%---------------------------Oddy---------------------------
\section{Oddy}

First we define the Oddy $O$ in terms of the Jacobian matrices $A_i$ from \S\ref{s:hex}:
\[
  O(A_i) = \frac{\left| A_i^t A_i \right|^2 - \frac {1}{3}\left|A_i\right|^4}{\alpha_i^{\frac{4}{3}}}.
\]
The metric value is then the maximum Oddy over all the corners and the element center
\[
  q = \max_{i\in\{0,1,\ldots,8\}}\left\{ O(A_i) \right\}.
\]
This can be interpreted as the maximum deviation of
the metric tensor ($A_i^tA_i$) from the identity matrix, evaluated at the corners and element center.

Note that if $\alpha_i \leq DBL\_MIN$ for any $i$, we set $q = DBL\_MAX$.

\hexmetrictable{Oddy}%
{$1$}%                                        Dimension
{$[0,0.5]$}%                                  Acceptable range
{$[0,DBL\_MAX]$}%                             Normal range
{$[0,DBL\_MAX]$}%                             Full range
{$0$}%                                        Cube
{Adapted from \cite{odd:88}}%                 Citation
{v\_hex\_oddy}%                               Verdict function name