From: spamless@Nil.nil Subject: Re: Cylindrical data Date: 24 Aug 2000 13:12:16 -0400 Newsgroups: sci.math Summary: [missing] Mr Unreliable wrote: > Clearly there are 5 degrees of freedom in the positioning of the cylinder. > The axis might meet z=0 at (x1, y1). > Two of its three direction cosines might be (c1, c2). > And the radius or the cylinder might be r. > Hence, in general, the cylinder will be determined by 5 of its points. That follows from, say, the submersion theorem in differential geometry. Suppose a surface is determined uniquely by an equation of the form f(x,y,z,p1,p2,...,pn)=0. Let me use

for the array of p vaules. Plug in values for points on such a surface. f(x1,y1,z1,

)=0, f(x2,y2,z2,

)=0, etc. Let me write these m (if you have m points) equations for

(you know x1,y1,z1 and x2,y2,z2,...etc. and are trying to solve for

) as: f1(

)=0, f2(

)=0, etc. Can you "solve for

"? Consider the equations: f1(

)=k1 f2(

)=k2, ... etc (m equations) As you vary the paramters of

(say, off from the values which determine the surface on which the points lie), k1,k2,...,km can vary from zero. Assume is a solution. IF this matrix from pi to kj is of full rank m at p_0 then by the submersion theorem there are parameters l1,...,l_(n-m) where there is a diffeomorphism in a neighbourhood of to ) (there is locally a 1-1 may so you can switch from p coordinates to coordinates). IF m coordinates where we put k1=k2=...=0 and let l_j vary (coordinates are ). As these vary, the p's vary (since this is a 1-1 map between the coordinates and the

coordinates) so locally there are n-m coordinates we can vary (an n-m dimensional submanifold in the coordinates and so locally an n-m submanifold of p-values) (varying l) over which all the points are on such surfaces (the k's are zero). That is, you can vary p's "n-m" dimensionally (there are various surfaces) satisfying f1(

)=0, f2(

)=0,...fm(

)=0 (the k's are all zero). This would happen if you don't have enough points (if you can vary the cylinder and still have the given points on the varied cylinder). If the rank is n, however, there is a 1-1 map from

to and so, locally, there is a unique set of p values which give a given (say, =<0> for the points having to be on a cylinder). (If the rank is m, you can just use the inverse function theorem. The submersion theorem tells you in how many "dimensions" you can vary your surface with the given points remaining on the surface if you don't have enough points to determine the surface.) With a cylinder determined by five parameters (try to write down a function of five p1,p2,p3,p4,p5 parameters for a cylinder: f(x,y,z,

)=0 being the

cylinder), if you have five points chosen with the Jacobean of the map from

to given by f1(

)=k1, f2(

)=k2, etc. (f1 obtained by putting in the coordinates of the first point, etc.) non-singular, then at least locally there is just one cylinder passing through those points. Since

has five coordinates, as long as you don't choose points where the map from

to is singular, you can do it (but showing there is no other cylinder with other coordinates, rather than small perturbations of the found solution is a bit more - that is not a "local" problem). (Of course you could choose points where the map is singular, for example, if you choose all the points as the same point! Then the rank would be one and there would be a four paramater family of cylinders passing through the point.) ============================================================================== From: Boudewijn Moonen Subject: Re: local diffeo & RP^n Date: Thu, 10 Aug 2000 15:01:06 +0200 Newsgroups: sci.math Summary: [missing] Ed Hook wrote: > > In article <3991BBAE.1D63B7DF@math.nwu.edu>, lena writes: > > > |> Ed Hook wrote: > > |> > |> Actually, since RP^n is compact and hausdorff, the local diffeo is indeed a > |> > |> covering map. > > |> > How do you prove that ?? It's clear that > |> > the map is finite-to-one everywhere, but > |> > I can (I think) visualize examples where > |> > such a map is _not_ a covering map. So I'd > |> > be interested in your proof (so that I can > |> > figure out where my intuition is going > |> > astray.) > > |> Let X and Y be smooth, compact, hausdorff manifolds of the same dimension, and let > |> f:X --> Y be the local diffeo. > |> Pick y in Y; this is a regular value because of the local diffeo. This implies > |> that f inverse (f^-1(y)) of y is a zero dimensional compact manifold--hence it is > |> a finite, discrete set, say x1, x2,...,xk. > |> Now, pick non-intersecting neighborhoods around each of these points such that f > |> restricted to these neighborhoods is a a homeo; this is easily accomplished > |> because of the local diffeo and the "hausdorfness" at hand. You might also need > |> the fact that a finite intersection of open sets is still open. > > You _do_ need to use that fact > to get "evenly-covered" neighborhoods > of all points in the base space. (Just > take f(U_1) \cap f(U_2) \cap ... \cap f(U_k), > where the U_i are the above-mentioned > neighborhoods of the x_i -- that's a > neighborhood of y that's evenly-covered > by f ... ) > > |> This shows that f is a covering map. > > Yep. It does, indeed :-) > > Thanks. That was (more or less) > the way that I convinced myself > that the map had to be finite-to-one. > But I stumbled over some sort of > mental block before I could get > all the way to the end ... > To put the result into its proper context, it is a special case of the famous "Ehresmann Fibration Theorem": THEOREM(Ehresmann) A proper submersion f : M --> N of connected manifolds is a locally trivial fibration. I commented on the definitions and outlined the idea of proof in some former postings: http://forum.swarthmore.edu/epigone/sci.math/cranspendlul/39577E65.7309EB47@ipb.uni-bonn.de http://forum.swarthmore.edu/epigone/sci.math/cranspendlul/39586B36.E3214CFF@ipb.uni-bonn.de Of course, in the case at hand (N [hence M] compact, dim M = dim N), the proof given by lena is the appropriate one, I just wanted to point at the broader context this result belongs to. Regards, -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712