Configuration space of three points in $\mathbb{R^{3}}$

Question

If $X,Y,Z$ are the distances between three points in $\mathbb{R}^{3}$ such that $X,Y,Z$ satisfy the triangle inequality. What will be the configuration space of the three points, given the translation symmetry (fix one point at origin).

Thanks

Answer 1

the question is not very clear, so I'm not sure whether this is an answer: If the distances $X,Y,Z$ are fixed and satisfy strict triangle inequality then the configuration space is (diffeomorphic to) $SO(3)\cong\mathbb{RP}^3$, as $SO(3)$ acts freely and transitively on this configuration space. If they are not fixed and you simply want to exclude configurations of collinear points then you get $SO(3)\times\{(X,Y,Z)\in\mathbb{R}_+^3;X,Y,Z$ satisfy strict triangle ineqality $\}$ which is diffeomorphic to $SO(3)\times\mathbb{R}^3$.