Quaternion math

[aldopareja]

Hi, I've been studying the definitions of quaternion math, and I'm not sure where all came from. In particular, where did this definition of rotation come from? Is there a textbook/paper you people would suggest to understand this better?

def rotate(vec: Vector3, quat: Quaternion):
  """Rotates a vector vec by a unit quaternion quat.

  Args:
    vec: (3,) a vector
    quat: (4,) a quaternion

  Returns:
    ndarray(3) containing vec rotated by quat.
  """
  if len(vec.shape) != 1:
    raise AssertionError('vec must have no batch dimensions.')
  s, u = quat[0], quat[1:]
  r = 2 * (jp.dot(u, vec) * u) + (s * s - jp.dot(u, u)) * vec
  r = r + 2 * s * jp.cross(u, vec)
  return r

[erikfrey]

This is a great question. I'm less familiar with textbooks in this space, maybe @cdfreeman-google can chime in. However I will say that I think transforms3d is a great repository for nicely documented, well commented code:

https://matthew-brett.github.io/transforms3d/index.html

For example, here is their implementation of the quaternion rotation op:

https://github.com/matthew-brett/transforms3d/blob/main/transforms3d/quaternions.py#L421

[cdfreeman-google]

Indeed--I've actually lost track of where I nabbed this particular identity from. This blogpost has a discussion of similar identities you can arrive at by playing with the vector triple product that appears when you fully write out all the components of rotating a vector by a quaternion: https://fgiesen.wordpress.com/2019/02/09/rotating-a-single-vector-using-a-quaternion/