Maths - 4D Multivector Functions

Inverse

Here we look at the multiplicative inverse of multivector 'm', that is '1/m'. On this page we saw that the inverse of a pure vector is the same vector with a scaling factor.

Inverse of Bivector

On the same page we saw that the inverse of a bivector blade is calculated by replacing the bivector bases with their reverse (with a scaling factor). Taking the reverse of a bivector is the same as changing its sign. There is a problem in 4D and above though, the non-scalar terms may not cancel out because bivectors like e12 and e34 don't have a common term. So do four dimensional bivectors always have a multiplicative inverse?

Note on terminology: its not a good idea to use the division symbol '/' or '÷' because order is important for multivector multiplication, so when we write: a/b, is not clear if we mean a*(b-1) or (b-1)*a. However I will use the '/' symbol here with the understanding that it means: a/b = a*(b-1).

Lets try inverting the bivector: a e12 + b e34, this multiplying top & bottom by the reverse gives:

bivectordivision

So the denominator still has a non-scalar term, we can make another attempt to cancel it out by multiplying by its reverse, so the numerator is,

(a e21 + b e43)(a² + b² + 2ab e1234)

= (a (a² + b²) - 2ab²) e21 + (b (a² + b²) - 2a²b) e43

= (a (a² - b²)) e21 + (b (b² - a²)) e43

= (a (a + b)(a - b)) e21 + (b (b - a)(b + a)) e43

and the denominator is:

(a² + b² - 2ab e1234)(a² + b² + 2ab e1234)

= (a + b)²(a² + b² - 2 a b)

so the result is:

= (a (a - b)/((a + b)(a² + b² - 2 a b))) e21

+ (b (b - a)/((a + b)(a² + b² - 2 a b))) e43

This looks very messy. Can it be simplified and further?

So how can we calculate the inverse for a general bivector based in 4D which contains up to 6 terms? The inverse of any two terms with a common term is the reverse, but if there is no common term the result is more complicated as we have seen.

The general case for the inverse of a 4D bivector:

1/ (a e12 + b e31 + c e23 + d e41 + f e42 + g e43)

is:

(-a e12 - b e31 - c e23 - d e41 - f e42 - g e43)(x - y e1234)/(x² - y²)

= ((g y -a x) e12
+ (f y - b x) e31
+ (d y - c x) e23
+ (c y - d x) e41
+ (b y - f x) e42
+ (a y - g x) e43)/(x² - y²)

where:

This is proved on this page.

Inverse of 4D Trivector

1/(a e123 + b e142 + c e134 + d e324)

is equal to

(-a e123 - b e142 - c e134 - d e324)/(a² + b² + c² + d²)

So we don't have the same problem for trivectors that we had for bivectors, in 4d we can just use the reverse of all the terms, with the appropriate scaling factor.

Inverse of 4D Quadvector

1/(a e1234) = (1/a) e1234

There is only one quadvector in 4D which is its own inverse.

Dual

This multiplies by e1234 , for general information about n-dimensional dual see this page.

This is the bottom row of the multiplication table here.

Ar
dual(Ar) = Ar* = e1234 Ar
e e1234
e1 e1234 e1= e324
e2 e1234 e2= e134
e3 e1234 e3= e142
e4 e1234 e4= e123
e12 e1234 e12= e43
e31 e1234 e31= e42
e23 e1234 e23= e41
e41 e1234 e41= e23
e42 e1234 e42= e31
e43 e1234 e43= e12
e123 e1234 e123= e4
e142 e1234 e142= e3
e134 e1234 e134= e2
e324 e1234 e324= e1
e1234 e1234 e1234= e

Reverse

The reverse function of a multivector reverses the order of its factors, including the order of the base values within a component. The reverse function is denoted by †, so the reversal of A is denoted by A†.

For general information about n-dimensional reverse see this page.

Ar
Ar
e 1
e1 e1
e2 e2
e3 e3
e4 e4
e12 e21 = -e12
e31 e41 = -e31
e23 e32 = -e23
e41 e24= -e41
e42 e32= -e42
e43 e34= -e43
e123 e321= -e123
e142 e241 = -e142
e134 e431 = -e134
e324 e423 = -e324
e1234 e4321= e1234

The reversal function is important for a number of reasons, one reason is that it can map a multiplication into another multiplication with the order of the multiplicands reversed:

(A * B)† = B†* A†

We can think of this as a morphism where † maps to an equivalent expression with order of multiplication reversed.

morphism

Another application for the reversal function is to specify a transformation from one vector field to another:

pout = A pin A†

In other words, if pin is a pure vector (i.e. real, bivector and tri-vector parts are all zero) then pout will also be a pure vector. I would appreciate any proof of this.If you have a proof, that I could add to this page, please let me know, In the 3D case I guess I could try the brute force approach of multiplying out the terms for a general expression A and p.

I think this may also apply to:

pout = A pin A -1

but not every multivector is invertible, one condition that should ensure that a multivector is invertible is:

A A† = 1

Conjugate

The conjugate of Ar is denoted Ar~ where: Ar~*Ar = I = psudoscalar, for general information about n-dimensional conjugate see this page.

If Ar represents a transformation then Ar~ reverses the transformation

Ar
Ar~=(Ar†)* Ar~*Ar
e (1)* = e1234 e1234
e1 (e1)* = -e234 e1234
e2 (e2 )* = e431 e1234
e3 (e3 )* = -e124 e1234
e4 (e4 )* = -e321 e1234
e12 (-e12 )* = e34 e1234
e31 (-e31 )* = e23 e1234
e23 (-e23 )* = e42 e1234
e41 (-e41 )* = e13 e1234
e42 ( -e42)* = e14 e1234
e43 ( -e43 )* = e12 e1234
e123 ( -e123 )* = -e4 e1234
e142 ( -e142 )* = e3 e1234
e134 ( -e134 )* = e2 e1234
e324 ( -e324 )* = e1 e1234
e1234 ( e1234 )* = 1 e1234

The last column confirms that Ar~*Ar = I = pseudoscalar.

Norm

under construction

Exponential

We define the exponential of multivector a as ea º åi=0¥ ai / i ! .
In particular:
ea is the traditional scalar exponential function
For any pure square multivector [ a2 = ±|a|2 ] we have ea = cos|a| + a~ sin|a| if a2 < 0 ;
cosh|a| + a~ sinh|a| if a2 > 0 ;
1+a if a2 = 0.

For unit multivector a:
eafeay = ea(f+y)
(d/df) elaf =lea(lf+p/2) =laelaf

Involution

a# = sum k=0(N (-1)k a <k> = a<+> - a<-> .



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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

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