%---------------------------Shape-----------------------------
\section{Shape\label{s:tet-shape}}

Let $J$ be the Jacobian as defined in \S\ref{s:tet-jacobian}.
We define the shape quality metric as
\begin{equation*}
q =  \frac{ 3 \left( J \sqrt{2} \right)^{2/3} } 
              { \frac {3}{2} \left( 
                      \vec L_0 \cdot  \vec L_0 +
                      \vec L_2 \cdot  \vec L_2 +
                      \vec L_3 \cdot  \vec L_3  \right) - 
               \left( \vec L_0 \cdot -\vec L_2 +
                      \vec L_0 \cdot  \vec L_3 +
                     -\vec L_2 \cdot  \vec L_3 \right) } 
\end{equation*}

Note that if $J < DBL\_MIN$, $q = 0$.
If $\frac {3}{2} \left( \vec L_0 \cdot \vec L_0 +
                      \vec L_2 \cdot \vec L_2 +
                      \vec L_3 \cdot \vec L_3  \right) - 
                 \left( \vec L_0 \cdot -\vec L_2 +
                      \vec L_0 \cdot \vec L_3 +
                      -\vec L_2 \cdot \vec L_3 \right) < DBL\_MIN$, we set $q = 0$.

\tetmetrictable{shape}%
{$1$}%                                        Dimension
{$[0.3,1]$}%                                  Acceptable range
{$[0,1]$}%                                    Normal range
{$[0,1]$}%                                    Full range
{1}%                                          Equilateral tet
{\cite{knu:03}}%                              Citation
{v\_tet\_shape}%                            Verdict function name