V_QuadMetric.cpp

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

  Module:    $RCSfile: V_QuadMetric.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.

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

/*
 *
 * QuadMetric.cpp contains quality calculations for Quads
 *
 * This file is part of VERDICT
 *
 */

#define VERDICT_EXPORTS

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

//! the average area of a quad
double verdict_quad_size = 0;

/*!
  weights based on the average size of a quad
*/
int get_weight( double& m11, double& m21, double& m12, double& m22 )
{

    m11 = 1;
    m21 = 0;
    m12 = 0;
    m22 = 1;

    double scale = sqrt( verdict_quad_size / ( m11 * m22 - m21 * m12 ) );

    m11 *= scale;
    m21 *= scale;
    m12 *= scale;
    m22 *= scale;

    return 1;
}

//! return the average area of a quad
C_FUNC_DEF void v_set_quad_size( double size )
{
    verdict_quad_size = size;
}

//! returns whether the quad is collapsed or not
VerdictBoolean is_collapsed_quad( double coordinates[][3] )
{
    if( coordinates[3][0] == coordinates[2][0] && coordinates[3][1] == coordinates[2][1] &&
        coordinates[3][2] == coordinates[2][2] )
        return VERDICT_TRUE;

    else
        return VERDICT_FALSE;
}

void make_quad_edges( VerdictVector edges[4], double coordinates[][3] )
{

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

void signed_corner_areas( double areas[4], double coordinates[][3] )
{
    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    VerdictVector corner_normals[4];
    corner_normals[0] = edges[3] * edges[0];
    corner_normals[1] = edges[0] * edges[1];
    corner_normals[2] = edges[1] * edges[2];
    corner_normals[3] = edges[2] * edges[3];

    // principal axes
    VerdictVector principal_axes[2];
    principal_axes[0] = edges[0] - edges[2];
    principal_axes[1] = edges[1] - edges[3];

    // quad center unit normal
    VerdictVector unit_center_normal;
    unit_center_normal = principal_axes[0] * principal_axes[1];
    unit_center_normal.normalize();

    areas[0] = unit_center_normal % corner_normals[0];
    areas[1] = unit_center_normal % corner_normals[1];
    areas[2] = unit_center_normal % corner_normals[2];
    areas[3] = unit_center_normal % corner_normals[3];
}

/*!
  localize the coordinates of a quad

  localizing puts the centriod of the quad
  at the orgin and also rotates the quad
  such that edge (0,1) is aligned with the x axis
  and the quad normal lines up with the y axis.

*/
void localize_quad_coordinates( VerdictVector nodes[4] )
{
    int i;
    VerdictVector global[4] = { nodes[0], nodes[1], nodes[2], nodes[3] };

    VerdictVector center = ( global[0] + global[1] + global[2] + global[3] ) / 4.0;
    for( i = 0; i < 4; i++ )
        global[i] -= center;

    VerdictVector vector_diff;
    VerdictVector vector_sum;
    VerdictVector ref_point( 0.0, 0.0, 0.0 );
    VerdictVector tmp_vector, normal( 0.0, 0.0, 0.0 );
    VerdictVector vector1, vector2;

    for( i = 0; i < 4; i++ )
    {
        vector1     = global[i];
        vector2     = global[( i + 1 ) % 4];
        vector_diff = vector2 - vector1;
        ref_point += vector1;
        vector_sum = vector1 + vector2;

        tmp_vector.set( vector_diff.y() * vector_sum.z(), vector_diff.z() * vector_sum.x(),
                        vector_diff.x() * vector_sum.y() );
        normal += tmp_vector;
    }

    normal.normalize();
    normal *= -1.0;

    VerdictVector local_x_axis = global[1] - global[0];
    local_x_axis.normalize();

    VerdictVector local_y_axis = normal * local_x_axis;
    local_y_axis.normalize();

    for( i = 0; i < 4; i++ )
    {
        nodes[i].x( global[i] % local_x_axis );
        nodes[i].y( global[i] % local_y_axis );
        nodes[i].z( global[i] % normal );
    }
}

/*!
  moves and rotates the quad such that it enables us to
  use components of ef's
*/
void localize_quad_for_ef( VerdictVector node_pos[4] )
{

    VerdictVector centroid( node_pos[0] );
    centroid += node_pos[1];
    centroid += node_pos[2];
    centroid += node_pos[3];

    centroid /= 4.0;

    node_pos[0] -= centroid;
    node_pos[1] -= centroid;
    node_pos[2] -= centroid;
    node_pos[3] -= centroid;

    VerdictVector rotate = node_pos[1] + node_pos[2] - node_pos[3] - node_pos[0];
    rotate.normalize();

    double cosine = rotate.x();
    double sine   = rotate.y();

    double xnew;

    for( int i = 0; i < 4; i++ )
    {
        xnew = cosine * node_pos[i].x() + sine * node_pos[i].y();
        node_pos[i].y( -sine * node_pos[i].x() + cosine * node_pos[i].y() );
        node_pos[i].x( xnew );
    }
}

/*!
  returns the normal vector of a quad
*/
VerdictVector quad_normal( double coordinates[][3] )
{
    // get normal at node 0
    VerdictVector edge0, edge1;

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

    edge1.set( coordinates[3][0] - coordinates[0][0], coordinates[3][1] - coordinates[0][1],
               coordinates[3][2] - coordinates[0][2] );

    VerdictVector norm0 = edge0 * edge1;
    norm0.normalize();

    // because some faces may have obtuse angles, check another normal at
    // node 2 for consistent sense

    edge0.set( coordinates[2][0] - coordinates[3][0], coordinates[2][1] - coordinates[3][1],
               coordinates[2][2] - coordinates[3][2] );

    edge1.set( coordinates[2][0] - coordinates[1][0], coordinates[2][1] - coordinates[1][1],
               coordinates[2][2] - coordinates[1][2] );

    VerdictVector norm2 = edge0 * edge1;
    norm2.normalize();

    // if these two agree, we are done, else test a third to decide

    if( ( norm0 % norm2 ) > 0.0 )
    {
        norm0 += norm2;
        norm0 *= 0.5;
        return norm0;
    }

    // test normal at node1

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

    edge1.set( coordinates[1][0] - coordinates[0][0], coordinates[1][1] - coordinates[0][1],
               coordinates[1][2] - coordinates[0][2] );

    VerdictVector norm1 = edge0 * edge1;
    norm1.normalize();

    if( ( norm0 % norm1 ) > 0.0 )
    {
        norm0 += norm1;
        norm0 *= 0.5;
        return norm0;
    }
    else
    {
        norm2 += norm1;
        norm2 *= 0.5;
        return norm2;
    }
}

/*!
   the edge ratio of a quad

   NB (P. Pebay 01/19/07):
     Hmax / Hmin where Hmax and Hmin are respectively the maximum and the
     minimum edge lengths
*/
C_FUNC_DEF double v_quad_edge_ratio( int /*num_nodes*/, double coordinates[][3] )
{
    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    double a2 = edges[0].length_squared();
    double b2 = edges[1].length_squared();
    double c2 = edges[2].length_squared();
    double d2 = edges[3].length_squared();

    double mab, Mab, mcd, Mcd, m2, M2;
    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;
    }
    m2 = mab < mcd ? mab : mcd;
    M2 = Mab > Mcd ? Mab : Mcd;

    if( m2 < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;
    else
    {
        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 );
    }
}

/*!
  maximum of edge ratio of a quad

  maximum edge length ratio at quad center
*/
C_FUNC_DEF double v_quad_max_edge_ratio( int /*num_nodes*/, double coordinates[][3] )
{
    VerdictVector quad_nodes[4];
    quad_nodes[0].set( coordinates[0][0], coordinates[0][1], coordinates[0][2] );
    quad_nodes[1].set( coordinates[1][0], coordinates[1][1], coordinates[1][2] );
    quad_nodes[2].set( coordinates[2][0], coordinates[2][1], coordinates[2][2] );
    quad_nodes[3].set( coordinates[3][0], coordinates[3][1], coordinates[3][2] );

    VerdictVector principal_axes[2];
    principal_axes[0] = quad_nodes[1] + quad_nodes[2] - quad_nodes[0] - quad_nodes[3];
    principal_axes[1] = quad_nodes[2] + quad_nodes[3] - quad_nodes[0] - quad_nodes[1];

    double len1 = principal_axes[0].length();
    double len2 = principal_axes[1].length();

    if( len1 < VERDICT_DBL_MIN || len2 < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;

    double max_edge_ratio = VERDICT_MAX( len1 / len2, len2 / len1 );

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

/*!
   the aspect ratio of a quad

   NB (P. Pebay 01/20/07):
     this is a generalization of the triangle aspect ratio
     using Heron's formula.
*/
C_FUNC_DEF double v_quad_aspect_ratio( int /*num_nodes*/, double coordinates[][3] )
{

    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    double a1 = edges[0].length();
    double b1 = edges[1].length();
    double c1 = edges[2].length();
    double d1 = edges[3].length();

    double ma = a1 > b1 ? a1 : b1;
    double mb = c1 > d1 ? c1 : d1;
    double hm = ma > mb ? ma : mb;

    VerdictVector ab   = edges[0] * edges[1];
    VerdictVector cd   = edges[2] * edges[3];
    double denominator = ab.length() + cd.length();

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

    double aspect_ratio = .5 * hm * ( a1 + b1 + c1 + d1 ) / denominator;

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

/*!
   the radius ratio of a quad

   NB (P. Pebay 01/19/07):
     this metric is called "radius ratio" by extension of a concept that does
     not exist in general with quads -- although a different name should probably
     be used in the future.
*/
C_FUNC_DEF double v_quad_radius_ratio( int /*num_nodes*/, double coordinates[][3] )
{
    static const double normal_coeff = 1. / ( 2. * sqrt( 2. ) );

    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    double a2 = edges[0].length_squared();
    double b2 = edges[1].length_squared();
    double c2 = edges[2].length_squared();
    double d2 = edges[3].length_squared();

    VerdictVector diag;
    diag.set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
              coordinates[2][2] - coordinates[0][2] );
    double m2 = diag.length_squared();

    diag.set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
              coordinates[3][2] - coordinates[1][2] );
    double n2 = diag.length_squared();

    double t0 = a2 > b2 ? a2 : b2;
    double t1 = c2 > d2 ? c2 : d2;
    double t2 = m2 > n2 ? m2 : n2;
    double h2 = t0 > t1 ? t0 : t1;
    h2        = h2 > t2 ? h2 : t2;

    VerdictVector ab = edges[0] * edges[1];
    VerdictVector bc = edges[1] * edges[2];
    VerdictVector cd = edges[2] * edges[3];
    VerdictVector da = edges[3] * edges[0];

    t0        = da.length();
    t1        = ab.length();
    t2        = bc.length();
    double t3 = cd.length();

    t0 = t0 < t1 ? t0 : t1;
    t2 = t2 < t3 ? t2 : t3;
    t0 = t0 < t2 ? t0 : t2;

    if( t0 < VERDICT_DBL_MIN ) return (double)VERDICT_DBL_MAX;

    double radius_ratio = normal_coeff * sqrt( ( a2 + b2 + c2 + d2 ) * h2 ) / t0;

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

/*!
   the average Frobenius aspect of a quad

   NB (P. Pebay 01/20/07):
     this metric is calculated by averaging the 4 Frobenius aspects at
     each corner of the quad, when the reference triangle is right isosceles.
*/
C_FUNC_DEF double v_quad_med_aspect_frobenius( int /*num_nodes*/, double coordinates[][3] )
{

    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    double a2 = edges[0].length_squared();
    double b2 = edges[1].length_squared();
    double c2 = edges[2].length_squared();
    double d2 = edges[3].length_squared();

    VerdictVector ab = edges[0] * edges[1];
    VerdictVector bc = edges[1] * edges[2];
    VerdictVector cd = edges[2] * edges[3];
    VerdictVector da = edges[3] * edges[0];

    double ab1 = ab.length();
    double bc1 = bc.length();
    double cd1 = cd.length();
    double da1 = da.length();

    if( ab1 < VERDICT_DBL_MIN || bc1 < VERDICT_DBL_MIN || cd1 < VERDICT_DBL_MIN || da1 < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;

    double qsum = ( a2 + b2 ) / ab1;
    qsum += ( b2 + c2 ) / bc1;
    qsum += ( c2 + d2 ) / cd1;
    qsum += ( d2 + a2 ) / da1;

    double med_aspect_frobenius = .125 * qsum;

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

/*!
   the maximum Frobenius aspect of a quad

   NB (P. Pebay 01/20/07):
     this metric is calculated by taking the maximum of the 4 Frobenius aspects at
     each corner of the quad, when the reference triangle is right isosceles.
*/
C_FUNC_DEF double v_quad_max_aspect_frobenius( int /*num_nodes*/, double coordinates[][3] )
{

    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    double a2 = edges[0].length_squared();
    double b2 = edges[1].length_squared();
    double c2 = edges[2].length_squared();
    double d2 = edges[3].length_squared();

    VerdictVector ab = edges[0] * edges[1];
    VerdictVector bc = edges[1] * edges[2];
    VerdictVector cd = edges[2] * edges[3];
    VerdictVector da = edges[3] * edges[0];

    double ab1 = ab.length();
    double bc1 = bc.length();
    double cd1 = cd.length();
    double da1 = da.length();

    if( ab1 < VERDICT_DBL_MIN || bc1 < VERDICT_DBL_MIN || cd1 < VERDICT_DBL_MIN || da1 < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;

    double qmax = ( a2 + b2 ) / ab1;

    double qcur = ( b2 + c2 ) / bc1;
    qmax        = qmax > qcur ? qmax : qcur;

    qcur = ( c2 + d2 ) / cd1;
    qmax = qmax > qcur ? qmax : qcur;

    qcur = ( d2 + a2 ) / da1;
    qmax = qmax > qcur ? qmax : qcur;

    double max_aspect_frobenius = .5 * qmax;

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

/*!
  skew of a quad

  maximum ||cos A|| where A is the angle between edges at quad center
*/
C_FUNC_DEF double v_quad_skew( int /*num_nodes*/, double coordinates[][3] )
{
    VerdictVector node_pos[4];
    for( int i = 0; i < 4; i++ )
        node_pos[i].set( coordinates[i][0], coordinates[i][1], coordinates[i][2] );

    VerdictVector principle_axes[2];
    principle_axes[0] = node_pos[1] + node_pos[2] - node_pos[3] - node_pos[0];
    principle_axes[1] = node_pos[2] + node_pos[3] - node_pos[0] - node_pos[1];

    if( principle_axes[0].normalize() < VERDICT_DBL_MIN ) return 0.0;
    if( principle_axes[1].normalize() < VERDICT_DBL_MIN ) return 0.0;

    double skew = fabs( principle_axes[0] % principle_axes[1] );

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

/*!
  taper of a quad

  maximum ratio of lengths derived from opposite edges
*/
C_FUNC_DEF double v_quad_taper( int /*num_nodes*/, double coordinates[][3] )
{
    VerdictVector node_pos[4];
    for( int i = 0; i < 4; i++ )
        node_pos[i].set( coordinates[i][0], coordinates[i][1], coordinates[i][2] );

    VerdictVector principle_axes[2];
    principle_axes[0] = node_pos[1] + node_pos[2] - node_pos[3] - node_pos[0];
    principle_axes[1] = node_pos[2] + node_pos[3] - node_pos[0] - node_pos[1];

    VerdictVector cross_derivative = node_pos[0] + node_pos[2] - node_pos[1] - node_pos[3];

    double lengths[2];
    lengths[0] = principle_axes[0].length();
    lengths[1] = principle_axes[1].length();

    // get min length
    lengths[0] = VERDICT_MIN( lengths[0], lengths[1] );

    if( lengths[0] < VERDICT_DBL_MIN ) return VERDICT_DBL_MAX;

    double taper = cross_derivative.length() / lengths[0];
    return (double)VERDICT_MIN( taper, VERDICT_DBL_MAX );
}

/*!
  warpage of a quad

  deviation of element from planarity
*/
C_FUNC_DEF double v_quad_warpage( int /*num_nodes*/, double coordinates[][3] )
{

    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    VerdictVector corner_normals[4];
    corner_normals[0] = edges[3] * edges[0];
    corner_normals[1] = edges[0] * edges[1];
    corner_normals[2] = edges[1] * edges[2];
    corner_normals[3] = edges[2] * edges[3];

    if( corner_normals[0].normalize() < VERDICT_DBL_MIN || corner_normals[1].normalize() < VERDICT_DBL_MIN ||
        corner_normals[2].normalize() < VERDICT_DBL_MIN || corner_normals[3].normalize() < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MIN;

    double warpage =
        pow( VERDICT_MIN( corner_normals[0] % corner_normals[2], corner_normals[1] % corner_normals[3] ), 3 );

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

/*!
  the area of a quad

  jacobian at quad center
*/
C_FUNC_DEF double v_quad_area( int /*num_nodes*/, double coordinates[][3] )
{

    double corner_areas[4];
    signed_corner_areas( corner_areas, coordinates );

    double area = 0.25 * ( corner_areas[0] + corner_areas[1] + corner_areas[2] + corner_areas[3] );

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

/*!
  the stretch of a quad

  sqrt(2) * minimum edge length / maximum diagonal length
*/
C_FUNC_DEF double v_quad_stretch( int /*num_nodes*/, double coordinates[][3] )
{
    VerdictVector edges[4], temp;
    make_quad_edges( edges, coordinates );

    double lengths_squared[4];
    lengths_squared[0] = edges[0].length_squared();
    lengths_squared[1] = edges[1].length_squared();
    lengths_squared[2] = edges[2].length_squared();
    lengths_squared[3] = edges[3].length_squared();

    temp.set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
              coordinates[2][2] - coordinates[0][2] );
    double diag02 = temp.length_squared();

    temp.set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
              coordinates[3][2] - coordinates[1][2] );
    double diag13 = temp.length_squared();

    static const double QUAD_STRETCH_FACTOR = sqrt( 2.0 );

    // 'diag02' is now the max diagonal of the quad
    diag02 = VERDICT_MAX( diag02, diag13 );

    if( diag02 < VERDICT_DBL_MIN )
        return (double)VERDICT_DBL_MAX;
    else
    {
        double stretch =
            (double)( QUAD_STRETCH_FACTOR * sqrt( VERDICT_MIN( VERDICT_MIN( lengths_squared[0], lengths_squared[1] ),
                                                               VERDICT_MIN( lengths_squared[2], lengths_squared[3] ) ) /
                                                  diag02 ) );

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

/*!
  the largest angle of a quad

  largest included quad area (degrees)
*/
C_FUNC_DEF double v_quad_maximum_angle( int /*num_nodes*/, double coordinates[][3] )
{

    // if this is a collapsed quad, just pass it on to
    // the tri_largest_angle routine
    if( is_collapsed_quad( coordinates ) == VERDICT_TRUE ) return v_tri_maximum_angle( 3, coordinates );

    double angle;
    double max_angle = 0.0;

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

    // go around each node and calculate the angle
    // at each node
    double length[4];
    length[0] = edges[0].length();
    length[1] = edges[1].length();
    length[2] = edges[2].length();
    length[3] = edges[3].length();

    if( length[0] <= VERDICT_DBL_MIN || length[1] <= VERDICT_DBL_MIN || length[2] <= VERDICT_DBL_MIN ||
        length[3] <= VERDICT_DBL_MIN )
        return 0.0;

    angle     = acos( -( edges[0] % edges[1] ) / ( length[0] * length[1] ) );
    max_angle = VERDICT_MAX( angle, max_angle );

    angle     = acos( -( edges[1] % edges[2] ) / ( length[1] * length[2] ) );
    max_angle = VERDICT_MAX( angle, max_angle );

    angle     = acos( -( edges[2] % edges[3] ) / ( length[2] * length[3] ) );
    max_angle = VERDICT_MAX( angle, max_angle );

    angle     = acos( -( edges[3] % edges[0] ) / ( length[3] * length[0] ) );
    max_angle = VERDICT_MAX( angle, max_angle );

    max_angle = max_angle * 180.0 / VERDICT_PI;

    // if any signed areas are < 0, then you are getting the wrong angle
    double areas[4];
    signed_corner_areas( areas, coordinates );

    if( areas[0] < 0 || areas[1] < 0 || areas[2] < 0 || areas[3] < 0 )
    {
        max_angle = 360 - max_angle;
    }

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

/*!
  the smallest angle of a quad

  smallest included quad angle (degrees)
*/
C_FUNC_DEF double v_quad_minimum_angle( int /*num_nodes*/, double coordinates[][3] )
{
    // if this quad is a collapsed quad, then just
    // send it to the tri_smallest_angle routine
    if( is_collapsed_quad( coordinates ) == VERDICT_TRUE ) return v_tri_minimum_angle( 3, coordinates );

    double angle;
    double min_angle = 360.0;

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

    // go around each node and calculate the angle
    // at each node
    double length[4];
    length[0] = edges[0].length();
    length[1] = edges[1].length();
    length[2] = edges[2].length();
    length[3] = edges[3].length();

    if( length[0] <= VERDICT_DBL_MIN || length[1] <= VERDICT_DBL_MIN || length[2] <= VERDICT_DBL_MIN ||
        length[3] <= VERDICT_DBL_MIN )
        return 360.0;

    angle     = acos( -( edges[0] % edges[1] ) / ( length[0] * length[1] ) );
    min_angle = VERDICT_MIN( angle, min_angle );

    angle     = acos( -( edges[1] % edges[2] ) / ( length[1] * length[2] ) );
    min_angle = VERDICT_MIN( angle, min_angle );

    angle     = acos( -( edges[2] % edges[3] ) / ( length[2] * length[3] ) );
    min_angle = VERDICT_MIN( angle, min_angle );

    angle     = acos( -( edges[3] % edges[0] ) / ( length[3] * length[0] ) );
    min_angle = VERDICT_MIN( angle, min_angle );

    min_angle = min_angle * 180.0 / VERDICT_PI;

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

/*!
  the oddy of a quad

  general distortion measure based on left Cauchy-Green Tensor
*/
C_FUNC_DEF double v_quad_oddy( int /*num_nodes*/, double coordinates[][3] )
{

    double max_oddy = 0.;

    VerdictVector first, second, node_pos[4];

    double g, g11, g12, g22, cur_oddy;
    int i;

    for( i = 0; i < 4; i++ )
        node_pos[i].set( coordinates[i][0], coordinates[i][1], coordinates[i][2] );

    for( i = 0; i < 4; i++ )
    {
        first  = node_pos[i] - node_pos[( i + 1 ) % 4];
        second = node_pos[i] - node_pos[( i + 3 ) % 4];

        g11 = first % first;
        g12 = first % second;
        g22 = second % second;
        g   = g11 * g22 - g12 * g12;

        if( g < VERDICT_DBL_MIN )
            cur_oddy = VERDICT_DBL_MAX;
        else
            cur_oddy = ( ( g11 - g22 ) * ( g11 - g22 ) + 4. * g12 * g12 ) / 2. / g;

        max_oddy = VERDICT_MAX( max_oddy, cur_oddy );
    }

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

/*!
  the condition of a quad

  maximum condition number of the Jacobian matrix at 4 corners
*/
C_FUNC_DEF double v_quad_condition( int /*num_nodes*/, double coordinates[][3] )
{

    if( is_collapsed_quad( coordinates ) == VERDICT_TRUE ) return v_tri_condition( 3, coordinates );

    double areas[4];
    signed_corner_areas( areas, coordinates );

    double max_condition = 0.;

    VerdictVector xxi, xet;

    double condition;

    for( int i = 0; i < 4; i++ )
    {

        xxi.set( coordinates[i][0] - coordinates[( i + 1 ) % 4][0], coordinates[i][1] - coordinates[( i + 1 ) % 4][1],
                 coordinates[i][2] - coordinates[( i + 1 ) % 4][2] );

        xet.set( coordinates[i][0] - coordinates[( i + 3 ) % 4][0], coordinates[i][1] - coordinates[( i + 3 ) % 4][1],
                 coordinates[i][2] - coordinates[( i + 3 ) % 4][2] );

        if( areas[i] < VERDICT_DBL_MIN )
            condition = VERDICT_DBL_MAX;
        else
            condition = ( xxi % xxi + xet % xet ) / areas[i];

        max_condition = VERDICT_MAX( max_condition, condition );
    }

    max_condition /= 2;

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

/*!
  the jacobian of a quad

  minimum pointwise volume of local map at 4 corners and center of quad
*/
C_FUNC_DEF double v_quad_jacobian( int /*num_nodes*/, double coordinates[][3] )
{

    if( is_collapsed_quad( coordinates ) == VERDICT_TRUE ) return (double)( v_tri_area( 3, coordinates ) * 2.0 );

    double areas[4];
    signed_corner_areas( areas, coordinates );

    double jacobian = VERDICT_MIN( VERDICT_MIN( areas[0], areas[1] ), VERDICT_MIN( areas[2], areas[3] ) );
    if( jacobian > 0 ) return (double)VERDICT_MIN( jacobian, VERDICT_DBL_MAX );
    return (double)VERDICT_MAX( jacobian, -VERDICT_DBL_MAX );
}

/*!
  scaled jacobian of a quad

  Minimum Jacobian divided by the lengths of the 2 edge vector
*/
C_FUNC_DEF double v_quad_scaled_jacobian( int /*num_nodes*/, double coordinates[][3] )
{
    if( is_collapsed_quad( coordinates ) == VERDICT_TRUE ) return v_tri_scaled_jacobian( 3, coordinates );

    double corner_areas[4], min_scaled_jac = VERDICT_DBL_MAX, scaled_jac;
    signed_corner_areas( corner_areas, coordinates );

    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    double length[4];
    length[0] = edges[0].length();
    length[1] = edges[1].length();
    length[2] = edges[2].length();
    length[3] = edges[3].length();

    if( length[0] < VERDICT_DBL_MIN || length[1] < VERDICT_DBL_MIN || length[2] < VERDICT_DBL_MIN ||
        length[3] < VERDICT_DBL_MIN )
        return 0.0;

    scaled_jac     = corner_areas[0] / ( length[0] * length[3] );
    min_scaled_jac = VERDICT_MIN( scaled_jac, min_scaled_jac );

    scaled_jac     = corner_areas[1] / ( length[1] * length[0] );
    min_scaled_jac = VERDICT_MIN( scaled_jac, min_scaled_jac );

    scaled_jac     = corner_areas[2] / ( length[2] * length[1] );
    min_scaled_jac = VERDICT_MIN( scaled_jac, min_scaled_jac );

    scaled_jac     = corner_areas[3] / ( length[3] * length[2] );
    min_scaled_jac = VERDICT_MIN( scaled_jac, min_scaled_jac );

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

/*!
  the shear of a quad

  2/Condition number of Jacobian Skew matrix
*/
C_FUNC_DEF double v_quad_shear( int /*num_nodes*/, double coordinates[][3] )
{
    double scaled_jacobian = v_quad_scaled_jacobian( 4, coordinates );

    if( scaled_jacobian <= VERDICT_DBL_MIN )
        return 0.0;
    else
        return (double)VERDICT_MIN( scaled_jacobian, VERDICT_DBL_MAX );
}

/*!
  the shape of a quad

   2/Condition number of weighted Jacobian matrix
*/
C_FUNC_DEF double v_quad_shape( int /*num_nodes*/, double coordinates[][3] )
{

    double corner_areas[4], min_shape = VERDICT_DBL_MAX, shape;
    signed_corner_areas( corner_areas, coordinates );

    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    double length_squared[4];
    length_squared[0] = edges[0].length_squared();
    length_squared[1] = edges[1].length_squared();
    length_squared[2] = edges[2].length_squared();
    length_squared[3] = edges[3].length_squared();

    if( length_squared[0] <= VERDICT_DBL_MIN || length_squared[1] <= VERDICT_DBL_MIN ||
        length_squared[2] <= VERDICT_DBL_MIN || length_squared[3] <= VERDICT_DBL_MIN )
        return 0.0;

    shape     = corner_areas[0] / ( length_squared[0] + length_squared[3] );
    min_shape = VERDICT_MIN( shape, min_shape );

    shape     = corner_areas[1] / ( length_squared[1] + length_squared[0] );
    min_shape = VERDICT_MIN( shape, min_shape );

    shape     = corner_areas[2] / ( length_squared[2] + length_squared[1] );
    min_shape = VERDICT_MIN( shape, min_shape );

    shape     = corner_areas[3] / ( length_squared[3] + length_squared[2] );
    min_shape = VERDICT_MIN( shape, min_shape );

    min_shape *= 2;

    if( min_shape < VERDICT_DBL_MIN ) min_shape = 0;

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

/*!
  the relative size of a quad

  Min( J, 1/J ), where J is determinant of weighted Jacobian matrix
*/
C_FUNC_DEF double v_quad_relative_size_squared( int /*num_nodes*/, double coordinates[][3] )
{

    double quad_area = v_quad_area( 4, coordinates );
    double rel_size  = 0;

    v_set_quad_size( quad_area );
    double w11, w21, w12, w22;
    get_weight( w11, w21, w12, w22 );
    double avg_area = determinant( w11, w21, w12, w22 );

    if( avg_area > VERDICT_DBL_MIN )
    {

        w11 = quad_area / avg_area;

        if( w11 > VERDICT_DBL_MIN )
        {
            rel_size = VERDICT_MIN( w11, 1 / w11 );
            rel_size *= rel_size;
        }
    }

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

/*!
  the relative shape and size of a quad

  Product of Shape and Relative Size
*/
C_FUNC_DEF double v_quad_shape_and_size( int num_nodes, double coordinates[][3] )
{
    double shape, size;
    size  = v_quad_relative_size_squared( num_nodes, coordinates );
    shape = v_quad_shape( num_nodes, coordinates );

    double shape_and_size = shape * size;

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

/*!
  the shear and size of a quad

  product of shear and relative size
*/
C_FUNC_DEF double v_quad_shear_and_size( int num_nodes, double coordinates[][3] )
{
    double shear, size;
    shear = v_quad_shear( num_nodes, coordinates );
    size  = v_quad_relative_size_squared( num_nodes, coordinates );

    double shear_and_size = shear * size;

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

/*!
  the distortion of a quad
*/
C_FUNC_DEF double v_quad_distortion( int num_nodes, double coordinates[][3] )
{
    // To calculate distortion for linear and 2nd order quads
    // distortion = {min(|J|)/actual area}*{parent area}
    // parent area = 4 for a quad.
    // min |J| is the minimum over nodes and gaussian integration points
    // created by Ling Pan, CAT on 4/30/01

    double element_area = 0.0, distrt, thickness_gauss;
    double cur_jacobian = 0., sign_jacobian, jacobian;
    VerdictVector aa, bb, cc, normal_at_point, xin;

    // use 2x2 gauss points for linear quads and 3x3 for 2nd order quads
    int number_of_gauss_points = 0;
    if( num_nodes == 4 )
    {  // 2x2 quadrature rule
        number_of_gauss_points = 2;
    }
    else if( num_nodes == 8 )
    {  // 3x3 quadrature rule
        number_of_gauss_points = 3;
    }

    int total_number_of_gauss_points = number_of_gauss_points * number_of_gauss_points;

    VerdictVector face_normal = quad_normal( coordinates );

    double distortion = VERDICT_DBL_MAX;

    VerdictVector first, second;

    int i;
    // Will work out the case for collapsed quad later
    if( is_collapsed_quad( coordinates ) == VERDICT_TRUE )
    {
        for( i = 0; i < 3; i++ )
        {

            first.set( coordinates[i][0] - coordinates[( i + 1 ) % 3][0],
                       coordinates[i][1] - coordinates[( i + 1 ) % 3][1],
                       coordinates[i][2] - coordinates[( i + 1 ) % 3][2] );

            second.set( coordinates[i][0] - coordinates[( i + 2 ) % 3][0],
                        coordinates[i][1] - coordinates[( i + 2 ) % 3][1],
                        coordinates[i][2] - coordinates[( i + 2 ) % 3][2] );

            sign_jacobian = ( face_normal % ( first * second ) ) > 0 ? 1. : -1.;
            cur_jacobian  = sign_jacobian * ( first * second ).length();
            distortion    = VERDICT_MIN( distortion, cur_jacobian );
        }
        element_area = ( first * second ).length() / 2.0;
        distortion /= element_area;
    }
    else
    {
        double shape_function[maxTotalNumberGaussPoints][maxNumberNodes];
        double dndy1[maxTotalNumberGaussPoints][maxNumberNodes];
        double dndy2[maxTotalNumberGaussPoints][maxNumberNodes];
        double weight[maxTotalNumberGaussPoints];

        // create an object of GaussIntegration
        GaussIntegration::initialize( number_of_gauss_points, num_nodes );
        GaussIntegration::calculate_shape_function_2d_quad();
        GaussIntegration::get_shape_func( shape_function[0], dndy1[0], dndy2[0], weight );

        // calculate element area
        int ife, ja;
        for( ife = 0; ife < total_number_of_gauss_points; ife++ )
        {
            aa.set( 0.0, 0.0, 0.0 );
            bb.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] );
                aa += dndy1[ife][ja] * xin;
                bb += dndy2[ife][ja] * xin;
            }
            normal_at_point = aa * bb;
            jacobian        = normal_at_point.length();
            element_area += weight[ife] * jacobian;
        }

        double dndy1_at_node[maxNumberNodes][maxNumberNodes];
        double dndy2_at_node[maxNumberNodes][maxNumberNodes];

        GaussIntegration::calculate_derivative_at_nodes( dndy1_at_node, dndy2_at_node );

        VerdictVector normal_at_nodes[9];

        // evaluate normal at nodes and distortion values at nodes
        int jai;
        for( ja = 0; ja < num_nodes; ja++ )
        {
            aa.set( 0.0, 0.0, 0.0 );
            bb.set( 0.0, 0.0, 0.0 );
            for( jai = 0; jai < num_nodes; jai++ )
            {
                xin.set( coordinates[jai][0], coordinates[jai][1], coordinates[jai][2] );
                aa += dndy1_at_node[ja][jai] * xin;
                bb += dndy2_at_node[ja][jai] * xin;
            }
            normal_at_nodes[ja] = aa * bb;
            normal_at_nodes[ja].normalize();
        }

        // determine if element is flat
        bool flat_element = true;
        double dot_product;

        for( ja = 0; ja < num_nodes; ja++ )
        {
            dot_product = normal_at_nodes[0] % normal_at_nodes[ja];
            if( fabs( dot_product ) < 0.99 )
            {
                flat_element = false;
                break;
            }
        }

        // take into consideration of the thickness of the element
        double thickness;
        // get_quad_thickness(face, element_area, thickness );
        thickness = 0.001 * sqrt( element_area );

        // set thickness gauss point location
        double zl = 0.5773502691896;
        if( flat_element ) zl = 0.0;

        int no_gauss_pts_z = ( flat_element ) ? 1 : 2;
        double thickness_z;
        int igz;
        // loop on Gauss points
        for( ife = 0; ife < total_number_of_gauss_points; ife++ )
        {
            // loop on the thickness direction gauss points
            for( igz = 0; igz < no_gauss_pts_z; igz++ )
            {
                zl          = -zl;
                thickness_z = zl * thickness / 2.0;

                aa.set( 0.0, 0.0, 0.0 );
                bb.set( 0.0, 0.0, 0.0 );
                cc.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] );
                    xin += thickness_z * normal_at_nodes[ja];
                    aa += dndy1[ife][ja] * xin;
                    bb += dndy2[ife][ja] * xin;
                    thickness_gauss = shape_function[ife][ja] * thickness / 2.0;
                    cc += thickness_gauss * normal_at_nodes[ja];
                }

                normal_at_point = aa * bb;
                // jacobian = normal_at_point.length();
                distrt = cc % normal_at_point;
                if( distrt < distortion ) distortion = distrt;
            }
        }

        // loop through nodal points
        for( ja = 0; ja < num_nodes; ja++ )
        {
            for( igz = 0; igz < no_gauss_pts_z; igz++ )
            {
                zl          = -zl;
                thickness_z = zl * thickness / 2.0;

                aa.set( 0.0, 0.0, 0.0 );
                bb.set( 0.0, 0.0, 0.0 );
                cc.set( 0.0, 0.0, 0.0 );

                for( jai = 0; jai < num_nodes; jai++ )
                {
                    xin.set( coordinates[jai][0], coordinates[jai][1], coordinates[jai][2] );
                    xin += thickness_z * normal_at_nodes[ja];
                    aa += dndy1_at_node[ja][jai] * xin;
                    bb += dndy2_at_node[ja][jai] * xin;
                    if( jai == ja )
                        thickness_gauss = thickness / 2.0;
                    else
                        thickness_gauss = 0.;
                    cc += thickness_gauss * normal_at_nodes[jai];
                }
            }
            normal_at_point = aa * bb;
            sign_jacobian   = ( face_normal % normal_at_point ) > 0 ? 1. : -1.;
            distrt          = sign_jacobian * ( cc % normal_at_point );

            if( distrt < distortion ) distortion = distrt;
        }

        if( element_area * thickness != 0 )
            distortion *= 8. / ( element_area * thickness );
        else
            distortion *= 8.;
    }

    return (double)distortion;
}

/*!
  multiple quality measures of a quad
*/
C_FUNC_DEF void v_quad_quality( int num_nodes,
                                double coordinates[][3],
                                unsigned int metrics_request_flag,
                                QuadMetricVals* metric_vals )
{

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

    // for starts, lets set up some basic and common information

    /*  node numbers and side numbers used below

                    2
              3 +--------- 2
               /         +
              /          |
           3 /           | 1
            /            |
           +             |
         0 -------------+ 1
               0
    */

    // vectors for each side
    VerdictVector edges[4];
    make_quad_edges( edges, coordinates );

    double areas[4];
    signed_corner_areas( areas, coordinates );

    double lengths[4];
    lengths[0] = edges[0].length();
    lengths[1] = edges[1].length();
    lengths[2] = edges[2].length();
    lengths[3] = edges[3].length();

    VerdictBoolean is_collapsed = is_collapsed_quad( coordinates );

    // handle collapsed quads metrics here
    if( is_collapsed == VERDICT_TRUE && metrics_request_flag & ( V_QUAD_MINIMUM_ANGLE | V_QUAD_MAXIMUM_ANGLE |
                                                                 V_QUAD_JACOBIAN | V_QUAD_SCALED_JACOBIAN ) )
    {
        if( metrics_request_flag & V_QUAD_MINIMUM_ANGLE )
            metric_vals->minimum_angle = v_tri_minimum_angle( 3, coordinates );
        if( metrics_request_flag & V_QUAD_MAXIMUM_ANGLE )
            metric_vals->maximum_angle = v_tri_maximum_angle( 3, coordinates );
        if( metrics_request_flag & V_QUAD_JACOBIAN )
            metric_vals->jacobian = (double)( v_tri_area( 3, coordinates ) * 2.0 );
        if( metrics_request_flag & V_QUAD_SCALED_JACOBIAN )
            metric_vals->jacobian = (double)( v_tri_scaled_jacobian( 3, coordinates ) * 2.0 );
    }

    // calculate both largest and smallest angles
    if( metrics_request_flag & ( V_QUAD_MINIMUM_ANGLE | V_QUAD_MAXIMUM_ANGLE ) && is_collapsed == VERDICT_FALSE )
    {
        // gather the angles
        double angles[4];
        angles[0] = acos( -( edges[0] % edges[1] ) / ( lengths[0] * lengths[1] ) );
        angles[1] = acos( -( edges[1] % edges[2] ) / ( lengths[1] * lengths[2] ) );
        angles[2] = acos( -( edges[2] % edges[3] ) / ( lengths[2] * lengths[3] ) );
        angles[3] = acos( -( edges[3] % edges[0] ) / ( lengths[3] * lengths[0] ) );

        if( lengths[0] <= VERDICT_DBL_MIN || lengths[1] <= VERDICT_DBL_MIN || lengths[2] <= VERDICT_DBL_MIN ||
            lengths[3] <= VERDICT_DBL_MIN )
        {
            metric_vals->minimum_angle = 360.0;
            metric_vals->maximum_angle = 0.0;
        }
        else
        {
            // if smallest angle, find the smallest angle
            if( metrics_request_flag & V_QUAD_MINIMUM_ANGLE )
            {
                metric_vals->minimum_angle = VERDICT_DBL_MAX;
                for( int i = 0; i < 4; i++ )
                    metric_vals->minimum_angle = VERDICT_MIN( angles[i], metric_vals->minimum_angle );
                metric_vals->minimum_angle *= 180.0 / VERDICT_PI;
            }
            // if largest angle, find the largest angle
            if( metrics_request_flag & V_QUAD_MAXIMUM_ANGLE )
            {
                metric_vals->maximum_angle = 0.0;
                for( int i = 0; i < 4; i++ )
                    metric_vals->maximum_angle = VERDICT_MAX( angles[i], metric_vals->maximum_angle );
                metric_vals->maximum_angle *= 180.0 / VERDICT_PI;

                if( areas[0] < 0 || areas[1] < 0 || areas[2] < 0 || areas[3] < 0 )
                    metric_vals->maximum_angle = 360 - metric_vals->maximum_angle;
            }
        }
    }

    // handle max_edge_ratio, skew, taper, and area together
    if( metrics_request_flag & ( V_QUAD_MAX_EDGE_RATIO | V_QUAD_SKEW | V_QUAD_TAPER ) )
    {
        // get principle axes
        VerdictVector principal_axes[2];
        principal_axes[0] = edges[0] - edges[2];
        principal_axes[1] = edges[1] - edges[3];

        if( metrics_request_flag & ( V_QUAD_MAX_EDGE_RATIO | V_QUAD_SKEW | V_QUAD_TAPER ) )
        {
            double len1 = principal_axes[0].length();
            double len2 = principal_axes[1].length();

            // calculate the max_edge_ratio ratio
            if( metrics_request_flag & V_QUAD_MAX_EDGE_RATIO )
            {
                if( len1 < VERDICT_DBL_MIN || len2 < VERDICT_DBL_MIN )
                    metric_vals->max_edge_ratio = VERDICT_DBL_MAX;
                else
                    metric_vals->max_edge_ratio = VERDICT_MAX( len1 / len2, len2 / len1 );
            }

            // calculate the taper
            if( metrics_request_flag & V_QUAD_TAPER )
            {
                double min_length = VERDICT_MIN( len1, len2 );

                VerdictVector cross_derivative = edges[1] + edges[3];

                if( min_length < VERDICT_DBL_MIN )
                    metric_vals->taper = VERDICT_DBL_MAX;
                else
                    metric_vals->taper = cross_derivative.length() / min_length;
            }

            // calculate the skew
            if( metrics_request_flag & V_QUAD_SKEW )
            {
                if( principal_axes[0].normalize() < VERDICT_DBL_MIN || principal_axes[1].normalize() < VERDICT_DBL_MIN )
                    metric_vals->skew = 0.0;
                else
                    metric_vals->skew = fabs( principal_axes[0] % principal_axes[1] );
            }
        }
    }

    // calculate the area
    if( metrics_request_flag & ( V_QUAD_AREA | V_QUAD_RELATIVE_SIZE_SQUARED ) )
    {
        metric_vals->area = 0.25 * ( areas[0] + areas[1] + areas[2] + areas[3] );
    }

    // calculate the relative size
    if( metrics_request_flag & ( V_QUAD_RELATIVE_SIZE_SQUARED | V_QUAD_SHAPE_AND_SIZE | V_QUAD_SHEAR_AND_SIZE ) )
    {
        double quad_area = v_quad_area( 4, coordinates );
        v_set_quad_size( quad_area );
        double w11, w21, w12, w22;
        get_weight( w11, w21, w12, w22 );
        double avg_area = determinant( w11, w21, w12, w22 );

        if( avg_area < VERDICT_DBL_MIN )
            metric_vals->relative_size_squared = 0.0;
        else
            metric_vals->relative_size_squared =
                pow( VERDICT_MIN( metric_vals->area / avg_area, avg_area / metric_vals->area ), 2 );
    }

    // calculate the jacobian
    if( metrics_request_flag & V_QUAD_JACOBIAN )
    {
        metric_vals->jacobian = VERDICT_MIN( VERDICT_MIN( areas[0], areas[1] ), VERDICT_MIN( areas[2], areas[3] ) );
    }

    if( metrics_request_flag & ( V_QUAD_SCALED_JACOBIAN | V_QUAD_SHEAR | V_QUAD_SHEAR_AND_SIZE ) )
    {
        double scaled_jac, min_scaled_jac = VERDICT_DBL_MAX;

        if( lengths[0] < VERDICT_DBL_MIN || lengths[1] < VERDICT_DBL_MIN || lengths[2] < VERDICT_DBL_MIN ||
            lengths[3] < VERDICT_DBL_MIN )
        {
            metric_vals->scaled_jacobian = 0.0;
            metric_vals->shear           = 0.0;
        }
        else
        {
            scaled_jac     = areas[0] / ( lengths[0] * lengths[3] );
            min_scaled_jac = VERDICT_MIN( scaled_jac, min_scaled_jac );

            scaled_jac     = areas[1] / ( lengths[1] * lengths[0] );
            min_scaled_jac = VERDICT_MIN( scaled_jac, min_scaled_jac );

            scaled_jac     = areas[2] / ( lengths[2] * lengths[1] );
            min_scaled_jac = VERDICT_MIN( scaled_jac, min_scaled_jac );

            scaled_jac     = areas[3] / ( lengths[3] * lengths[2] );
            min_scaled_jac = VERDICT_MIN( scaled_jac, min_scaled_jac );

            metric_vals->scaled_jacobian = min_scaled_jac;

            // what the heck...set shear as well
            if( min_scaled_jac <= VERDICT_DBL_MIN )
                metric_vals->shear = 0.0;
            else
                metric_vals->shear = min_scaled_jac;
        }
    }

    if( metrics_request_flag & ( V_QUAD_WARPAGE | V_QUAD_ODDY ) )
    {
        VerdictVector corner_normals[4];
        corner_normals[0] = edges[3] * edges[0];
        corner_normals[1] = edges[0] * edges[1];
        corner_normals[2] = edges[1] * edges[2];
        corner_normals[3] = edges[2] * edges[3];

        if( metrics_request_flag & V_QUAD_ODDY )
        {
            double oddy, max_oddy = 0.0;

            double diff, dot_prod;

            double length_squared[4];
            length_squared[0] = corner_normals[0].length_squared();
            length_squared[1] = corner_normals[1].length_squared();
            length_squared[2] = corner_normals[2].length_squared();
            length_squared[3] = corner_normals[3].length_squared();

            if( length_squared[0] < VERDICT_DBL_MIN || length_squared[1] < VERDICT_DBL_MIN ||
                length_squared[2] < VERDICT_DBL_MIN || length_squared[3] < VERDICT_DBL_MIN )
                metric_vals->oddy = VERDICT_DBL_MAX;
            else
            {
                diff     = ( lengths[0] * lengths[0] ) - ( lengths[1] * lengths[1] );
                dot_prod = edges[0] % edges[1];
                oddy     = ( ( diff * diff ) + 4 * dot_prod * dot_prod ) / ( 2 * length_squared[1] );
                max_oddy = VERDICT_MAX( oddy, max_oddy );

                diff     = ( lengths[1] * lengths[1] ) - ( lengths[2] * lengths[2] );
                dot_prod = edges[1] % edges[2];
                oddy     = ( ( diff * diff ) + 4 * dot_prod * dot_prod ) / ( 2 * length_squared[2] );
                max_oddy = VERDICT_MAX( oddy, max_oddy );

                diff     = ( lengths[2] * lengths[2] ) - ( lengths[3] * lengths[3] );
                dot_prod = edges[2] % edges[3];
                oddy     = ( ( diff * diff ) + 4 * dot_prod * dot_prod ) / ( 2 * length_squared[3] );
                max_oddy = VERDICT_MAX( oddy, max_oddy );

                diff     = ( lengths[3] * lengths[3] ) - ( lengths[0] * lengths[0] );
                dot_prod = edges[3] % edges[0];
                oddy     = ( ( diff * diff ) + 4 * dot_prod * dot_prod ) / ( 2 * length_squared[0] );
                max_oddy = VERDICT_MAX( oddy, max_oddy );

                metric_vals->oddy = max_oddy;
            }
        }

        if( metrics_request_flag & V_QUAD_WARPAGE )
        {
            if( corner_normals[0].normalize() < VERDICT_DBL_MIN || corner_normals[1].normalize() < VERDICT_DBL_MIN ||
                corner_normals[2].normalize() < VERDICT_DBL_MIN || corner_normals[3].normalize() < VERDICT_DBL_MIN )
                metric_vals->warpage = VERDICT_DBL_MAX;
            else
            {
                metric_vals->warpage =
                    pow( VERDICT_MIN( corner_normals[0] % corner_normals[2], corner_normals[1] % corner_normals[3] ),
                         3 );
            }
        }
    }

    if( metrics_request_flag & V_QUAD_STRETCH )
    {
        VerdictVector temp;

        temp.set( coordinates[2][0] - coordinates[0][0], coordinates[2][1] - coordinates[0][1],
                  coordinates[2][2] - coordinates[0][2] );
        double diag02 = temp.length_squared();

        temp.set( coordinates[3][0] - coordinates[1][0], coordinates[3][1] - coordinates[1][1],
                  coordinates[3][2] - coordinates[1][2] );
        double diag13 = temp.length_squared();

        static const double QUAD_STRETCH_FACTOR = sqrt( 2.0 );

        // 'diag02' is now the max diagonal of the quad
        diag02 = VERDICT_MAX( diag02, diag13 );

        if( diag02 < VERDICT_DBL_MIN )
            metric_vals->stretch = VERDICT_DBL_MAX;
        else
            metric_vals->stretch =
                QUAD_STRETCH_FACTOR *
                VERDICT_MIN( VERDICT_MIN( lengths[0], lengths[1] ), VERDICT_MIN( lengths[2], lengths[3] ) ) /
                sqrt( diag02 );
    }

    if( metrics_request_flag & ( V_QUAD_CONDITION | V_QUAD_SHAPE | V_QUAD_SHAPE_AND_SIZE ) )
    {
        double lengths_squared[4];
        lengths_squared[0] = edges[0].length_squared();
        lengths_squared[1] = edges[1].length_squared();
        lengths_squared[2] = edges[2].length_squared();
        lengths_squared[3] = edges[3].length_squared();

        if( areas[0] < VERDICT_DBL_MIN || areas[1] < VERDICT_DBL_MIN || areas[2] < VERDICT_DBL_MIN ||
            areas[3] < VERDICT_DBL_MIN )
        {
            metric_vals->condition = VERDICT_DBL_MAX;
            metric_vals->shape     = 0.0;
        }
        else
        {
            double max_condition = 0.0, condition;

            condition     = ( lengths_squared[0] + lengths_squared[3] ) / areas[0];
            max_condition = VERDICT_MAX( max_condition, condition );

            condition     = ( lengths_squared[1] + lengths_squared[0] ) / areas[1];
            max_condition = VERDICT_MAX( max_condition, condition );

            condition     = ( lengths_squared[2] + lengths_squared[1] ) / areas[2];
            max_condition = VERDICT_MAX( max_condition, condition );

            condition     = ( lengths_squared[3] + lengths_squared[2] ) / areas[3];
            max_condition = VERDICT_MAX( max_condition, condition );

            metric_vals->condition = 0.5 * max_condition;
            metric_vals->shape     = 2 / max_condition;
        }
    }

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

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

    if( metrics_request_flag & V_QUAD_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 );
    }

    // calculate distortion
    if( metrics_request_flag & V_QUAD_DISTORTION )
    {
        metric_vals->distortion = v_quad_distortion( num_nodes, coordinates );

        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_QUAD_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_QUAD_MAXIMUM_ANGLE )
    {
        if( metric_vals->maximum_angle > 0 )
            metric_vals->maximum_angle = (double)VERDICT_MIN( metric_vals->maximum_angle, VERDICT_DBL_MAX );
        metric_vals->maximum_angle = (double)VERDICT_MAX( metric_vals->maximum_angle, -VERDICT_DBL_MAX );
    }

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

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

    if( metrics_request_flag & V_QUAD_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_QUAD_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_QUAD_SHEAR )
    {
        if( metric_vals->shear > 0 ) metric_vals->shear = (double)VERDICT_MIN( metric_vals->shear, VERDICT_DBL_MAX );
        metric_vals->shear = (double)VERDICT_MAX( metric_vals->shear, -VERDICT_DBL_MAX );
    }

    // calculate shear and size
    // reuse values from above
    if( metrics_request_flag & V_QUAD_SHEAR_AND_SIZE )
    {
        metric_vals->shear_and_size = metric_vals->shear * metric_vals->relative_size_squared;

        if( metric_vals->shear_and_size > 0 )
            metric_vals->shear_and_size = (double)VERDICT_MIN( metric_vals->shear_and_size, VERDICT_DBL_MAX );
        metric_vals->shear_and_size = (double)VERDICT_MAX( metric_vals->shear_and_size, -VERDICT_DBL_MAX );
    }

    if( metrics_request_flag & V_QUAD_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 );
    }

    // calculate shape and size
    // reuse values from above
    if( metrics_request_flag & V_QUAD_SHAPE_AND_SIZE )
    {
        metric_vals->shape_and_size = metric_vals->shape * metric_vals->relative_size_squared;

        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_QUAD_SKEW )
    {
        if( metric_vals->skew > 0 ) metric_vals->skew = (double)VERDICT_MIN( metric_vals->skew, VERDICT_DBL_MAX );
        metric_vals->skew = (double)VERDICT_MAX( metric_vals->skew, -VERDICT_DBL_MAX );
    }

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

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

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

    if( metrics_request_flag & V_QUAD_MED_ASPECT_FROBENIUS )
        metric_vals->med_aspect_frobenius = v_quad_med_aspect_frobenius( 4, coordinates );
    if( metrics_request_flag & V_QUAD_MAX_ASPECT_FROBENIUS )
        metric_vals->max_aspect_frobenius = v_quad_max_aspect_frobenius( 4, coordinates );
    if( metrics_request_flag & V_QUAD_EDGE_RATIO ) metric_vals->edge_ratio = v_quad_edge_ratio( 4, coordinates );
    if( metrics_request_flag & V_QUAD_ASPECT_RATIO ) metric_vals->aspect_ratio = v_quad_aspect_ratio( 4, coordinates );
    if( metrics_request_flag & V_QUAD_RADIUS_RATIO ) metric_vals->radius_ratio = v_quad_radius_ratio( 4, coordinates );
}