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MathGroup Archive 2006

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Re: need mathematica's help for exploring a certain type of mapping

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68578] Re: need mathematica's help for exploring a certain type of mapping
  • From: "Nabeel Butt" <nabeel.butt at gmail.com>
  • Date: Wed, 9 Aug 2006 23:57:40 -0400 (EDT)
  • References: <200608090820.EAA21373@smc.vnet.net> <NDBBJGNHKLMPLILOIPPOMELCFBAA.djmp@earthlink.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Dear David,
             Keep the metric Eucilidean(easy to visualise in mathematica).
    Also,i want a map that preserves unit distances but not necessarily all
distances.
   It may happen that a mapping from R^2->R^3 that preserves unit distances
preserves all distances and hence,is an isometry.
   It is proven in mathematical literature that a unit preserving map from
R^3->R^2 is an isometry.
   However, for f:R^2->R^3 this is still an open question.

         regards,
             Nabeel
On 8/9/06, David Park <djmp at earthlink.net> wrote:
>
> Nabeel,
>
> I thought that the definition of an isometry was that it preserved
> distances.
>
> A mapping that did a 3D rotation of the xy-plane, plus translations in 3D
> space, plus reflections in a plane would cover the isometries if we use
> the
> Euclidean metric in both 2D and 3D.
>
> But suppose you wanted to map onto a specific curved surface. Then are you
> going to allow a different distance measuring function on the surface, a
> different metric? Are we allowed to design a metric for any given mapping?
> Is it possible to have an isometry then? I don't know.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
>
> From: Nabeel Butt [mailto:nabeel.butt at gmail.com]
To: mathgroup at smc.vnet.net
>
>
> Dear Users,
>               I need to use mathematica's graphics to explore a certain
> kind of problem.The following theorem is not yet proven nor disproven and
> mathematica might proof  useful in disproving it though.
> Hypothesis:If a mapping from R^2->R^3 is unit distance preserving then it
> must be an isometry.
>     The real issue at hand is for mathematica to generate a mapping that
> preserves unit distance but is not an isometry so in the process
> disproving
> the theorem.
>     The real problem is that R^2 consists of infinite points and it might
> not be possible to check all of them.What i suggest is that you apply the
> unit preserving maps to special type of figures in R^2 like the
> circumfrence
> of the circle,square,isoceles triangle etc.
>     Any ideas are welcome.Thanks in advance.
>         regards,
>           Nabeel
>
> --
> Nabeel Butt
> LUMS,Lahore
>
>
>


-- 
Nabeel Butt
LUMS,Lahore



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