Group Action of $SO(3)$ on unit sphere

Question

We know that $SO(3)$ acts transitively on $S^2$. That is, given any two points $p_1,p_2\in S^2$, one can find an element in $SO(3)$ mapping $p_1$ to $p_2$. My question is, given two sets of points $(a_1,a_2)$ and $(b_1,b_2)$ on $S^2$ such that the induced distance on sphere are the same, can one necessarily find an element in $SO(3)$ mapping one set to the other?

Answer 1

Yes; this is just elementary geometry. We may first choose a rotation sending $a_1$ to $b_1$, and so may assume $a_1=b_1$. Then we can choose a rotation about the diameter of the sphere through $a_1$ that maps $a_2$ to $b_2$. (Explicitly, rotate by the angle between the geodesic from $a_1$ to $a_2$ and the geodesic from $a_1$ to $b_2$.)

Answer 2

Yes. Take $a_3 = a_1 \times a_2$ and $b_3 = b_1 \times b_2$. Define the map $T: \mathbb R^3 \to \mathbb R^3$ by the formula $T(a_i) = b_i$. This map is a oriented isometry of $\mathbb R^3$ and therefore belongs to $SO(3)$.