/*! 
  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);
  }
}

/*!
   the radius ratio of a triangle

   NB (P. Pebay 01/13/07): 
     CR / (3.0*IR) where CR is the circumradius and IR is the inradius

     this quality metric is also known to VERDICT, for tetrahedral elements only,
     a the "aspect beta"
   
*/
C_FUNC_DEF double v_tri_radius_ratio( int /*num_nodes*/, double coordinates[][3] )
{

  // three vectors for each side 
  VerdictVector a( coordinates[1][0] - coordinates[0][0],
                   coordinates[1][1] - coordinates[0][1],
                   coordinates[1][2] - coordinates[0][2] );
  
  VerdictVector b( coordinates[2][0] - coordinates[1][0],
                   coordinates[2][1] - coordinates[1][1],
                   coordinates[2][2] - coordinates[1][2] );
  
  VerdictVector c( coordinates[0][0] - coordinates[2][0],
                   coordinates[0][1] - coordinates[2][1],
                   coordinates[0][2] - coordinates[2][2] );

  double a2 = a.length_squared();
  double b2 = b.length_squared();
  double c2 = c.length_squared();
 
  VerdictVector ab = a * b;
  double denominator = ab.length_squared();

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

  double radius_ratio;
  radius_ratio = .25 * a2 * b2 * c2 * ( a2 + b2 + c2 ) / denominator;
  
  if( radius_ratio > 0 )
    return (double) VERDICT_MIN( radius_ratio, VERDICT_DBL_MAX );
  return (double) VERDICT_MAX( radius_ratio, -VERDICT_DBL_MAX );
}

/*!
  The scaled jacobian of a tri

  minimum of the jacobian divided by the lengths of 2 edge vectors
*/
C_FUNC_DEF double v_tri_scaled_jacobian( int /*num_nodes*/, double coordinates[][3])
{
  static const double detw = 2./sqrt(3.0);
  VerdictVector first, second;
  double jacobian; 
  
  VerdictVector edge[3];
  edge[0].set(coordinates[1][0] - coordinates[0][0],
              coordinates[1][1] - coordinates[0][1],
              coordinates[1][2] - coordinates[0][2]);

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

  edge[2].set(coordinates[2][0] - coordinates[1][0],
              coordinates[2][1] - coordinates[1][1],
              coordinates[2][2] - coordinates[1][2]);
  first = edge[1]-edge[0];
  second = edge[2]-edge[0];

  VerdictVector cross = first * second;
  jacobian = cross.length();

  double max_edge_length_product;
  max_edge_length_product = VERDICT_MAX( edge[0].length()*edge[1].length(),
                            VERDICT_MAX( edge[1].length()*edge[2].length(), 
                                         edge[0].length()*edge[2].length() ) ); 

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

  jacobian *= detw;
  jacobian /= max_edge_length_product; 

  if( compute_normal )
  {
    //center of tri
    double point[3], surf_normal[3];
    point[0] =  (coordinates[0][0] + coordinates[1][0] + coordinates[2][0]) / 3;
    point[1] =  (coordinates[0][1] + coordinates[1][1] + coordinates[2][1]) / 3;
    point[2] =  (coordinates[0][2] + coordinates[1][2] + coordinates[2][2]) / 3;

    //dot product
    compute_normal( point, surf_normal ); 
    if( (cross.x()*surf_normal[0] + 
         cross.y()*surf_normal[1] +
         cross.z()*surf_normal[2] ) < 0 )
      jacobian *= -1; 
  }

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

}

/*!
  the aspect of a tet

  CR / (3.0*IR) where CR is the circumsphere radius and IR is the inscribed sphere radius
*/
C_FUNC_DEF VERDICT_REAL v_tet_aspect_beta( int /*num_nodes*/, VERDICT_REAL 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 = 0.0;
  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 (VERDICT_REAL)VERDICT_DBL_MAX;
  else
  {
    double aspect_ratio;
    aspect_ratio = numerator.length() * area_sum / (108*volume*volume); 
    
    if( aspect_ratio > 0 )
      return (VERDICT_REAL) VERDICT_MIN( aspect_ratio, VERDICT_DBL_MAX );
    return (VERDICT_REAL) VERDICT_MAX( aspect_ratio, -VERDICT_DBL_MAX );
  }

}

inline void form_Q( const VerdictVector& v1,
                    const VerdictVector& v2,
                    const VerdictVector& v3,
                    VerdictVector& q1,
                    VerdictVector& q2,
                    VerdictVector& q3 )
{

    double g11, g12, g13, g22, g23, g33;

    g11 = v1 % v1;
    g12 = v1 % v2;
    g13 = v1 % v3;
    g22 = v2 % v2;
    g23 = v2 % v3;
    g33 = v3 % v3;

    double rtg11 = sqrt(g11);
    double rtg22 = sqrt(g22);
    double rtg33 = sqrt(g33);
    VerdictVector temp1;

    temp1 = v1 * v2;

    double cross = sqrt( temp1 % temp1 );

    double q11,q21,q31;
    double q12,q22,q32;
    double q13,q23,q33;

    q11=1;
    q21=0;
    q31=0;

    q12 = g12 / rtg11 / rtg22;
    q22 = cross / rtg11 / rtg22;
    q32 = 0;

    q13 = g13 / rtg11 / rtg33;
    q23 = ( g11*g23-g12*g13 )/ rtg11 / rtg33 / cross;
    temp1 = v2 * v3;
    q33 = ( v1 % temp1  ) / rtg33 / cross;

    q1.set( q11, q21, q31 );
    q2.set( q12, q22, q32 );
    q3.set( q13, q23, q33 );

}

/*!
  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 aspect ratio of a triangle

   NB (P. Pebay 01/14/07): 
     Hmax / ( 2.0 * sqrt(3.0) * IR) where Hmax is the maximum edge length 
     and IR is the inradius

     note that previous incarnations of verdict used "v_tri_aspect_ratio" to denote
     what is now called "v_tri_aspect_frobenius"
   
*/
C_FUNC_DEF double v_tri_aspect_ratio( int /*num_nodes*/, double coordinates[][3] )
{
  static const double normal_coeff = sqrt( 3. ) / 6.;

  // three vectors for each side 
  VerdictVector a( coordinates[1][0] - coordinates[0][0],
                   coordinates[1][1] - coordinates[0][1],
                   coordinates[1][2] - coordinates[0][2] );
  
  VerdictVector b( coordinates[2][0] - coordinates[1][0],
                   coordinates[2][1] - coordinates[1][1],
                   coordinates[2][2] - coordinates[1][2] );
  
  VerdictVector c( coordinates[0][0] - coordinates[2][0],
                   coordinates[0][1] - coordinates[2][1],
                   coordinates[0][2] - coordinates[2][2] );

  double a1 = a.length();
  double b1 = b.length();
  double c1 = c.length();
 
  double hm = a1 > b1 ? a1 : b1;
  hm = hm > c1 ? hm : c1;

  VerdictVector ab = a * b;
  double denominator = ab.length();

  if( denominator < VERDICT_DBL_MIN ) 
    return (double)VERDICT_DBL_MAX;
  else
  {
    double aspect_ratio;
    aspect_ratio = normal_coeff * hm * (a1 + b1 + c1) / denominator;
    
    if( aspect_ratio > 0 )
      return (double) VERDICT_MIN( aspect_ratio, VERDICT_DBL_MAX );
    return (double) VERDICT_MAX( aspect_ratio, -VERDICT_DBL_MAX );
  }

}

/*!
  The area of a tri

  0.5 * jacobian at a node
*/
C_FUNC_DEF double v_tri_area( int /*num_nodes*/, double coordinates[][3] )
{
  // two vectors for two sides
  VerdictVector side1( coordinates[1][0] - coordinates[0][0],
                       coordinates[1][1] - coordinates[0][1],
                       coordinates[1][2] - coordinates[0][2] );
  
  VerdictVector side3( coordinates[2][0] - coordinates[0][0],
                       coordinates[2][1] - coordinates[0][1],
                       coordinates[2][2] - coordinates[0][2] );
 
  // the cross product of the two vectors representing two sides of the
  // triangle 
  VerdictVector tmp = side1 * side3;
  
  // return the magnitude of the vector divided by two
  double area = 0.5 * tmp.length();
  if( area > 0 )
    return (double) VERDICT_MIN( area, VERDICT_DBL_MAX );
  return (double) VERDICT_MAX( area, -VERDICT_DBL_MAX );
  
}

inline double normalize_jacobian( double jacobi,
                                  VerdictVector& v1,
                                  VerdictVector& v2,
                                  VerdictVector& v3,
                                  int tet_flag = 0 )
{
    double return_value = 0.0;

    if ( jacobi != 0.0 )
    {

        double l1, l2, l3, length_product;
        // Note: there may be numerical problems if one is a lot shorter
        // than the others this way. But scaling each vector before the
        // triple product would involve 3 square roots instead of just
        // one.
        l1 = v1.length_squared();
        l2 = v2.length_squared();
        l3 = v3.length_squared();
        length_product = sqrt( l1 * l2 * l3 );

        // if some numerical scaling problem, or just plain roundoff,
        // then push back into range [-1,1].
        if ( length_product < fabs(jacobi) )
        {
            length_product = fabs(jacobi);
        }

        if( tet_flag == 1)
            return_value = v_sqrt_2 * jacobi / length_product;
        else
            return_value = jacobi / length_product;

    }
    return return_value;

}

inline void inverse(VerdictVector x1,
                    VerdictVector x2,
                    VerdictVector x3,
                    VerdictVector& u1,
                    VerdictVector& u2,
                    VerdictVector& u3 )
{
    double  detx = v_determinant(x1, x2, x3);
    VerdictVector rx1, rx2, rx3;

    rx1.set(x1.x(), x2.x(), x3.x());
    rx2.set(x1.y(), x2.y(), x3.y());
    rx3.set(x1.z(), x2.z(), x3.z());

    u1 = rx2 * rx3;
    u2 = rx3 * rx1;
    u3 = rx1 * rx2;

    u1 /= detx;
    u2 /= detx;
    u3 /= detx;
}

/*!
  the quality metrics of a tet
*/
C_FUNC_DEF void v_tet_quality( int num_nodes, VERDICT_REAL 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 = (VERDICT_REAL)(edges[3] % (edges[2] * edges[0]));
  }
 
  // calculate the volume 
  if(metrics_request_flag & V_TET_VOLUME)
  {
    metric_vals->volume = (VERDICT_REAL)(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 = (VERDICT_REAL)(VERDICT_DBL_MAX);
    else
      metric_vals->aspect_beta = 
        (VERDICT_REAL)( 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 ) 
//.........这里部分代码省略.........

inline void product( VerdictVector& a1,
                     VerdictVector& a2,
                     VerdictVector& a3,
                     VerdictVector& b1,
                     VerdictVector& b2,
                     VerdictVector& b3,
                     VerdictVector& c1,
                     VerdictVector& c2,
                     VerdictVector& c3 )
{

    VerdictVector x1, x2, x3;

    x1.set( a1.x(), a2.x(), a3.x() );
    x2.set( a1.y(), a2.y(), a3.y() );
    x3.set( a1.z(), a2.z(), a3.z() );

    c1.set( x1 % b1, x2 % b1, x3 % b1 );
    c2.set( x1 % b2, x2 % b2, x3 % b2 );
    c3.set( x1 % b3, x2 % b3, x3 % b3 );
}

/*! 
  tri_quality for calculating multiple tri functions at once

  using this method is generally faster than using the individual 
  method multiple times.

*/
C_FUNC_DEF void v_tri_quality( int num_nodes, double coordinates[][3], 
    unsigned int metrics_request_flag, TriMetricVals *metric_vals ) 
{

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

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

  /*  node numbers and side numbers used below

             2
             ++
            /  \ 
         2 /    \ 1
          /      \
         /        \
       0 ---------+ 1
             0
  */
  
  // vectors for each side
  VerdictVector sides[3];
  sides[0].set(
      coordinates[1][0] - coordinates[0][0],
      coordinates[1][1] - coordinates[0][1],
      coordinates[1][2] - coordinates[0][2]
      );
  sides[1].set(
      coordinates[2][0] - coordinates[1][0],
      coordinates[2][1] - coordinates[1][1],
      coordinates[2][2] - coordinates[1][2]
      );
  sides[2].set(
      coordinates[2][0] - coordinates[0][0],
      coordinates[2][1] - coordinates[0][1],
      coordinates[2][2] - coordinates[0][2]
      );
  VerdictVector tri_normal = sides[0] * sides[2];
    //if we have access to normal information, check to see if the
    //element is inverted.  If we don't have the normal information
    //that we need for this, assume the element is not inverted.
    //This flag will be used for condition number, jacobian, shape,
    //and size and shape.
  bool is_inverted = false;
  if( compute_normal )
  {
      //center of tri
    double point[3], surf_normal[3];
    point[0] =  (coordinates[0][0] + coordinates[1][0] + coordinates[2][0]) / 3;
    point[1] =  (coordinates[0][1] + coordinates[1][1] + coordinates[2][1]) / 3;
    point[2] =  (coordinates[0][2] + coordinates[1][2] + coordinates[2][2]) / 3;
      //dot product
    compute_normal( point, surf_normal ); 
    if( (tri_normal.x()*surf_normal[0] + 
         tri_normal.y()*surf_normal[1] +
         tri_normal.z()*surf_normal[2] ) < 0 )
      is_inverted=true; 
  }
  
  // lengths squared of each side
  double sides_lengths_squared[3];
  sides_lengths_squared[0] = sides[0].length_squared();
  sides_lengths_squared[1] = sides[1].length_squared();
  sides_lengths_squared[2] = sides[2].length_squared();
 

  // if we are doing angle calcuations
  if( metrics_request_flag & (V_TRI_MINIMUM_ANGLE | V_TRI_MAXIMUM_ANGLE) )
  {
    // which is short and long side
    int short_side=0, long_side=0;

    if(sides_lengths_squared[1] < sides_lengths_squared[0])
      short_side = 1;
    if(sides_lengths_squared[2] < sides_lengths_squared[short_side])
      short_side = 2;
  
    if(sides_lengths_squared[1] > sides_lengths_squared[0])
      long_side = 1;
    if(sides_lengths_squared[2] > sides_lengths_squared[long_side])
      long_side = 2;


    // calculate the minimum angle of the tri
    if( metrics_request_flag & V_TRI_MINIMUM_ANGLE )
    {
      if(sides_lengths_squared[0] == 0.0 || 
        sides_lengths_squared[1] == 0.0 ||
        sides_lengths_squared[2] == 0.0)
      {
        metric_vals->minimum_angle = 0.0;
      }        
      else if(short_side == 0)
//.........这里部分代码省略.........

/*!
  multiple quality measures of a quad
*/
C_FUNC_DEF void v_quad_quality( int num_nodes, VERDICT_REAL 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 = (VERDICT_REAL)(v_tri_area(3, coordinates) * 2.0);
    if(metrics_request_flag & V_QUAD_SCALED_JACOBIAN)
      metric_vals->jacobian = (VERDICT_REAL)(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 aspect, skew, taper, and area together
  if( metrics_request_flag & ( V_QUAD_ASPECT | V_QUAD_SKEW | V_QUAD_TAPER ) )
//.........这里部分代码省略.........