%---------------------------Maximum Edge Ratio---------------------------
\section{Maximum Edge Ratio}

Given principal axes with lengths $L_f$ and $L_g$,
the aspect ratio is defined as the largest ratio of those lengths
\[
   A_{fg} = \max\left\{ \frac{L_f}{L_g}, \frac{L_g}{L_f} \right\}.
\]
Since a hexahedron has 3 principal axes, we take the largest of all pairwise combinations of axes.
\[
  q  = \max\left\{
    A_{\normvec{ X_1 }\normvec{ X_2 }},
    A_{\normvec{ X_1 }\normvec{ X_3 }},
    A_{\normvec{ X_2 }\normvec{ X_3 }}
  \right\}
\]

Note that if $\normvec{X_1}$ or $\normvec{X_2}$ or $\normvec{X_3} < DBL\_MIN$, we set $q = DBL\_MAX$.

\hexmetrictable{maximum edge ratio}%
{$1$}%                                      Dimension
{$[1,1.3]$}%                                Acceptable range
{$[1,DBL\_MAX]$}%                           Normal range
{$[1,DBL\_MAX]$}%                           Full range
{$1$}%                                      Unit square
{Adapted from \cite{tf:89}}%                Citation
{v\_hex\_max\_edge\_ratio}%                      Verdict function name
