From: Ed Gerck Newsgroups: sci.math Subject: Grassmann Geometry Date: Tue, 07 Apr 1998 23:15:31 -0600 I am looking for sites that carry technical information on Grassmann's Geometry, what he called Ausdehnungslehre (extension principle). Which is not very much used today, even though it provides a very powerful abstraction. For example, where people still today compare a plane segment to a lake surface, Grassmann 150 years ago already thought of it as a river... Not finding much material, and needing it as a reference for modelling the technical concept of trust in communication systems (such as in cyptographic certificates X.509 or PGP), I have prepared a very short summary for a general audience, based directly on Grassmann's original works in German, which can be seen at (long URL): http://mcg.org.br/cgi-bin/lwg-mcg/MCG-TALK/archives/mcg/date/article-411.html and in the discussion threads. The discussion would benefit from other sources on Grassmann`s Geometry besides his own books (which are very good but use old terminology), possibly with figures, application examples, etc. Help is appreciated on that. (BTW, not Grassmann's Algebra... not Clifford's Algebra.. but Geometry) Thanks, Ed Gerck _____________________________________________________________________ Dr.rer.nat. E. Gerck egerck@novaware.cps.softex.br http://novaware.cps.softex.br --- Visit the Meta-Certificate Group at http://www.mcg.org.br --- -----== Posted via Deja News, The Leader in Internet Discussion ==----- http://www.dejanews.com/ Now offering spam-free web-based newsreading ============================================================================== From: Steven T. Smith Newsgroups: sci.math Subject: Re: Grassmann Geometry Date: 08 Apr 1998 11:57:26 GMT Ed Gerck writes: > I am looking for sites that carry technical information on Grassmann's > Geometry, what he called Ausdehnungslehre (extension principle). Which is > not very much used today, even though it provides a very powerful abstraction. > For example, where people still today compare a plane segment to a lake > surface, Grassmann 150 years ago already thought of it as a river... See: http://www-math.mit.edu/~edelman ftp://theory.lcs.mit.edu/pub/people/edelman/scicom/grass.ps ============================================================================== From: Chris Hillman Newsgroups: sci.math Subject: Re: Grassmann Geometry Date: Wed, 8 Apr 1998 02:20:25 -0700 On Tue, 7 Apr 1998, Ed Gerck wrote: > I am looking for sites that carry technical information on Grassmann's > Geometry, what he called Ausdehnungslehre (extension principle). Desmond Fearnley-Sander (Math, University of Tasmania, Hobart, Australia) has a web site http://www.maths.utas.edu.au/People/dfs/dfs.html which is sure to interest you :-) Ladnor Geissinger (Math, Univ. of North Carolina) posted something on Grasmmann geometry to the Geometry Forum; see http://forum.swarthmore.edu/epigone/geometry-college/penberdquim You might also look at the page of Hans Havlicek (Inst. fur Geometrie, Vienna): http://picasso.geometrie.tuwien.ac.at/havlicek/proj302.html Of course, many more mathematicians are working with Grassmann algebras (multivectors, exterior product, etc.), and this is certainly part of the socalled Grassmann geometry. Random examples: Neil Sloane (AT&T Bell Labs): http://www.research.att.com/~njas/grass/index.html Bernd Sturmfels (Math, Cornell): http://www.can.nl/SAC_Newsletter/Invariant.html Frank Sottile (Math, Univ. of Toronto): http://www.math.toronto.edu/~sottile/ Kequan Ding (IAS, Princeton): http://dimacs.rutgers.edu/People/Postdocs/Ding.html Patrick Coulton (Math, Eastern Illinois): http://www.ux1.eiu.edu/~cfprc/vita/vc.html I have myself used Grassmann algebras in my work on "generalized Penrose tilings" or "Sturmian systems". I have (re?)discovered some simple but useful tricks concerning angles between Pluecker lines in Grassmann algebras. In particular, if you naively put the obvious euclidean inner product on the subspace of p-multivectors, and look at the operators induced on that subspace by operators on the original vector space, you find that isometries induces isometries and the orthoprojection on the subspace W = span{v_1... v_p} induces the orthoprojection on the Pluecker line, the one dimensional subspace P(W) spanned by the simple multivector v_1 ^ v_2 ^ ... v_p This means that (for instance) you can talk about Euler angles for p-flats, although if p > 1 there will be relations (Pluecker relations) among the corresponding direction cosines. This sounds like the sort of thing which might well have been known (in different language) to Grassmann himself, in any case it is probably folklore in mathematics but perhaps little known outside of math. > The discussion would benefit from other sources on Grassmann`s Geometry > besides his own books (which are very good but use old terminology), possibly > with figures, application examples, etc. Help is appreciated on that. Coquereaux is a physicist who has published a great deal on applications in physics of Kleinian, projective, and Grassmann geometries, e.g. try the review paper: Differential and Integral Geometry of Grassmann Algebras (R Coquereaux, A Jadczyk & D Kastler), Reviews in Mathematical Physics vol 3, no. 1 Also, there seem to be lots of applications in robotics and computer vision, e.g.: Singular Configurations of Parallel Manipulators and Grassmann Geometry M.G. Mohamed, J. Duffy An Application of Grassmann Geometry to a Problem in Robotics (I have no further information on this one, but you might be able to find it by searching on the title in MathSci net). Prof. Olivier Faugeras (Electrical Eng, MIT) taught a course in 1995 on Grassmann-Cayley algebras in robot vision. Hope this helps, Chris Hillman