# Maths - clifford / Geometric Algebra - Discussion with Fred Lunnon

 By: Nobody/Anonymous - nobody Geometric (Clifford) Algebra   2005-11-09 21:13 In a previous posting elsewhere (sci.math.research?) in 2003,   Martin Baker enquired about the possible application of GA   methods to affine transformations and dynamics in 3-D   Euclidean space, a topic which I happen to have begun to   study at around the same time.     There he made a remark which I think worth examining, to the   effect that devoting 16 components to represent an isometry   seemed an inefficient use of resources. This articulates an   attitude I have encountered frequently among my own   students, who are frequently unable to come to terms even   with using 4 rather than 3 components to represent a point.     I wonder why this mysterious parsimony is so widespread.   It would not occur to most of these individuals to agonise   over whether their vectors were represented as arrays or as   doubly-linked lists (using 3 times the storage), yet the notion   of expending extra components in the cause of simplifying   drastically their algorithmic design seems to arouse in them   a positively moral outrage.     The fact is that many of the elementary difficulties expressed   in postings here and elsewhere regarding 3-D geometry  become trivial if attacked with a suitably powerful notation. Furthermore, in the event that speed or storage are   significant concerns, it is usually possible to streamline   the resulting GA algorithm by specialising multivectors back   to vectors, matrices, etc.     A typical example is the representation and composition   of general helical (screw) isometries about arbitrary skew axes   in space. Because they can (usually) manage to avoid having   to tackle such situations head-on, programmers working in   geometric computation universally wallow in a morass of   partial solutions more appropriate to hand computation,   attended by a predictable plague of bugs resulting from the   attempt to interface them.     Unfortunately, there does not at present appear to be a   treatment of GA methods in a form which programmers   (rather than mathematicians) might be expected to find   digestible and applicable. But rest assured, I'm on the case ...     Fred Lunnon
 By: Martin Baker - martinbaker RE: Geometric (Clifford) Algebra   2005-11-10 02:32 Hi Fred,    I apologise for my parsimony (well I looked it up and a simplified definition of parsimony is "a principle that states that the simplest explanation that explains the greatest number of observations is preferred to more complex explanations" which seems like a good principle to me so perhaps I don't apologise). However I like to think that I am quite prepared to overthrow established way of doing things if necessary.    Anyway I'm in danger of becoming flippant which I don't mean to be, I share your belief that GA would be a very powerful an general way to represent affine transformations and therefore solid body dynamics in 3D, however although the mathematics is extremely powerful its general application seems just out of reach. I would welcome any thoughts you might have on the following issues:    How to represent in programs - The GA libraries that I have seen seem to have generators that generate GA algebras in any arbitrary dimension with the scalar, vector, bivector, tri-vector... parts all held separately. If we are mainly interested in representing affine transformations in 3D then it seems simpler to me to hold every 3d multivector as 8 scalar numbers even if all parts are not used?    Inverse of multivector ? - In general multivectors don't always have an inverse, but affine transformations always have inverses, therefore there must be a subset of 3d multivectors that can represent affine transformations and always have inverses, so what is it? In the same way that we define a subset of matricies: transpose(M) = inverse(M) or quaternions: conjugate(q) = inverse(q) to represent orthogonal transformations.    How to represent inertia tensor in GA ? - If we want to relate motion of solid bodies to forces, or impulses in the case of solid body collisions we need to represent inertia tensor. Can we use multivectors to represent an inertia tensor? It seems to me that multivectors are better at representing spinors than tensors? Or do we have to mix multivectors with matrices? How shold we do this?    I have put some pages about GA here:  https://www.euclideanspace.com/maths/algebra/clifford/    I would welcome thoughts from anyone about how to take this forward.    Thanks,    Martin
 By: Nobody/Anonymous - nobody RE: Geometric (Clifford) Algebra   2005-11-10 19:03 Correction: a combination of proportional spacing, non-functioning tab key, and nannying automatic editor has so far frustrated my attempts to typeset an intelligible table. One last try :---     In detail, the mapping of a   real with freedom 1 requires 1 component of grade 4;   point ----------------- 3 ----------- 4 ----------------------------- 3;   line ------------------- 4 ----------- 6 ----------------------------- 2;   plane ---------------- 3 ----------- 4 ----------------------------- 1;   scalar volume ---- 1 ------------1 ---------------------------- 0;   proper isometry -- 6 ----------- 8 ------------------------- 0,2,4;   improper isometry 6 ----------- 8 -------------------------- 1,3.   Notice that the number of "redundant" components never exceeds 2.     Fred Lunnon

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation.

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