Maths - Multivector Exponent

Here we calculate the exponent of a multivector. When we looked at the result for pure vectors (on this page) we saw that it depends on whether the dimensions commute or anti-commute and whether the dimensions square to positive or negative. A summary of the results is given in the following table:

commutative square to result example derivation
anti-commute all positive exp(v) = cosh(|v|) + norm(v)*sinh(|v|) Euclidean vector see below
anti-commute all negative exp(v) = cos(|v|) + norm(v)*sin(|v|) bivector see below
commute two positive exp(x + Dy) = cosh(|v|) + D sinh(|v|) double number double number pages
commute one positive, one negative r e = r (cos(θ) + i sin(θ)) complex number complex number pages

where:

To summarise and interpret the results, if the dimensions commute (as they do for complex numbers for example) then the result is a pure vector but, if the dimensions anti-commute (as they do for vectors in euclidean space for example) then the result is a scalar plus a vector.

in order to generalise this result to any multivector we will use infinite series:

Infinite Series

The exponent is given by the series:

e(m) =
n=0
(m)n
n!

Where:

We have to be careful with multivectors because they are not in general commutative. I think the above series applies but I'm not absolutely sure.

We now need to plug in a value for (m)n which we have calculated on this page.

     
sin(x) x - x3/3! + x5/5! ... +(-1)rx2r+1/(2r+1)! all values of x
cos(x) 1 - x2/2! + x4/4! ... +(-1)rx2r/(2r)! all values of x
ln(1+x) x - x2/2! + x3/3! ... +(-1)r+1xr/(r)! -1 < x <= 1
exp(x) 1 + x1/1! + x2/2! + x3/3! ... + xr/(r)! all values of x
exp(-x) 1 - x1/1! + x2/2! - x3/3! ... all values of x
e 1 + 1/1! + 2/2! + 3/3! =2.718281828
sinh(x) x + x3/3! + x5/5! ... +x2r+1/(2r+1)! all values of x
cosh(x) 1 + x2/2! + x4/4! ... +x2r/(2r)! all values of x

We start with the series:

exp(m) = 1 + m 1/1! + m 2/2! + m 3/3!+ m 4/4! + m 5/5! + …

When we look at powers of multivectors (here) then we made the assumption that the multivector squares to a pure scalar value (does not apply if scalar part is non zero so be warned: these results are not general):

m2 = s = positive or negative scalar

so substituting gives:

exp(m) = 1 + m + s/2! + s*m/3!+ s2/4! + s2*m/5! + …

splitting up into real and vector parts gives:

exp(m) = 1 + s/2! + s2/4! +…+ m*( 1+ s/3! + s2/5! + …)

In order to fit to the series above we will express this in terms of √s:

exp(m) = 1 + (√s)2/2! + (√s)4/4! +…+ m/(√s)*( √s+ (√s)3/3! + (√s)5/5! + …)

So the match to the series depends on the sign of √s as follows:

√s exp(m)
if √s = +ve then: exp(m) = cosh(√s) + m/(√s)*sinh(√s)
if √s = -ve then: exp(m) = cos(√s) + m/(√s)*sin(√s)
if √s = 0 then: exp(m) = 1 + m

 


metadata block
see also:

 

Correspondence about this page

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.

flag flag flag flag flag flag 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.

 

Terminology and Notation

Specific to this page here:

 

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.