Consider the space $\left(\mathbb{R}^3\right)^N$ of configurations of $N$ three-dimensional points. Given for each point a "mass" $m_n$ and a reference configuration $x^o_n$, is it always possible given an arbitrary configuration $x_n$ to find a decomposition $$ x_n = X + O \left( x^o_n + \Delta x_n \right) $$ where: $$ X = \frac{\sum_{m=1}^N m_m x_m}{\sum_{m'=1}^N m_{m'}} $$ is the center-of-mass, $O$ is a matrix representing a proper rotation, and the $\Delta x_n$ are a set of vectors representing "distortion" of the framework $x^o_n$ in the sense that $$ \sum_{m=1}^N m_m \Delta x_m = 0 $$ (i.e. $\Delta x_n$ does not change center of mass) and $$ \sum_{m=1}^N m_m \left( x^o_m \times \Delta x_n \right) = 0 $$ (i.e. $\lambda \Delta x_n$ does not generate rotation about center of mass as $\lambda \to 0$)?

If so, is this decomposition unique?



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