4 ms·
Yes, you are referring to SO(3) but SE(3) includes translations.
by choonway 8y ago
Yes, you are referring to SO(3) but SE(3) includes translations.
- edflsafoiewq 8y agoNo, I was referring to the group of non-zero quaternions. There's a double-cover from the group U of unit quaternions to SO(3) though. If you care about SE(3), there's an obvious surjection U x R^3 -> SE(3) where the domain is "a translation following a rotation". This is a homomorphism of Lie groups. I just don't get what you were going for with "use Lie groups instead". Everything here appears to already be a Lie group. How am I supposed to use that?
- choonway 8y agoI was careful to put the Algebra after the Groups. :) Look up Matrix Exponentials and Logarithms, or just follow the link I gave above.
- chombier 8y agoDual quaternions offer a cheap projection operator by means of dual normalization (very much like quaternion normalization), which the Lie group perspective does not provide. In fact this is probably the only reason to use dual quaternions at all.
- choonway 8y agoQuaternions are more efficient yes, but I would prefer the more general Screw theory.
- chombier 8y agoReally both are the same thing up to group homomorphism (double covering in this case). Dual quaternions only add a cheap way of blending several rigid transformations together, which is useful in computer graphics for skinning meshes around articulated rigid bodies.