From: Ronald Bruck Subject: Re: References on classifications of isometries in R^n??? Date: Thu, 23 Nov 2000 08:03:13 -0800 Newsgroups: sci.math Summary: [missing] In article <8vhcn6$iia$1@nntp.itservices.ubc.ca>, israel@math.ubc.ca (Robert Israel) wrote: :In article <8vgshe$5ng$1@panix6.panix.com>, :George Baloglou wrote: :>I am writing a short note on an "elementary" way of classifying :>isometries :>of R^3, and I would like to know of other proofs, beyond the one found :>in, :>say, George Martin's "Geometric Transformations" (a great book, by the :>way). : :>Moreover, and even though that was not my main goal, I am now exploring :>the field beyond n = 3 a bit... I am sure that isometries of R^n have :>been :>classified decades ago, but, in addition to the result itself, I would be :>interested in any "constructive/intuitive" ways you might be aware of. : :Any isometry of R^n is of the form x -> a + T x where a is a constant :vector and T is an orthogonal matrix. Is that what you're looking for? :In fact, an isometry of any strictly convex linear space is an affine map. :The proof is as follows... [Proof omitted] Banach proved that an isometry of ANY Banach space is an affine map. It's not hard--look in Day's little book on normed linear spaces. --Ron Bruck -- Due to University fiscal constraints, .sigs may not be exceed one line. ============================================================================== From: renfrod@central.edu (Dave L. Renfro) Subject: Re: References on classifications of isometries in R^n??? Date: 23 Nov 2000 10:50:46 -0500 Newsgroups: sci.math George Baloglou [sci.math 22 Nov 2000 11:33:50 -0500] wrote > I am writing a short note on an "elementary" way of classifying > isometries of R^3, and I would like to know of other proofs, > beyond the one found in, say, George Martin's "Geometric > Transformations" (a great book, by the way). > > Moreover, and even though that was not my main goal, I am now > exploring the field beyond n = 3 a bit... I am sure that > isometries of R^n have been classified decades ago, but, in > addition to the result itself, I would be interested in any > "constructive/intuitive" ways you might be aware of. Patrick Morandi (New Mexico State University) has a nice 7 page essay titled "Isometries of R^n)" [150 K .pdf file; 26 K .dvi file] that you might want to look at. It's the 6'th essay from the bottom at . Dave L. Renfro