Let q be a unit quaternion which represents a rotation. Let p be a coordinate in R3
which we map to H by setting the scalar to zero. Then the equation:
p' = qpq-1
is p rotated by q. Multiply q by non zero scalar s.
) = (s/s)qpq-1
scalars commute so the terms s and 1/s cancel so it yields the same result.
If we change the equation to using the conjugate then we get the same for unit since q-1
in that case.
p' = qpq*
and for non unit the scale factors compose yielding the rotation plus a uniform scaling by s2
p' = (sq)p(sq)*
If we change the mapping of p to setting the scalar to one then we are back to an unscaled rotation.
- All quaternions except zero can represent a rotation.
- The set of all quaternion that represent the same rotation fall on a line through the origin.
- There are only two points the line intersects the unit sphere: q and -q.
- If two quaternions don't fall on the same line through the origin then the represent different rotations.