Let $x_1,\dots,x_n$ be $n$ points forming a rigid body in $\mathbb{R}^3$. The distance between each pair of points is constant. Let $R\in SO(3)$ be a rotation and $T\in\mathbb{R}^3$ be a translation. Then $\{R,T\}$ actually is a rigid body transformation. If I apply $\{R,T\}$ to $\{x_i\}_{i=1}^N$, then I obtain a \{x'_i=Rx_i+T\}_{i=1}^N, which is different from the original one by a rigid body transformation.
My question is: Since $\{R,T\}$ can be arbitrary, all these \{x'_i\}_{i=1}^N form a set. If I define x'=[x'_1^T,\dots,x'_n^T]^T\in\mathbb{R}^{3n}, all these \{x'_i\}_{i=1}^N form a set $\Omega$ in $\mathbb{R}^{3n}$. Then what does this set $\Omega$ look like? Is it connected? compact? or a manifold? I'm not very familiar with the set theory or differential geometry, can someone give me a hint on how to analyze the problem? Thanks.