#ifndef Curvature3D_H
#define Curvature3D_H

/* Last edited on DATE TIME by USER */

#include <Energy.h>

typedef struct Curvature3D_data_t { } Curvature3D_data_t;
  /* Ideally, the 3-manifold realized as $\phi(\CT)$ should be as smooth as
    possible.

    Each tetrahedron of $\phi({\CT})$ is contained in some 3-di\-men\-sio\
    -nal affine subspace of $\Real^m$. In general, if $t_1,t_2$ are two ad-
    jacent tetrahedra, their images $\phi(t_1)$ and $\phi(t_2)$ will lie in
    two distinct 3D spaces $V_1,V_2$ of $\Real^{m}$, whose intersection  is
    the plane containing the shared wall $f$. In order to flatten out the mo-
    del $\phi(\CT)$ at that spot, we need to minimize the angle $\theta_f$
    between $V_1$ and $V_2$. This requirement is captured by the {\em curvatu-
    re energy}, defined as

    \begin{equation}
       \Ecurv == \sum_{f \in \WALL{\CT}}
         \left(1+\cos\theta_{f}\right)
       \label{eq.curvature}
    \end{equation}

    \noindent Note that $1-\cos\theta_{f}$ is approximately $\frac{1}{2}
    \,\theta_{f}^{2}$ when $\theta_{f}$ is small. Therefore, minimizing
    $\Ecurv$ tends to make the angles $\theta_{f}$ as small as possible. The
    value of $\cos \hspace{0.10cm} \theta_{f}$ can be computed by the formula
    $\cos \theta_f == - r_1 \dotpr r_2/(\abs{r_1}\,\abs{r_2})$, where $r_i$ is a
    vector in $U_i$ perpendicular to the wall $f$, and pointing into the tetrahe-
    dron $t_i$, for $i==1,2$.
  */

Energy_t Curvature3D_new(void);
  
#endif