Number of components needed for 3D rotation

Question

Using Euler angles, a 3D rotation can be expressed using 3 real numbers. Using quaternions, 4 are needed and using rotation matrices 9. Is it possible to express a 3D rotation using less than 3 real numbers, or why are 3 needed?

Answer 1

No, it's not possible, at least not possible in a bi-continuous way.

Because by a generalization of a Peano curve, there's a map from the unit interval onto the unit cube in 3-space, the euler-angle parameterization can be reduced to a single-number parameterization...but that's completely useless in practice.

Now...why, if you want a "nice" parameterization, do you need three coordinates? Well, start at the "no motion" rotation, and consider the three independent rotations about the x, y, and z axes. What I mean by "independent" is that no small rotation about x -- say, less than 10 degrees -- can be expressed as a combination of small rotations about y and z.

On the other hand, every small rotation (e.g., less than 5 degrees) can be expressed as a combinations of small-ish (less than 10 degrees) rotations about x, y, and z.

In short: there's a 3-parameter nonsingular parameterization of a small region of rotation space. Let's call this $$ H: I \times I \times I \to SO(3) $$

If you want to parameterize rotations near some rotation $R$, you can use $$ H_R (a, b, c) = R \circ H(a, b, c). $$

So there's a nonsingular 3-parameter parameterization of a neighborhood of any rotation $R$.

That makes the set of rotations -- let me call it $SO(3)$ -- into a topological 3-manifold. And the Brouwer theorem on the invariance of domain then shows that any other "nice" parameterization that's 1-1 (as $H$ is) must also have three parameters.