Intuition for quaternion

Going back to 2D and complex space

• - 2D rotations have 1 dof. (rotation angle)
• - 2D rotation represented as a point on a unit circle (\(S^1\)).
• • - 1D structure embedded in a 2D space.
  • - Complex space has algebraic structure adapted to model 2D rotation.

Moving to 3D

• - 3D rotations have 3 dof.
• • - (point on a unit 3D sphere allows only 2 dof : not sufficient ex. orientation that can twist)
  • - point on a unit 4D sphere allows 3 dof
• - 3D rotation are represented as point on the unit sphere in 4D (\(S^3\)).
• • - 3D structure embedded in a 4D space.
  • - Complex quaternion space has algebraic structure adapted to model 3D rotation.