%---------------------------Aspect Beta----------------------------------
\section{Aspect Beta}

This metric measures the radius ratio (\emph{cf.}
\S\ref{s:tet-radius-ratio}) of a positively-oriented tetrahedron. Note
that it is equal to the tetrahedral radius ratio.

For a positively-oriented tetrahedron, the aspect $\beta$ is the
quotient of these two radii normalized by $\frac{1}{3}$ so 
that an equilateral tetrahedron has quality of~$1$:
\begin{eqnarray*}
q & = & \frac{R}{3 r} \nonumber \\
  & = & \frac { \left| 
   \normvec{L_3}^2 \left( \vec L_2 \times \vec L_0 \right) + 
   \normvec{L_2}^2 \left( \vec L_3 \times \vec L_0 \right) + 
   \normvec{L_0}^2 \left( \vec L_3 \times \vec L_2 \right)
   \right| A}{108 V^2}.
\end{eqnarray*}

Note that if the tetrahedron has negative orientation, we set $q = DBL\_MAX$.

\tetmetrictable{aspect $\beta$}%
{$1$}%                  Dimension
{$[1,3]$}%              Acceptable range
{$[1,DBL\_MAX]$}%       Normal range
{$[1,DBL\_MAX]$}%       Full range
{$1$}%                  Equilateral tet
{\cite{par:93}}%        Citation
{v\_tet\_aspect\_beta}% Verdict function name