V_TetMetric.cpp

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/*=========================================================================

  Module:    $RCSfile: V_TetMetric.cpp,v $

  Copyright (c) 2006 Sandia Corporation.
  All rights reserved.
  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.

     This software is distributed WITHOUT ANY WARRANTY; without even
     the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
     PURPOSE.  See the above copyright notice for more information.

=========================================================================*/

/*
 *
 * TetMetric.cpp contains quality calculations for Tets
 *
 * This file is part of VERDICT
 *
 */

#define VERDICT_EXPORTS

#include "moab/verdict.h"
#include "verdict_defines.hpp"
#include "VerdictVector.hpp"
#include "V_GaussIntegration.hpp"
#include 

//! the average volume of a tet
double verdict_tet_size = 0;

/*!
  set the average volume of a tet
*/
C_FUNC_DEF void v_set_tet_size( double size )
{
    verdict_tet_size = size;
}

/*!
  get the weights based on the average size
  of a tet
*/
int get_weight( VerdictVector& w1, VerdictVector& w2, VerdictVector& w3 )
{
    static const double rt3       = sqrt( 3.0 );
    static const double root_of_2 = sqrt( 2.0 );

    w1.set( 1, 0, 0 );
    w2.set( 0.5, 0.5 * rt3, 0 );
    w3.set( 0.5, rt3 / 6.0, root_of_2 / rt3 );

    double scale = pow( 6. * verdict_tet_size / determinant( w1, w2, w3 ), 0.3333333333333 );

    w1 *= scale;
    w2 *= scale;
    w3 *= scale;

    return 1;
}

/*!
   the edge ratio of a tet

   NB (P. Pebay 01/22/07):
     Hmax / Hmin where Hmax and Hmin are respectively the maximum and the
     minimum edge lengths
*/
C_FUNC_DEF double v_tet_edge_ratio( int /*num_nodes*/, double coordinates[][3] )
{
    VerdictVector a, b, c, d, e, f;

    a.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
           coordinates[1][2] - coordinates[0][2] );

    b.set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
           coordinates[2][2] - coordinates[1][2] );

    c.set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
           coordinates[0][2] - coordinates[2][2] );

    d.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
           coordinates[3][2] - coordinates[0][2] );

    e.set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
           coordinates[3][2] - coordinates[1][2] );

    f.set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
           coordinates[3][2] - coordinates[2][2] );

    double a2 = a.length_squared();
    double b2 = b.length_squared();
    double c2 = c.length_squared();
    double d2 = d.length_squared();
    double e2 = e.length_squared();
    double f2 = f.length_squared();

    double m2, M2, mab, mcd, mef, Mab, Mcd, Mef;

    if( a2 < b2 )
    {
        mab = a2;
        Mab = b2;
    }
    else  // b2 <= a2
    {
        mab = b2;
        Mab = a2;
    }
    if( c2 < d2 )
    {
        mcd = c2;
        Mcd = d2;
    }
    else  // d2 <= c2
    {
        mcd = d2;
        Mcd = c2;
    }
    if( e2 < f2 )
    {
        mef = e2;
        Mef = f2;
    }
    else  // f2 <= e2
    {
        mef = f2;
        Mef = e2;
    }

    m2 = mab < mcd ? mab : mcd;
    m2 = m2 < mef ? m2 : mef;

    if( m2 < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;

    M2 = Mab > Mcd ? Mab : Mcd;
    M2 = M2 > Mef ? M2 : Mef;

    double edge_ratio = sqrt( M2 / m2 );

    if( edge_ratio > 0 ) return (double)VERDICT_MIN( edge_ratio, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( edge_ratio, -VERDICT_DBL_MAX );
}

/*!
  the scaled jacobian of a tet

  minimum of the jacobian divided by the lengths of 3 edge vectors

*/
C_FUNC_DEF double v_tet_scaled_jacobian( int /*num_nodes*/, double coordinates[][3] )
{

    VerdictVector side0, side1, side2, side3, side4, side5;

    side0.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
               coordinates[1][2] - coordinates[0][2] );

    side1.set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
               coordinates[2][2] - coordinates[1][2] );

    side2.set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
               coordinates[0][2] - coordinates[2][2] );

    side3.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
               coordinates[3][2] - coordinates[0][2] );

    side4.set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
               coordinates[3][2] - coordinates[1][2] );

    side5.set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
               coordinates[3][2] - coordinates[2][2] );

    double jacobi;

    jacobi = side3 % ( side2 * side0 );

    // products of lengths squared of each edge attached to a node.
    double length_squared[4] = { side0.length_squared() * side2.length_squared() * side3.length_squared(),
                                 side0.length_squared() * side1.length_squared() * side4.length_squared(),
                                 side1.length_squared() * side2.length_squared() * side5.length_squared(),
                                 side3.length_squared() * side4.length_squared() * side5.length_squared() };
    int which_node           = 0;
    if( length_squared[1] > length_squared[which_node] ) which_node = 1;
    if( length_squared[2] > length_squared[which_node] ) which_node = 2;
    if( length_squared[3] > length_squared[which_node] ) which_node = 3;

    double length_product = sqrt( length_squared[which_node] );
    if( length_product < fabs( jacobi ) ) length_product = fabs( jacobi );

    if( length_product < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;

    static const double root_of_2 = sqrt( 2.0 );

    return (double)( root_of_2 * jacobi / length_product );
}

/*!
  The radius ratio of a tet

  NB (P. Pebay 04/16/07):
    CR / (3.0 * IR) where CR is the circumsphere radius and IR is the inscribed
    sphere radius.
    Note that this metric is similar to the aspect beta of a tet, except that
    it does not return VERDICT_DBL_MAX if the element has negative orientation.
*/
C_FUNC_DEF double v_tet_radius_ratio( int /*num_nodes*/, double coordinates[][3] )
{

    // Determine side vectors
    VerdictVector side[6];

    side[0].set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                 coordinates[1][2] - coordinates[0][2] );

    side[1].set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                 coordinates[2][2] - coordinates[1][2] );

    side[2].set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
                 coordinates[0][2] - coordinates[2][2] );

    side[3].set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
                 coordinates[3][2] - coordinates[0][2] );

    side[4].set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
                 coordinates[3][2] - coordinates[1][2] );

    side[5].set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
                 coordinates[3][2] - coordinates[2][2] );

    VerdictVector numerator = side[3].length_squared() * ( side[2] * side[0] ) +
                              side[2].length_squared() * ( side[3] * side[0] ) +
                              side[0].length_squared() * ( side[3] * side[2] );

    double area_sum;
    area_sum = ( ( side[2] * side[0] ).length() + ( side[3] * side[0] ).length() + ( side[4] * side[1] ).length() +
                 ( side[3] * side[2] ).length() ) *
               0.5;

    double volume = v_tet_volume( 4, coordinates );

    if( fabs( volume ) < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;
    else
    {
        double radius_ratio;
        radius_ratio = numerator.length() * area_sum / ( 108 * volume * volume );

        return (double)VERDICT_MIN( radius_ratio, VERDICT_DBL_MAX );
    }
}

/*!
  The radius ratio of a positively-oriented tet, a.k.a. "aspect beta"

  NB (P. Pebay 04/16/07):
    CR / (3.0 * IR) where CR is the circumsphere radius and IR is the inscribed
    sphere radius if the element has positive orientation.
    Note that this metric is similar to the radius ratio of a tet, except that
    it returns VERDICT_DBL_MAX if the element has negative orientation.

*/
C_FUNC_DEF double v_tet_aspect_beta( int /*num_nodes*/, double coordinates[][3] )
{

    // Determine side vectors
    VerdictVector side[6];

    side[0].set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                 coordinates[1][2] - coordinates[0][2] );

    side[1].set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                 coordinates[2][2] - coordinates[1][2] );

    side[2].set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
                 coordinates[0][2] - coordinates[2][2] );

    side[3].set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
                 coordinates[3][2] - coordinates[0][2] );

    side[4].set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
                 coordinates[3][2] - coordinates[1][2] );

    side[5].set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
                 coordinates[3][2] - coordinates[2][2] );

    VerdictVector numerator = side[3].length_squared() * ( side[2] * side[0] ) +
                              side[2].length_squared() * ( side[3] * side[0] ) +
                              side[0].length_squared() * ( side[3] * side[2] );

    double area_sum;
    area_sum = ( ( side[2] * side[0] ).length() + ( side[3] * side[0] ).length() + ( side[4] * side[1] ).length() +
                 ( side[3] * side[2] ).length() ) *
               0.5;

    double volume = v_tet_volume( 4, coordinates );

    if( volume < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;
    else
    {
        double radius_ratio;
        radius_ratio = numerator.length() * area_sum / ( 108 * volume * volume );

        return (double)VERDICT_MIN( radius_ratio, VERDICT_DBL_MAX );
    }
}

/*!
  The aspect ratio of a tet

  NB (P. Pebay 01/22/07):
    Hmax / (2 sqrt(6) r) where Hmax and r respectively denote the greatest edge
    length and the inradius of the tetrahedron
*/
C_FUNC_DEF double v_tet_aspect_ratio( int /*num_nodes*/, double coordinates[][3] )
{
    static const double normal_coeff = sqrt( 6. ) / 12.;

    // Determine side vectors
    VerdictVector ab, bc, ac, ad, bd, cd;

    ab.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
            coordinates[1][2] - coordinates[0][2] );

    ac.set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
            coordinates[2][2] - coordinates[0][2] );

    ad.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
            coordinates[3][2] - coordinates[0][2] );

    double detTet = ab % ( ac * ad );

    if( detTet < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;

    bc.set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
            coordinates[2][2] - coordinates[1][2] );

    bd.set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
            coordinates[3][2] - coordinates[1][2] );

    cd.set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
            coordinates[3][2] - coordinates[2][2] );

    double ab2 = ab.length_squared();
    double bc2 = bc.length_squared();
    double ac2 = ac.length_squared();
    double ad2 = ad.length_squared();
    double bd2 = bd.length_squared();
    double cd2 = cd.length_squared();

    double A  = ab2 > bc2 ? ab2 : bc2;
    double B  = ac2 > ad2 ? ac2 : ad2;
    double C  = bd2 > cd2 ? bd2 : cd2;
    double D  = A > B ? A : B;
    double hm = D > C ? sqrt( D ) : sqrt( C );

    bd = ab * bc;
    A  = bd.length();
    bd = ab * ad;
    B  = bd.length();
    bd = ac * ad;
    C  = bd.length();
    bd = bc * cd;
    D  = bd.length();

    double aspect_ratio;
    aspect_ratio = normal_coeff * hm * ( A + B + C + D ) / fabs( detTet );

    if( aspect_ratio > 0 ) return (double)VERDICT_MIN( aspect_ratio, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( aspect_ratio, -VERDICT_DBL_MAX );
}

/*!
  the aspect gamma of a tet

  srms^3 / (8.48528137423857*V) where srms = sqrt(sum(Si^2)/6), where Si is the edge length
*/
C_FUNC_DEF double v_tet_aspect_gamma( int /*num_nodes*/, double coordinates[][3] )
{

    // Determine side vectors
    VerdictVector side0, side1, side2, side3, side4, side5;

    side0.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
               coordinates[1][2] - coordinates[0][2] );

    side1.set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
               coordinates[2][2] - coordinates[1][2] );

    side2.set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
               coordinates[0][2] - coordinates[2][2] );

    side3.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
               coordinates[3][2] - coordinates[0][2] );

    side4.set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
               coordinates[3][2] - coordinates[1][2] );

    side5.set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
               coordinates[3][2] - coordinates[2][2] );

    double volume = fabs( v_tet_volume( 4, coordinates ) );

    if( volume < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;
    else
    {
        double srms = sqrt( ( side0.length_squared() + side1.length_squared() + side2.length_squared() +
                              side3.length_squared() + side4.length_squared() + side5.length_squared() ) /
                            6.0 );

        double aspect_ratio_gamma = pow( srms, 3 ) / ( 8.48528137423857 * volume );
        return (double)aspect_ratio_gamma;
    }
}

/*!
  The aspect frobenius of a tet

  NB (P. Pebay 01/22/07):
    Frobenius condition number when the reference element is regular
*/
C_FUNC_DEF double v_tet_aspect_frobenius( int /*num_nodes*/, double coordinates[][3] )
{
    static const double normal_exp = 1. / 3.;

    VerdictVector ab, ac, ad;

    ab.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
            coordinates[1][2] - coordinates[0][2] );

    ac.set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
            coordinates[2][2] - coordinates[0][2] );

    ad.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
            coordinates[3][2] - coordinates[0][2] );

    double denominator = ab % ( ac * ad );
    denominator *= denominator;
    denominator *= 2.;
    denominator = 3. * pow( denominator, normal_exp );

    if( denominator < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;

    double u[3];
    ab.get_xyz( u );
    double v[3];
    ac.get_xyz( v );
    double w[3];
    ad.get_xyz( w );

    double numerator = u[0] * u[0] + u[1] * u[1] + u[2] * u[2];
    numerator += v[0] * v[0] + v[1] * v[1] + v[2] * v[2];
    numerator += w[0] * w[0] + w[1] * w[1] + w[2] * w[2];
    numerator *= 1.5;
    numerator -= v[0] * u[0] + v[1] * u[1] + v[2] * u[2];
    numerator -= w[0] * u[0] + w[1] * u[1] + w[2] * u[2];
    numerator -= w[0] * v[0] + w[1] * v[1] + w[2] * v[2];

    double aspect_frobenius = numerator / denominator;

    if( aspect_frobenius > 0 ) return (double)VERDICT_MIN( aspect_frobenius, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( aspect_frobenius, -VERDICT_DBL_MAX );
}

/*!
  The minimum angle of a tet

  NB (P. Pebay 01/22/07):
    minimum nonoriented dihedral angle
*/
C_FUNC_DEF double v_tet_minimum_angle( int /*num_nodes*/, double coordinates[][3] )
{
    static const double normal_coeff = 180. * .3183098861837906715377675267450287;

    // Determine side vectors
    VerdictVector ab, bc, ad, cd;

    ab.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
            coordinates[1][2] - coordinates[0][2] );

    ad.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
            coordinates[3][2] - coordinates[0][2] );

    bc.set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
            coordinates[2][2] - coordinates[1][2] );

    cd.set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
            coordinates[3][2] - coordinates[2][2] );

    VerdictVector abc = ab * bc;
    double nabc       = abc.length();
    VerdictVector abd = ab * ad;
    double nabd       = abd.length();
    VerdictVector acd = ad * cd;
    double nacd       = acd.length();
    VerdictVector bcd = bc * cd;
    double nbcd       = bcd.length();

    double alpha   = acos( ( abc % abd ) / ( nabc * nabd ) );
    double beta    = acos( ( abc % acd ) / ( nabc * nacd ) );
    double gamma   = acos( ( abc % bcd ) / ( nabc * nbcd ) );
    double delta   = acos( ( abd % acd ) / ( nabd * nacd ) );
    double epsilon = acos( ( abd % bcd ) / ( nabd * nbcd ) );
    double zeta    = acos( ( acd % bcd ) / ( nacd * nbcd ) );

    alpha = alpha < beta ? alpha : beta;
    alpha = alpha < gamma ? alpha : gamma;
    alpha = alpha < delta ? alpha : delta;
    alpha = alpha < epsilon ? alpha : epsilon;
    alpha = alpha < zeta ? alpha : zeta;
    alpha *= normal_coeff;

    if( alpha < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;

    if( alpha > 0 ) return (double)VERDICT_MIN( alpha, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( alpha, -VERDICT_DBL_MAX );
}

/*!
  The collapse ratio of a tet
*/
C_FUNC_DEF double v_tet_collapse_ratio( int /*num_nodes*/, double coordinates[][3] )
{
    // Determine side vectors
    VerdictVector e01, e02, e03, e12, e13, e23;

    e01.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
             coordinates[1][2] - coordinates[0][2] );

    e02.set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
             coordinates[2][2] - coordinates[0][2] );

    e03.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
             coordinates[3][2] - coordinates[0][2] );

    e12.set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
             coordinates[2][2] - coordinates[1][2] );

    e13.set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
             coordinates[3][2] - coordinates[1][2] );

    e23.set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
             coordinates[3][2] - coordinates[2][2] );

    double l[6];
    l[0] = e01.length();
    l[1] = e02.length();
    l[2] = e03.length();
    l[3] = e12.length();
    l[4] = e13.length();
    l[5] = e23.length();

    // Find longest edge for each bounding triangle of tetrahedron
    double l012 = l[4] > l[0] ? l[4] : l[0];
    l012        = l[1] > l012 ? l[1] : l012;
    double l031 = l[0] > l[2] ? l[0] : l[2];
    l031        = l[3] > l031 ? l[3] : l031;
    double l023 = l[2] > l[1] ? l[2] : l[1];
    l023        = l[5] > l023 ? l[5] : l023;
    double l132 = l[4] > l[3] ? l[4] : l[3];
    l132        = l[5] > l132 ? l[5] : l132;

    // Compute collapse ratio for each vertex/triangle pair
    VerdictVector N;
    double h, magN;
    double cr;
    double crMin;

    N     = e01 * e02;
    magN  = N.length();
    h     = ( e03 % N ) / magN;  // height of vertex 3 above 0-1-2
    crMin = h / l012;            // ratio of height to longest edge of 0-1-2

    N    = e03 * e01;
    magN = N.length();
    h    = ( e02 % N ) / magN;  // height of vertex 2 above 0-3-1
    cr   = h / l031;            // ratio of height to longest edge of 0-3-1
    if( cr < crMin ) crMin = cr;

    N    = e02 * e03;
    magN = N.length();
    h    = ( e01 % N ) / magN;  // height of vertex 1 above 0-2-3
    cr   = h / l023;            // ratio of height to longest edge of 0-2-3
    if( cr < crMin ) crMin = cr;

    N    = e12 * e13;
    magN = N.length();
    h    = ( e01 % N ) / magN;  // height of vertex 0 above 1-3-2
    cr   = h / l132;            // ratio of height to longest edge of 1-3-2
    if( cr < crMin ) crMin = cr;

    if( crMin < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;
    if( crMin > 0 ) return (double)VERDICT_MIN( crMin, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( crMin, -VERDICT_DBL_MAX );
}

/*!
  the volume of a tet

  1/6 * jacobian at a corner node
*/
C_FUNC_DEF double v_tet_volume( int /*num_nodes*/, double coordinates[][3] )
{

    // Determine side vectors
    VerdictVector side0, side2, side3;

    side0.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
               coordinates[1][2] - coordinates[0][2] );

    side2.set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
               coordinates[0][2] - coordinates[2][2] );

    side3.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
               coordinates[3][2] - coordinates[0][2] );

    return (double)( ( side3 % ( side2 * side0 ) ) / 6.0 );
}

/*!
  the condition of a tet

  condition number of the jacobian matrix at any corner
*/
C_FUNC_DEF double v_tet_condition( int /*num_nodes*/, double coordinates[][3] )
{

    double condition, term1, term2, det;
    double rt3 = sqrt( 3.0 );
    double rt6 = sqrt( 6.0 );

    VerdictVector side0, side2, side3;

    side0.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
               coordinates[1][2] - coordinates[0][2] );

    side2.set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
               coordinates[0][2] - coordinates[2][2] );

    side3.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
               coordinates[3][2] - coordinates[0][2] );

    VerdictVector c_1, c_2, c_3;

    c_1 = side0;
    c_2 = ( -2 * side2 - side0 ) / rt3;
    c_3 = ( 3 * side3 + side2 - side0 ) / rt6;

    term1 = c_1 % c_1 + c_2 % c_2 + c_3 % c_3;
    term2 = ( c_1 * c_2 ) % ( c_1 * c_2 ) + ( c_2 * c_3 ) % ( c_2 * c_3 ) + ( c_1 * c_3 ) % ( c_1 * c_3 );
    det   = c_1 % ( c_2 * c_3 );

    if( det <= VERDICT_DBL_MIN )
        return VERDICT_DBL_MAX;
    else
        condition = sqrt( term1 * term2 ) / ( 3.0 * det );

    return (double)condition;
}

/*!
  the jacobian of a tet

  TODO
*/
C_FUNC_DEF double v_tet_jacobian( int /*num_nodes*/, double coordinates[][3] )
{
    VerdictVector side0, side2, side3;

    side0.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
               coordinates[1][2] - coordinates[0][2] );

    side2.set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
               coordinates[0][2] - coordinates[2][2] );

    side3.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
               coordinates[3][2] - coordinates[0][2] );

    return (double)( side3 % ( side2 * side0 ) );
}

/*!
  the shape of a tet

  3/ condition number of weighted jacobian matrix
*/
C_FUNC_DEF double v_tet_shape( int /*num_nodes*/, double coordinates[][3] )
{

    static const double two_thirds = 2.0 / 3.0;
    static const double root_of_2  = sqrt( 2.0 );

    VerdictVector edge0, edge2, edge3;

    edge0.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
               coordinates[1][2] - coordinates[0][2] );

    edge2.set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
               coordinates[0][2] - coordinates[2][2] );

    edge3.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
               coordinates[3][2] - coordinates[0][2] );

    double jacobian = edge3 % ( edge2 * edge0 );
    if( jacobian < VERDICT_DBL_MIN )
    {
        return (double)0.0;
    }
    double num = 3 * pow( root_of_2 * jacobian, two_thirds );
    double den =
        1.5 * ( edge0 % edge0 + edge2 % edge2 + edge3 % edge3 ) - ( edge0 % -edge2 + -edge2 % edge3 + edge3 % edge0 );

    if( den < VERDICT_DBL_MIN ) return (double)0.0;

    return (double)VERDICT_MAX( num / den, 0 );
}

/*!
  the relative size of a tet

  Min(J,1/J), where J is the determinant of the weighted Jacobian matrix
*/
C_FUNC_DEF double v_tet_relative_size_squared( int /*num_nodes*/, double coordinates[][3] )
{
    double size;
    VerdictVector w1, w2, w3;
    get_weight( w1, w2, w3 );
    double avg_volume = ( w1 % ( w2 * w3 ) ) / 6.0;

    double volume = v_tet_volume( 4, coordinates );

    if( avg_volume < VERDICT_DBL_MIN )
        return 0.0;
    else
    {
        size = volume / avg_volume;
        if( size <= VERDICT_DBL_MIN ) return 0.0;
        if( size > 1 ) size = (double)( 1 ) / size;
    }
    return (double)( size * size );
}

/*!
  the shape and size of a tet

  Product of the shape and relative size
*/
C_FUNC_DEF double v_tet_shape_and_size( int num_nodes, double coordinates[][3] )
{

    double shape, size;
    shape = v_tet_shape( num_nodes, coordinates );
    size  = v_tet_relative_size_squared( num_nodes, coordinates );

    return (double)( shape * size );
}

/*!
  the distortion of a tet
*/
C_FUNC_DEF double v_tet_distortion( int num_nodes, double coordinates[][3] )
{

    double distortion          = VERDICT_DBL_MAX;
    int number_of_gauss_points = 0;
    if( num_nodes == 4 )
        // for linear tet, the distortion is always 1 because
        // straight edge tets are the target shape for tet
        return 1.0;

    else if( num_nodes == 10 )
        // use four integration points for quadratic tet
        number_of_gauss_points = 4;

    int number_dims                  = 3;
    int total_number_of_gauss_points = number_of_gauss_points;
    // use is_tri=1 to indicate this is for tet in 3D
    int is_tri = 1;

    double shape_function[maxTotalNumberGaussPoints][maxNumberNodes];
    double dndy1[maxTotalNumberGaussPoints][maxNumberNodes];
    double dndy2[maxTotalNumberGaussPoints][maxNumberNodes];
    double dndy3[maxTotalNumberGaussPoints][maxNumberNodes];
    double weight[maxTotalNumberGaussPoints];

    // create an object of GaussIntegration for tet
    GaussIntegration::initialize( number_of_gauss_points, num_nodes, number_dims, is_tri );
    GaussIntegration::calculate_shape_function_3d_tet();
    GaussIntegration::get_shape_func( shape_function[0], dndy1[0], dndy2[0], dndy3[0], weight );

    // vector xxi is the derivative vector of coordinates w.r.t local xi coordinate in the
    // computation space
    // vector xet is the derivative vector of coordinates w.r.t local et coordinate in the
    // computation space
    // vector xze is the derivative vector of coordinates w.r.t local ze coordinate in the
    // computation space
    VerdictVector xxi, xet, xze, xin;

    double jacobian, minimum_jacobian;
    double element_volume = 0.0;
    minimum_jacobian      = VERDICT_DBL_MAX;

    // calculate element volume
    int ife, ja;
    for( ife = 0; ife < total_number_of_gauss_points; ife++ )
    {
        xxi.set( 0.0, 0.0, 0.0 );
        xet.set( 0.0, 0.0, 0.0 );
        xze.set( 0.0, 0.0, 0.0 );

        for( ja = 0; ja < num_nodes; ja++ )
        {
            xin.set( coordinates[ja][0], coordinates[ja][1], coordinates[ja][2] );
            xxi += dndy1[ife][ja] * xin;
            xet += dndy2[ife][ja] * xin;
            xze += dndy3[ife][ja] * xin;
        }

        // determinant
        jacobian = xxi % ( xet * xze );
        if( minimum_jacobian > jacobian ) minimum_jacobian = jacobian;

        element_volume += weight[ife] * jacobian;
    }  // element_volume is 6 times the actual volume

    // loop through all nodes
    double dndy1_at_node[maxNumberNodes][maxNumberNodes];
    double dndy2_at_node[maxNumberNodes][maxNumberNodes];
    double dndy3_at_node[maxNumberNodes][maxNumberNodes];

    GaussIntegration::calculate_derivative_at_nodes_3d_tet( dndy1_at_node, dndy2_at_node, dndy3_at_node );
    int node_id;
    for( node_id = 0; node_id < num_nodes; node_id++ )
    {
        xxi.set( 0.0, 0.0, 0.0 );
        xet.set( 0.0, 0.0, 0.0 );
        xze.set( 0.0, 0.0, 0.0 );

        for( ja = 0; ja < num_nodes; ja++ )
        {
            xin.set( coordinates[ja][0], coordinates[ja][1], coordinates[ja][2] );
            xxi += dndy1_at_node[node_id][ja] * xin;
            xet += dndy2_at_node[node_id][ja] * xin;
            xze += dndy3_at_node[node_id][ja] * xin;
        }

        jacobian = xxi % ( xet * xze );
        if( minimum_jacobian > jacobian ) minimum_jacobian = jacobian;
    }
    distortion = minimum_jacobian / element_volume;

    return (double)distortion;
}

/*!
  the quality metrics of a tet
*/
C_FUNC_DEF void v_tet_quality( int num_nodes,
                               double coordinates[][3],
                               unsigned int metrics_request_flag,
                               TetMetricVals* metric_vals )
{

    memset( metric_vals, 0, sizeof( TetMetricVals ) );

    /*

      node numbers and edge numbers below



               3
               +            edge 0 is node 0 to 1
              +|+           edge 1 is node 1 to 2
            3/ | \5         edge 2 is node 0 to 2
            / 4|  \         edge 3 is node 0 to 3
          0 - -|- + 2       edge 4 is node 1 to 3
            \  |  +         edge 5 is node 2 to 3
            0\ | /1
              +|/           edge 2 is behind edge 4
               1


    */

    // lets start with making the vectors
    VerdictVector edges[6];
    edges[0].set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
                  coordinates[1][2] - coordinates[0][2] );

    edges[1].set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
                  coordinates[2][2] - coordinates[1][2] );

    edges[2].set( coordinates[0][0] - coordinates[2][0], coordinates[0][1] - coordinates[2][1],
                  coordinates[0][2] - coordinates[2][2] );

    edges[3].set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
                  coordinates[3][2] - coordinates[0][2] );

    edges[4].set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
                  coordinates[3][2] - coordinates[1][2] );

    edges[5].set( coordinates[3][0] - coordinates[2][0], coordinates[3][1] - coordinates[2][1],
                  coordinates[3][2] - coordinates[2][2] );

    // common numbers
    static const double root_of_2 = sqrt( 2.0 );

    // calculate the jacobian
    static const int do_jacobian = V_TET_JACOBIAN | V_TET_VOLUME | V_TET_ASPECT_BETA | V_TET_ASPECT_GAMMA |
                                   V_TET_SHAPE | V_TET_RELATIVE_SIZE_SQUARED | V_TET_SHAPE_AND_SIZE |
                                   V_TET_SCALED_JACOBIAN | V_TET_CONDITION;
    if( metrics_request_flag & do_jacobian )
    {
        metric_vals->jacobian = (double)( edges[3] % ( edges[2] * edges[0] ) );
    }

    // calculate the volume
    if( metrics_request_flag & V_TET_VOLUME )
    {
        metric_vals->volume = (double)( metric_vals->jacobian / 6.0 );
    }

    // calculate aspect ratio
    if( metrics_request_flag & V_TET_ASPECT_BETA )
    {
        double surface_area = ( ( edges[2] * edges[0] ).length() + ( edges[3] * edges[0] ).length() +
                                ( edges[4] * edges[1] ).length() + ( edges[3] * edges[2] ).length() ) *
                              0.5;

        VerdictVector numerator = edges[3].length_squared() * ( edges[2] * edges[0] ) +
                                  edges[2].length_squared() * ( edges[3] * edges[0] ) +
                                  edges[0].length_squared() * ( edges[3] * edges[2] );

        double volume = metric_vals->jacobian / 6.0;

        if( volume < VERDICT_DBL_MIN )
            metric_vals->aspect_beta = (double)( VERDICT_DBL_MAX );
        else
            metric_vals->aspect_beta = (double)( numerator.length() * surface_area / ( 108 * volume * volume ) );
    }

    // calculate the aspect gamma
    if( metrics_request_flag & V_TET_ASPECT_GAMMA )
    {
        double volume = fabs( metric_vals->jacobian / 6.0 );
        if( fabs( volume ) < VERDICT_DBL_MIN )
            metric_vals->aspect_gamma = VERDICT_DBL_MAX;
        else
        {
            double srms = sqrt( ( edges[0].length_squared() + edges[1].length_squared() + edges[2].length_squared() +
                                  edges[3].length_squared() + edges[4].length_squared() + edges[5].length_squared() ) /
                                6.0 );

            // cube the srms
            srms *= ( srms * srms );
            metric_vals->aspect_gamma = (double)( srms / ( 8.48528137423857 * volume ) );
        }
    }

    // calculate the shape of the tet
    if( metrics_request_flag & ( V_TET_SHAPE | V_TET_SHAPE_AND_SIZE ) )
    {
        // if the jacobian is non-positive, the shape is 0
        if( metric_vals->jacobian < VERDICT_DBL_MIN )
        {
            metric_vals->shape = (double)0.0;
        }
        else
        {
            static const double two_thirds = 2.0 / 3.0;
            double num                     = 3.0 * pow( root_of_2 * metric_vals->jacobian, two_thirds );
            double den                     = 1.5 * ( edges[0] % edges[0] + edges[2] % edges[2] + edges[3] % edges[3] ) -
                         ( edges[0] % -edges[2] + -edges[2] % edges[3] + edges[3] % edges[0] );

            if( den < VERDICT_DBL_MIN )
                metric_vals->shape = (double)0.0;
            else
                metric_vals->shape = (double)VERDICT_MAX( num / den, 0 );
        }
    }

    // calculate the relative size of the tet
    if( metrics_request_flag & ( V_TET_RELATIVE_SIZE_SQUARED | V_TET_SHAPE_AND_SIZE ) )
    {
        VerdictVector w1, w2, w3;
        get_weight( w1, w2, w3 );
        double avg_vol = ( w1 % ( w2 * w3 ) ) / 6;

        if( avg_vol < VERDICT_DBL_MIN )
            metric_vals->relative_size_squared = 0.0;
        else
        {
            double tmp = metric_vals->jacobian / ( 6 * avg_vol );
            if( tmp < VERDICT_DBL_MIN )
                metric_vals->relative_size_squared = 0.0;
            else
            {
                tmp *= tmp;
                metric_vals->relative_size_squared = (double)VERDICT_MIN( tmp, 1 / tmp );
            }
        }
    }

    // calculate the shape and size
    if( metrics_request_flag & V_TET_SHAPE_AND_SIZE )
    {
        metric_vals->shape_and_size = (double)( metric_vals->shape * metric_vals->relative_size_squared );
    }

    // calculate the scaled jacobian
    if( metrics_request_flag & V_TET_SCALED_JACOBIAN )
    {
        // find out which node the normalized jacobian can be calculated at
        // and it will be the smaller than at other nodes
        double length_squared[4] = { edges[0].length_squared() * edges[2].length_squared() * edges[3].length_squared(),
                                     edges[0].length_squared() * edges[1].length_squared() * edges[4].length_squared(),
                                     edges[1].length_squared() * edges[2].length_squared() * edges[5].length_squared(),
                                     edges[3].length_squared() * edges[4].length_squared() *
                                         edges[5].length_squared() };

        int which_node = 0;
        if( length_squared[1] > length_squared[which_node] ) which_node = 1;
        if( length_squared[2] > length_squared[which_node] ) which_node = 2;
        if( length_squared[3] > length_squared[which_node] ) which_node = 3;

        // find the scaled jacobian at this node
        double length_product = sqrt( length_squared[which_node] );
        if( length_product < fabs( metric_vals->jacobian ) ) length_product = fabs( metric_vals->jacobian );

        if( length_product < VERDICT_DBL_MIN )
            metric_vals->scaled_jacobian = (double)VERDICT_DBL_MAX;
        else
            metric_vals->scaled_jacobian = (double)( root_of_2 * metric_vals->jacobian / length_product );
    }

    // calculate the condition number
    if( metrics_request_flag & V_TET_CONDITION )
    {
        static const double root_of_3 = sqrt( 3.0 );
        static const double root_of_6 = sqrt( 6.0 );

        VerdictVector c_1, c_2, c_3;
        c_1 = edges[0];
        c_2 = ( -2 * edges[2] - edges[0] ) / root_of_3;
        c_3 = ( 3 * edges[3] + edges[2] - edges[0] ) / root_of_6;

        double term1 = c_1 % c_1 + c_2 % c_2 + c_3 % c_3;
        double term2 = ( c_1 * c_2 ) % ( c_1 * c_2 ) + ( c_2 * c_3 ) % ( c_2 * c_3 ) + ( c_3 * c_1 ) % ( c_3 * c_1 );

        double det = c_1 % ( c_2 * c_3 );

        if( det <= VERDICT_DBL_MIN )
            metric_vals->condition = (double)VERDICT_DBL_MAX;
        else
            metric_vals->condition = (double)( sqrt( term1 * term2 ) / ( 3.0 * det ) );
    }

    // calculate the distortion
    if( metrics_request_flag & V_TET_DISTORTION )
    {
        metric_vals->distortion = v_tet_distortion( num_nodes, coordinates );
    }

    // check for overflow
    if( metrics_request_flag & V_TET_ASPECT_BETA )
    {
        if( metric_vals->aspect_beta > 0 )
            metric_vals->aspect_beta = (double)VERDICT_MIN( metric_vals->aspect_beta, VERDICT_DBL_MAX );
        metric_vals->aspect_beta = (double)VERDICT_MAX( metric_vals->aspect_beta, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_ASPECT_GAMMA )
    {
        if( metric_vals->aspect_gamma > 0 )
            metric_vals->aspect_gamma = (double)VERDICT_MIN( metric_vals->aspect_gamma, VERDICT_DBL_MAX );
        metric_vals->aspect_gamma = (double)VERDICT_MAX( metric_vals->aspect_gamma, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_VOLUME )
    {
        if( metric_vals->volume > 0 ) metric_vals->volume = (double)VERDICT_MIN( metric_vals->volume, VERDICT_DBL_MAX );
        metric_vals->volume = (double)VERDICT_MAX( metric_vals->volume, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_CONDITION )
    {
        if( metric_vals->condition > 0 )
            metric_vals->condition = (double)VERDICT_MIN( metric_vals->condition, VERDICT_DBL_MAX );
        metric_vals->condition = (double)VERDICT_MAX( metric_vals->condition, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_JACOBIAN )
    {
        if( metric_vals->jacobian > 0 )
            metric_vals->jacobian = (double)VERDICT_MIN( metric_vals->jacobian, VERDICT_DBL_MAX );
        metric_vals->jacobian = (double)VERDICT_MAX( metric_vals->jacobian, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_SCALED_JACOBIAN )
    {
        if( metric_vals->scaled_jacobian > 0 )
            metric_vals->scaled_jacobian = (double)VERDICT_MIN( metric_vals->scaled_jacobian, VERDICT_DBL_MAX );
        metric_vals->scaled_jacobian = (double)VERDICT_MAX( metric_vals->scaled_jacobian, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_SHAPE )
    {
        if( metric_vals->shape > 0 ) metric_vals->shape = (double)VERDICT_MIN( metric_vals->shape, VERDICT_DBL_MAX );
        metric_vals->shape = (double)VERDICT_MAX( metric_vals->shape, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_RELATIVE_SIZE_SQUARED )
    {
        if( metric_vals->relative_size_squared > 0 )
            metric_vals->relative_size_squared =
                (double)VERDICT_MIN( metric_vals->relative_size_squared, VERDICT_DBL_MAX );
        metric_vals->relative_size_squared =
            (double)VERDICT_MAX( metric_vals->relative_size_squared, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_SHAPE_AND_SIZE )
    {
        if( metric_vals->shape_and_size > 0 )
            metric_vals->shape_and_size = (double)VERDICT_MIN( metric_vals->shape_and_size, VERDICT_DBL_MAX );
        metric_vals->shape_and_size = (double)VERDICT_MAX( metric_vals->shape_and_size, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_DISTORTION )
    {
        if( metric_vals->distortion > 0 )
            metric_vals->distortion = (double)VERDICT_MIN( metric_vals->distortion, VERDICT_DBL_MAX );
        metric_vals->distortion = (double)VERDICT_MAX( metric_vals->distortion, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_TET_ASPECT_RATIO ) metric_vals->aspect_ratio = v_tet_aspect_ratio( 4, coordinates );

    if( metrics_request_flag & V_TET_ASPECT_FROBENIUS )
        metric_vals->aspect_frobenius = v_tet_aspect_frobenius( 4, coordinates );

    if( metrics_request_flag & V_TET_MINIMUM_ANGLE ) metric_vals->minimum_angle = v_tet_minimum_angle( 4, coordinates );

    if( metrics_request_flag & V_TET_COLLAPSE_RATIO )
        metric_vals->collapse_ratio = v_tet_collapse_ratio( 4, coordinates );

    if( metrics_request_flag & V_TET_RADIUS_RATIO ) metric_vals->radius_ratio = v_tet_radius_ratio( 4, coordinates );
}