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

• To: mathgroup at smc.vnet.net
• Subject: [mg68576] RE: need mathematica's help for exploring a certain type of mapping
• From: "David Park" <djmp at earthlink.net>
• Date: Wed, 9 Aug 2006 23:57:37 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

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