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0 pointstwic7y ago0 comments

Thanks for this - i have studied very little maths, so even this perhaps quite basic stuff is like magic for me!

I love the revelation that log/exp shifts between Cartesian and (log-)polar meanings of complex numbers. That's enormously satisfying for some reason.

I note that the pattern in multiplying rotations also comes up if you do it with matrices:

    a -b     c -d     ac-bd -(ad+bc)
    b  a  .  d  c  =  ad+bc    ac-bd

Does something similar happen with quaternions / geometric products in 3D?

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1 comments · 1 top-level

jacobolus7y ago

Taking 3D rotations into log space gets tricky. 3D rotation is not commutative, so if you have two quaternions Q and R, you can’t in general say that QR = exp(log(Q))exp(log(R)) = exp(log(Q) + log(R)). That identity only holds if the bivector (“imaginary”) parts of Q and R are oriented the same way in space, i.e. are scalar multiples of each-other. Otherwise in general exp(log(Q))exp(log(R)) ≠ exp(log(R))exp(log(Q)).

The log space version is still useful though if you want to e.g. take the nth root of a spatial rotation.

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