Despite the costs of using a “too big” symmetry group (I’d say overparameterization), I bet it’s hard to beat the performance of 4*4 matrix multiplications, since the computation is so uniform, vectorizable, and perfectly sized for e.g. SIMD. Even if there are fewer math operations with another representation.
It depends. For composing multiple transforms, composing versors aka quaternions/dual quaternions is cheaper than a full 4x4 multiply. For a single application, a 4x4 will be cheaper.
Quaternions are used in animation for blending and composing transforms for a reason. You don't have to trust me, but yes... tried for maybe ten years and counting from doing graphics and animation work.
> If the quaternion is used to rotate several points (>1) then it is much more efficient to first convert it to a 3x3 Matrix. Comparison of the operation cost for n transformations:
> - Quaternion2: 30n
> - Via a Matrix3: 24 + 15n
So really it depends on what you're doing (admittedly here it is 3x3 not 4x4)
