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