%---------------------------Scaled Jacobian-----------------------------
\section{Scaled Jacobian\label{s:tri-scaled-jacobian}}

First, let $L_{\max}$ be the product of the lengths of the 2 longest edges:
\[
  L_{\max} = \max\left\{
    \normvec{ L_0} \normvec{ L_1},
    \normvec{ L_0} \normvec{ L_2},
    \normvec{ L_1} \normvec{ L_2}
  \right\}
\]
Let $J^{\prime}$ be the Jacobian of the triangle.
If the triangle surface normal $\hat n$ is evaluated at the center of the triangle
and $\hat n\cdot\left(\vec L_2\times\vec L_1\right) < 0$, then take $J = -J^{\prime}$.
Otherwise take $J = J^{\prime}$.
The scaled Jacobian is then
\[
  q = \frac{2\sqrt{3}}{3} \frac{J}{L_{\max}}
\]
which is normalized so that a unit equilateral triangle has value $1$.

Note that if $L_{\max} \leq DBL\_MIN$, we set $q = 0$.

\trimetrictable{scaled Jacobian}%
{$1$}%                                                Dimension
{$[0.5,\frac{2\sqrt{3}}{3}]$}%                        Acceptable range
{$[-\frac{2\sqrt{3}}{3},\frac{2\sqrt{3}}{3}]$}%       Normal range
{$[-DBL\_MAX,DBL\_MAX]$}%                             Full range
{$1$}%                                                Unit equilateral triangle value
{\cite{knu:00}}%                                      Reference(s)                   
{v\_tri\_scaled\_jacobian}%                            Verdict function name