%---------------------------Distortion-----------------------------
\section{Distortion\label{s:tri-distortion}}

Let $A$ be the area as defined in \S\ref{s:tri-area}
and $A_m = \sqrt{3}$ be the area of a ``master'' triangle with vertices
\[
\begin{array}{lcrcrcrl}
  \vec P_0 &= (&-1&,& -\frac{ \sqrt{3}}{3}&,& 0&)\\
  \vec P_1 &= (& 1&,& -\frac{ \sqrt{3}}{3}&,& 0&)\\
  \vec P_2 &= (& 0&,&  \frac{2\sqrt{3}}{3}&,& 0&).
\end{array}
\]
Now define $|J|$ as the minimum value of the
determinant of the Jacobian evaluated at all Gauss points of the element.
The distortion is then
\[
q = \frac{|J| A_m}{A} = \frac{|J|\sqrt{3}}{A}.
\]
Distortion is a measure of how well-behaved the mapping from
parameter space to world coordinates is.

Note that this metric is currently unsupported.

\trimetrictable{distortion}%
{$1$}%                                                Dimension
{$[0.5,1]$}%                                          Acceptable range
{$[0,1]$}%                                            Normal range
{$[-DBL\_MAX,DBL\_MAX]$}%                             Full range
{$1$}%                                                Unit equilateral triangle value
{Adapted from \cite{ideas:xx}}%                       Reference(s)                   
{v\_tri\_distortion}%                            Verdict function name