Definition of rotations

Question

The naive way I think about rotations is by imagining a finger sliding the sphere and in doing so it traces a spherical curve. This processes is then possibly repeated starting from a different point and so on. Let's start now with a spherical curve which I assume to be rectifiable. Can I recover the rotation I performed to produce the curve? I think we should impose the condition that when the curve stops the sphere doesn't rotate, to have unicity of solution. I would like the answer to make possible to define rotations.

Answer 1

Touching the sphere with your finger on a point of the sphere and move it tangentially, makes 2 degrees of freedom in the Lie algebra of small deviations from unity in the group manifold.

The third degree of freedom is to rotate the sphere around with the finger, so that the point remains fixed.

As one learns from navigating a car sideways into a parking lot, the third direction is the commutator of the two generators.

With 3x3 unit matrices with infinitesimal entries for roations in 1,2 and 2,3 - planes making a non-closed rectangular path, we get

$$\left( \begin{array}{ccc} 1 & a & 0 \\ -a & 1 & 0 \\ 0 & 0 & 1 \end{array} \right).\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & b \\ 0 & -b & 1 \end{array} \right).\left( \begin{array}{ccc} 1 & -a & 0 \\ a & 1 & 0 \\ 0 & 0 & 1 \end{array} \right).\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -b \\ 0 & b & 1 \end{array} \right)= \left( \begin{array}{ccc} 1 & 0 & a b \\ 0 & 1 & 0 \\ -a b & 0 & 1 \end{array} \right)$$

up to first order, the path will be closed by a fixed point rotation in the 1,3-plane

$$\left( \begin{array}{ccc} 1 & 0 & -(a b)^{-1} \\ 0 & 1 & 0 \\ (a b)^{-1} & 0 & 1 \end{array} \right)$$

Answer 2

If we have two spherical curves, and $v_0\in S^2$ is the starting point of the first curve, $v_1\in S^2$ is the end point of the first curve, $w_0\in S^2$ is the starting point of the second curve and $w_1\in S^2$ is the end point of the second curve, and we assume the points are located on $S^2$ such that there is a rotation $R$ that maps $v_0$ to $v_1$ and $w_0$ to $w_1$ (The distance between the two starting points is the same as the distance between the two end points, $\langle v_0,w_0\rangle =\langle v_1,w_1\rangle$), then this rotation obvioulsy also maps $v_0\times w_0$ to $v_1\times w_1.$ Therefore, the rotation matrix is $$ \begin{pmatrix} | & | & | \\ v_1 & w_1 & v_1 \times w_1 \\ | & | & | \end{pmatrix} \begin{pmatrix} | & | & | \\ v_0 & w_0 & v_0 \times w_0 \\ | & | & | \end{pmatrix} ^{-1} $$