
3D kinematics using dual quaternions: applications in neuroscience (2013) - adamnemecek
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3576712/
======
chombier
In my experience, dual quaternions have little to no practical interest unless
you need to blend rigid transformations, in which case they provide a cheap
projection operator similar to quaternion normalization. (See that SIGGRAPH
paper by Kavan on dual quaternion skinning)

On the theoretical side however, it is quite magic to see how rigid
transformations arise from the unit quaternion tangent bundle.

~~~
layoutIfNeeded
You seem to be knowledgeable on the subject. Could you point me to material
which gives insight on _why_ dual quaternions work the way they do?

I know that unit complex numbers can represent 2d rotations, dual complex
numbers can represent 2d rigid transformations, and similarly unit quaternions
can represent 3d rotations and dual quaternions can represent 3d rigid
transformations... I can derive their properties algebraically and indeed they
work the way they do. Still I want to see the "big picture".

~~~
chombier
Here is a tentative explanation:

Dual numbers introduce a constant `eps` whose purpose is to cancel out
anything second-order and more. A dual number a + eps * b can be thought of as
encoding a tangent vector at position a, with velocity b. Everything plays
nicely so that one can extend regular functions to dual numbers using
derivatives: f(a + eps * b) = f(a) + eps * df(a).b (try it out on product,
inverses and so on to see how it works)

The same idea can be applied to rotation matrices to get dual rotation
matrices R + eps * dR, where dR is the derivative of a rotation matrix at R.
As such, dR is a matrix that can be decomposed using the spatial velocity or
the body-fixed velocity, which are both 3-vectors.

Now when you compose rotations R1, R2 with spatial angular velocities w1, w2,
the spatial angular velocity of the composition is w1 + R1w2, which is the
translation you get when composing rigid transformations. In other words, if
we _encode_ a rigid transformation (R, t) as a tangent vector R + eps * dR
where dR has spatial velocity t, then we can compose the encoded rigid
transformations by multiplying dual rotation matrices.

Dual unit quaternions are simply the implementation of this idea using unit
quaternions instead of rotations, but the principle remains the same.

In the end it all boils down to how the Euclidean group somehow arises from
the tangent bundle of the rotation group (both 2d or 3d). It still looks kind
of magic to me, but maybe the construction above always gives the semi-direct
product of the Lie group by its Lie algebra.

~~~
chombier
Apparently this construction always provides the (outer) semi-direct product
of the Lie group by its Lie algebra with respect to the group adjoint.

This also means we should get another one considering body-fixed velocities
instead, which I suspect will be some form of outer semi-direct product with
respect to the coadjoint representation of the group.

------
cat199
(2013)

~~~
adamnemecek
I tried to put it in but was over character limit.

