Say you have a set of $N$ points in $\mathbb{R}^3$ with the centroid at origin with fixed distance between the points (assume a rigid body constraint). Assuming the centroid to be fixed at the origin, can one always get back to this original configuration from any other configuration by using a single rotation and a single reflection?
What I am trying to understand mainly is how reflection plays a role in all this. For example, it is pretty easy to see that for the $\mathbb{R}^2$ case, again keeping the centroid at origin, you would only need a rotation or at most a rotation and a reflection to go back to the original configuration. This reflection can be along any line that passes through the origin. The rotation required would change depending upon the choice of the line used for reflection. Can the same be said when considering a point set in $\mathbb{R}^3$? i.e. will a single reflection along any plane followed by an appropriate rotation lead to the original configuration?
Any references will be appreciated. Do let me know if the question need more clarity.
A rigid motion that preserves position of the centroid is always a combination of rotation and reflection. Such a transformation is represented by an orthogonal operator $R$, which produces a rotation if $\det R = 1$ and produces a reflection (possibly accompanied with rotation) if $\det R = -1$.
Source: https://en.wikipedia.org/wiki/Rigid_transformation
See also: Michael Artin, Algebra, Chapter 6.