%---------------------------Aspect Ratio-----------------------------
\section{Aspect Ratio\label{s:quad-aspect-ratio}}

The aspect ratio of a quadrilateral is: 
\[
q = \frac{L_{\max}(L_0+L_1+L_2+L_3)}{4A},
\]
where $A$ is the area of the quadrilateral. 

Note that, strictly speaking, the aspect ratio is usually defined for
simplicial elements as the ratio of the maximum edge length to the
inradius (\emph{cf.}~\S\ref{s:tri-aspect-ratio} and
~\S\ref{s:tet-aspect-ratio}). However, a planar quadrilateral does not
have, in general, an inscribed circle: such an incircle exists if and
only if $L_0+L_2=L_1+L_3$. Nonetheless, using the expression
of the triangle aspect ratio as given
in~\eqref{eq:triangle_aspect_ratio}, that is, with no explicit
reference to the inradius but only to the perimeter and the area, one
can then directly extrapolate to obtain a meaningful definition of the
quadrangle aspect ratio.

\trimetrictable{aspect ratio}%
{$1$}%                                                Dimension
{$[1,1.3]$}%                                          Acceptable range
{$[1,DBL\_MAX]$}%                                     Normal range
{$[1,DBL\_MAX]$}%                                     Full range
{$1$}%                                                Unit square
{\cite{pebay:04}}%                                    Reference(s)                   
{v\_quad\_aspect\_ratio}%                             Verdict function name