-direction, though?  Is "direction" reserved for lines rather than rays?Yes, at least in my usage. These conventions change from field to field and from country to country.Quote:Quote:What are you trying to do with this space?It has to do with an optimization problem which may or may not be bounded.  If it's not bounded, it still makes sense to ask what the "best direction" is -- and it'd be nice to be able to treat the bounded and unbounded cases in the same way simultaneously.Hmmm... That sounds reasonable. But why is the metric important?   ;Quote:Original post by alvaroHmmm... That sounds reasonable. But why is the metric important?All the quantities in the optimization problem are distances.  But I see how this doesn't necessarily matter... so maybe I should just treat it this way:Say you want to find    argmin_x f(x)    s.t.  g(x) >= 0and it turns out that the solution is unbounded.  Then what you might want to find instead is,    argmin_u lim_{t-->infty} f(t u)/||t u||    s.t.  lim_{t-->infty}g(t u)/||t u|| >= 0 .Suppose also that f(x), g(x) ~ O(||x||), so that this is defined.What would be sort of cool would be a way to treat these two cases simultaneously.My enthusiasm for this idea is waning a little bit though, as I begin to simply be happy with leaving these as separate problems, a la  try    solve first problem  catch(unbounded)    solve second problem .;  I think your idea of mapping to a ball is good. You basically convert the search space into a compact subset of R^n and that way you can treat both cases more uniformly. And I would not worry about the metric. Chances are the Euclidean metric in the original space is not all that relevant if picking parameters at infinity is feasible.   