%---------------------------Distortion-----------------------------
\section{Distortion}

Let $A$ be the area as defined in \S\ref{s:quad-area}
and $A_m = 4$ be the area of a ``master'' quadrilateral with vertices
\[
\begin{array}{lcrcrcrl}
  \vec P_0 &= (&-1&,&-1&,& 0&)\\
  \vec P_1 &= (& 1&,&-1&,& 0&)\\
  \vec P_2 &= (& 1&,& 1&,& 0&)\\
  \vec P_3 &= (&-1&,& 1&,& 0&).
\end{array}
\]
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
\[
q = \frac{|J| A_m}{A} = \frac{4|J|}{A}.
\]
Distortion is a measure of how well-behaved the mapping from
parameter space to world coordinates is.

\quadmetrictable{distortion}%
{$1$}%                                      Dimension
{$[0.5,1]$}%                                Acceptable range
{$[0,1]$}%                                  Normal range
{$[-DBL\_MAX,DBL\_MAX]$}%                   Full range
{$1$}%                                      Unit square
{\cite{ideas:xx}}%                          Citation
{v\_quad\_distortion}%                      Verdict function name