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 #algebra, 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
ε² = 0
It makes sense to think of ε as a first approximation for infinitesimals: smaller than any non-infinitesimal, but still greater than zero.
You might also notice that Grassmann's exterior algebra works in a similar way.
Grassmann's exterior algebra also appears as the wedge product in differential algebra. That smells like a kind of unification of the two interpretations above, but I don't know how to make that precise...
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