[eestrada]

My (poor) understanding of quaternions is they are like a vector with a rotation angle around the axis of the vector. This is probably incorrect on many levels. But it was a simple enough explanation that it made sense to my brain why this would work better than simple SRT transform parameters for avoiding gimbal lock. It's been several years since I needed to deal with 3D transformations of any sort, so I'm a bit rusty on all this.

[xeonmc]

Ultimately, the “canonical” representation of rotation state is still axis-angles, thanks to Euler’s rotation theorem (any combo of rotation results in just one rotation around some final axis)

(normalized) Quaternions are just the intermediate representation of axis-angles, they describe the component-wise algebra of combining axis-angle rotations [Euler-Rodrigues formula](https://en.wikipedia.org/wiki/Euler–Rodrigues_formula)

Game engines just leaves them as quaternions for performance reasons.

A 2D analogy would be

    “angle” <-> [quaternion]
    “45deg” <-> [sqrt2 , sqrt2]

The [ Re , Im ] form is convenient for manipulating by coordinates, but the “final” succinct representation is still “angle”, because the component-wise form has more degrees of freedom than necessary.

Summary:

    Quaternion = sqrt( exp(axis_angle) )
               =       exp(axis_angle/2)
               = cos(angle/2) + axis*sin(angle/2)

And

    (lw,lx,ly,lz)*(rw,rx,ry,rz) = (w,x,y,z) 
    where
      w = ww-xx-yy-zz 
      x = wx+xw+yz-zy 
      y = wy+yw+zx-xz 
      z = wz+zw+xy-yx 
      where
        ww,wx,wy,wz = lw*rw,lw*rx,lw*ry,lw*rz 
        xw,xx,xy,xz = lx*rw,lx*rx,lx*ry,lx*rz
        yw,yx,yy,yz = ly*rw,ly*rx,ly*ry,ly*rz
        zw,zx,zy,zz = lz*rw,lz*rx,lz*ry,lz*rz