Maths - 2D multivector Exponent

Infinite Series

The exponent is given by the series:

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

Where:

We start with the series:

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

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

exp(m) = 1 + m + (s + 2a*m)/2!
+ (2a*s + (4a²+s)*m) /3!
+ ((s² + 4*a²*s) + (4*s*a + 8*a³)*m) /4!
+ ( (4*s²*a + 8*s*a³) + (s² + 12*a²*s + 16*a4)*m) /5! + …

splitting up the scalars and powers of m gives:

exp(m) = 1 + s/2!
+ (2a*s ) /3!
+ (s² + 4*a²*s) /4!
+ (4*s²*a + 8*s*a³) /5! + …

+ m*( 1 + 2a)/2!
+ (4a²+s) /3!
+ (4*s*a + 8*a³) /4!
+ (s² + 12*a²*s + 16*a4)/5! + … )

 

 

     
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

When we look at powers of multivectors (here) then we made the assumption that the multivector squares to a pure scalar value:

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

 


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Book Shop - Further reading.

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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.

 

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