Dual quarternions are an 8-dimensional algebra that can be used to represent not only rotation and scaling, but also translations in R³.


Using mere quarternions for rotations in R³ is a well-known trick. But translations work differently, so we need a new kind of , octonions simply won't work here.



Recall that quarternions are like complex numbers on steroids. They have three imaginary units i, j, k each of which squares to minus one, just like the complex unit:

i² = -1

To get dual quarternions, take two quarternions A and B and think of one of them as carrying a new kind of symbol ε:


ε also reacts in a funny way when squared:

ε² = 0


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