§ Why quaternions work better
- We want to manipuate . Imagine it like .
- Unfortunately, . This is a pain, much like rotations of a circle need to be concatenated with modulo, which is a pain.
- idea for why is : is sphere with antipodal points identified. So a path from the north pole to the south pole on the sphere is a "loop" in . Concatenate this loop with itself (make another trip from the south pole to the north pole) to get a full loop around the sphere, which can be shrunk into nothing as is trivial. So , where is the north-south path in which is a loop in ).
- Key idea: deloop the space! How? find univesal cover. Lucikly, universal cover of is / quaternions, just as universal cover of is .
- Universal cover also explains why is a double cover. Since is , we need to deloop "once" to get the delooped space.
- No more redundancy now! Just store a bloch sphere representation, or a quaternion (store ). Just like we can just store a real number for angle and add it.
- How to go back to or ? Move down the universal cover map or .
- This is strange though. Why is both the lie algebra and the covering space of ? What about in general?
- In general, the original lie group and the universal cover both have the same lie algebra . It is only that the lie group has less or more fundamental group.