%---------------------------Radius Ratio-----------------------------
\section{Radius Ratio}

Let $h_{\max}$ be the maximum length of all edges and diagonals
\[
  h_{\max} = \max\left(L_{\max},D_{\max}\right)
\]
and ${\cal L}_2$ be the sum of the squares of all edge lengths
\[
{\cal L}_2 = \sum_{i=0}^3\normvec{L_i}^2
\]
and ${\cal A}_i$ be the area of one of the 4 triangles formed by pairs of quadrilateral neighboring edges
\[
  {\cal A}_i = \left|\frac{\alpha_i}{2}\right|.
\]
Then the radius ratio of a planar quadrilateral is
\[
  q = \frac{{\cal L}_2 h_{\max}}{\min_{i\in\{0,1,2,3\}}{\cal A}_i}.
\]

\quadmetrictable{radius ratio}%
{$1$}%                                      Dimension
{$[1,1.3]$}%                                Acceptable range
{$[1,DBL\_MAX]$}%                           Normal range
{$[1,DBL\_MAX]$}%                           Full range
{$1$}%                                      Square
{\cite{pebay:04}}%                          Citation
{v\_quad\_radius\_ratio}%                   Verdict function name