Maths - Dual Complex Arithmetic

Adding Dual Complex numbers

Just add the real and imaginary components independently as follows:

(a1 + i b1 + ε c1 +iε d1)+(a2 + i b2+ ε c2 +iε d2) = (a1+a2) + i (b1+b2) + ε (c1+c2) +iε (d1+d2)

Subtracting Dual Complex numbers

Just subtract the real and imaginary components independently as follows:

(a1 + i b1 + ε c1 +iε d1)-(a2 + i b2+ ε c2 +iε d2) = (a1-a2) + i (b1-b2) + ε (c1-c2) +iε (d1-d2)

Multiplying Dual Complex numbers

We put each quaternion in brackets and multiply out all the terms: (a1 + i b1 + ε c1 +iε d1)*(a2 + i b2+ ε c2 +iε d2). When we are multiplying the imaginary operators we use the multipication table:

a*b
b.1 b.i b.ε b.iε
a.1 1 i ε
a.i i -1
a.ε ε -iε 0 0
a.iε ε 0 0

Note that the order of multiplication is significant, in other words dc1 * dc2 is not necessarily equal to dc2 * dc1.

So back to the general case of multiplying any two quaternions, we can just expand out the terms and group as follows:

(a1 + i b1 + ε c1 +iε d1)*(a2 + i b2+ ε c2 +iε d2) =a1*a2 - b1*b2
+ i (a1*b2 + b1*a2)
+ ε (c1*a2 + d1*b2 + a1*c2 - b1*d2)
+iε (d1*a2 - c1*b2 + b1*c2 + a1*d2)

so the result is:

dc1 * dc2= a1*a2 - b1*b2 + i (a1*b2 + b1*a2) + ε (c1*a2 + d1*b2 + a1*c2 - b1*d2) +iε (d1*a2 - c1*b2 + b1*c2 + a1*d2)

Division

We don't tend to use the notation for division, since quaternion multiplication is not commutative we need to be able to distinguish between dc1*dc2-1 and dc2-1*dc1. So instead of a divide operation we multiply by the inverse.

Due to the zero terms in the multiplication table I don't think there is a general multiplicative inverse because information is lost by multiplication.

However, if we restrict ourselves to the linear rotation-displacement discussed above, then there must be an inverse because the operations they represent have an inverse. The complex number is always normalised so its inverse will be its conjugate and the inverse of the displacement is minus its value.

So, in this case, the inverse of:

a + b i + c ε + d iε

is:

a - b i - c ε - d iε

Dual Complex Calculator

The following calculator allows you to calculate quaternion arithmetic. Enter the values into the top two quaternion and then press "+ - or * " to display the result in the bottom quaternion:

real i ε
real i ε
=
real i ε

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.

 

cover us uk de jp fr ca Quaternions and Rotation Sequences.

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.