Haven't dual quaternions taken over for their ability to represent all rigid body motions?
Quaternions are still important for building intuition for dual quaternions.
Quaternions are still important for building intuition for dual quaternions.
Dual quaternions haven’t really taken over. While there’s no universal way that people end up handling rigid body transforms, SO(3) + R3 for rigid body transforms are the norm in robotics.
Some robotics orgs choose a single representation for SO(3) — usually quaternions or matrices — and use it everywhere. Note that quaternions are more space efficient and easier to normalize, but require 2x more operations to rotate a vector [1]. So you might see a quaternion shop use matrices in a hot loop for transforming 3D points. And a matrix shop might use quaternions in a hot loop for composing many transformations. Usually you have many fewer rigid body links than points in an e.g. pointcloud so it is performance favorable to use matrices.
A final comment: Most users in a codebase should interact with rotations through a rotation library like Sophus or Manif. That library should implement left/right lerp, splines, “rotate a vector” and “compose a transformation”. There is basically never a need for a developer to know what rotation representation is being used to compute the rotation.
If such a library is correctly implemented, you can even use Euler Angles as the underlying representation and never experience gimbal lock.
[1] https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...