Representation of Rotations

The table below summarizes the six methods as the representation of rotations.

Representation	Expression	# of Params	Singularity	Topology	Fundamental Group
Rotation matrices	R_3x3	9	No	RP^3	Z/2Z
Exponential coordinates	ω ̂θ	3	Yes	S^2×S^1	Z
Axis-angle	ω ̂, θ	4	Yes	S^2×S^1	Z
Euler angles	α, β, γ	3	Yes	T^3	Z x Z x Z
Roll-pitch-yaw angles	α, β, γ	3	Yes	T^3	Z x Z x Z
Unit quaternions	q=[q_0, q_1, q_2, q_3]	4	No	S^3	trivial

Fundamental Group

• Imagine you have a loop (a path that starts and ends at the same point) in a space. The fundamental group of a space at a point is the set of all such "loops" at that point, where two loops are considered the same if you can deform one into the other without breaking or lifting them off the space. The operation of this group is "concatenation" of loops, which means doing one loop after another.

• In topology, a basic theorem is that if two spaces are homeomorphic (i.e., they can be continuously deformed into each other), then their fundamental groups are isomorphic (i.e., there is a bijective map between them that preserves the group structure).

Examples

• The fundamental group of S^1 is Z, because each integer counts the number of times a loop winds around the circle.

• The fundamental group of S^n (n>2) is trivial, because any loop on the sphere can be shrunk to a point without leaving the sphere. In group theory, a "trivial" group is one that contains only a single element. This is the smallest possible group.

• The fundamental group of a torus T^2 is Z x Z, because there are two independent ways to loop around a torus. These loops are independent in the sense that you can't deform loop A into loop B without leaving the surface of the torus. The set of all possible "combinations" of loops A and B forms a group that is isomorphic to Z x Z.

• The fundamental group of SO(3) is isomorphic to Z/2Z, the group of integers modulo 2. This is a deep fact about SO(3) that requires some sophisticated machinery to prove, but intuitively, it comes from the fact that a 180-degree rotation in 3-dimensional space cannot be continuously deformed to the identity rotation, but a 360-degree rotation can.