Maths - Clifford Algebra - 3D Arithmetic

Maths - 3D Clifford Algebra - Calculator Correction from Paolo

Geometric product calculator

By: deleva (deleva) - 2007-05-30 03:05

          http://www.euclideanspace.com/maths/algebra/clifford/d3/arithmetic/index.htm#calc 

        The calculator does not seem to perform correctly a simple geometric
        product ab where a and b are orthogonal(which should be numerically
        equivalent to an outer product a ^ b and to a cross product a x b).  

        The calculator gives ab = - a x b.  

        However, every book and web page that I have seen agree that (a x b) * e123 = a ^ b, which means that  

        the coefficients c1  c2  c3  of a x b = c are the same as  

        the coefficients d23 d31 d12 of a ^ b = d 

        because 

        e1 * e123 = e23 

        e2 * e123 = e31 

        e3 * e123 = e12 

        Do you agree? 

        Paolo

RE: Geometric product calculator

By: Martin Baker (martinbaker

) - 2007-06-15 01:34

          Paolo, 

          Yes I agree, I tried it myself and got, 

          e1 * e2 = -e12 

          I'll correct it to agree with the multiplication table on the same page: 

          e1 * e2 = e12 

          Thanks very much for pointing this out I'll correct it as soon as I can. 

          Cheers, 

          Martin

RE: Geometric product calculator

By: Martin Baker (martinbaker

) - 2007-06-18 11:03

          Paolo, 

          I've fixed it, it took me a while as some of the 'a' and 'b' operands
          were reversed, both in the javascript and some of the web page content. 

          Hopefully its all correct now, I've tested it as much as I can, but if
          anyone else would like to try it please let me know what you find. 

          Thanks again for letting me know about this. 

          Martin

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Book Shop - Further reading.

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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