For a volumetric mapping $\boldsymbol{X}:\,U\to \mathbb{R}^3$ with $U\subset\mathbb{R}^3$, we can define a normalized measure for orthogonality of the mapping $(\boldsymbol{\xi}=(\xi_1,\xi_2,\xi_3)\mapsto\boldsymbol{X}(\xi_1,\xi_2,\xi_3)$), with \begin{equation} J_{\mathrm{r}} = \frac{\det(\boldsymbol{J})}{\prod_{i=1}^3\left\|\frac{\partial \boldsymbol{X}}{\partial \xi_i}\right\|} = \frac{\frac{\partial \boldsymbol{X}}{\partial \xi_1}\cdot\left(\frac{\partial \boldsymbol{X}}{\partial \xi_2}\times\frac{\partial \boldsymbol{X}}{\partial \xi_3}\right)}{\left\|\frac{\partial \boldsymbol{X}}{\partial \xi_1}\right\|\left\|\frac{\partial \boldsymbol{X}}{\partial \xi_2}\right\|\left\|\frac{\partial \boldsymbol{X}}{\partial \xi_3}\right\|}, \end{equation} such that $|J_{\mathrm{r}}|\leq 1$. Here, $J_{\mathrm{r}}$ gives a measure for the orthogonality of the parametric directions $\xi_1$, $\xi_2$ and $\xi_3$ (having $J_{\mathrm{r}}=1$ for positively oriented orthogonal mappings).
Can we construct a similar measure, $J_{\mathrm{c}}$, for deviation from volumetric affinity (defined by a mapping of the form $\boldsymbol{\xi}\mapsto \boldsymbol{A}\boldsymbol{\xi}+\boldsymbol{b}$ for some constant matrix $\boldsymbol{A}\in\mathbb{R}^{3\times3}$ and vector $\boldsymbol{b}\in\mathbb{R}^3$)? The masure should satisfy $J_{\mathrm{c}} = 0$ when $\boldsymbol{X}(\boldsymbol{\xi}) = \boldsymbol{A}\boldsymbol{\xi}+\boldsymbol{b}$. It is not clear to me if we can have a normalized measure s.t. $|J_{\mathrm{c}}|\leq 1$.