# include <cmath>
# include <cstdlib>
# include <cstring>
# include <ctime>
# include <fstream>
# include <iomanip>
# include <iostream>

using namespace std;

# include "polygon_triangulate.hpp"

//****************************************************************************80

double angle_degree ( double x1, double y1, double x2, double y2, double x3, 
  double y3 )

//****************************************************************************80
//
//  Purpose:
//
//    ANGLE_DEGREE returns the degree angle defined by three points.
//
//  Discussion:
//
//        P1
//        /
//       /
//      /
//     /
//    P2--------->P3
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    28 August 2016
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X1, Y1, X2, Y2, X3, Y3, the coordinates of the points
//    P1, P2, P3.
//
//    Output, double ANGLE_DEGREE, the angle swept out by the rays, measured
//    in degrees.  0 <= VALUE < 360.  If either ray has zero length,
//    then VALUE is set to 0.
//
{
  double r8_pi = 3.141592653589793;
  double value;
  double x;
  double y;

  x = ( x3 - x2 ) * ( x1 - x2 ) + ( y3 - y2 ) * ( y1 - y2 );

  y = ( x3 - x2 ) * ( y1 - y2 ) - ( y3 - y2 ) * ( x1 - x2 );

  if ( x == 0.0 && y == 0.0 )
  {
    value = 0.0;
  }
  else
  {
    value = atan2 ( y, x );

    if ( value < 0.0 )
    {
      value = value + 2.0 * r8_pi;
    }
    value = 180.0 * value / r8_pi;
  }

  return value;
}
//****************************************************************************80

bool between ( double xa, double ya, double xb, double yb, double xc, 
  double yc )

//****************************************************************************80
//
//  Purpose:
//
//    BETWEEN is TRUE if vertex C is between vertices A and B.
//
//  Discussion:
//
//    The points must be (numerically) collinear.
//
//    Given that condition, we take the greater of XA - XB and YA - YB
//    as a "scale" and check where C's value lies.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 May 2014
//
//  Author:
//
//    Original C version by Joseph ORourke.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Joseph ORourke,
//    Computational Geometry in C,
//    Cambridge, 1998,
//    ISBN: 0521649765,
//    LC: QA448.D38.
//
//  Parameters:
//
//    Input, double XA, YA, XB, YB, XC, YC, the coordinates of 
//    the vertices.
//
//    Output, bool BETWEEN, is TRUE if C is between A and B.
//
{
  bool value;
  double xmax;
  double xmin;
  double ymax;
  double ymin;

  if ( ! collinear ( xa, ya, xb, yb, xc, yc ) )
  {
    value = false;
  }
  else if ( fabs ( ya - yb ) < fabs ( xa - xb ) )
  {
    xmax = fmax ( xa, xb );
    xmin = fmin ( xa, xb );
    value = ( xmin <= xc && xc <= xmax );
  }
  else
  {
    ymax = fmax ( ya, yb );
    ymin = fmin ( ya, yb );
    value = ( ymin <= yc && yc <= ymax );
  }

  return value;
}
//****************************************************************************80

char ch_cap ( char ch )

//****************************************************************************80
//
//  Purpose:
//
//    CH_CAP capitalizes a single character.
//
//  Discussion:
//
//    This routine should be equivalent to the library "toupper" function.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    19 July 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH, the character to capitalize.
//
//    Output, char CH_CAP, the capitalized character.
//
{
  if ( 97 <= ch && ch <= 122 )
  {
    ch = ch - 32;
  }

  return ch;
}
//****************************************************************************80

bool ch_eqi ( char ch1, char ch2 )

//****************************************************************************80
//
//  Purpose:
//
//    CH_EQI is true if two characters are equal, disregarding case.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    13 June 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH1, CH2, the characters to compare.
//
//    Output, bool CH_EQI, is true if the two characters are equal,
//    disregarding case.
//
{
  if ( 97 <= ch1 && ch1 <= 122 )
  {
    ch1 = ch1 - 32;
  }
  if ( 97 <= ch2 && ch2 <= 122 )
  {
    ch2 = ch2 - 32;
  }

  return ( ch1 == ch2 );
}
//****************************************************************************80

int ch_to_digit ( char ch )

//****************************************************************************80
//
//  Purpose:
//
//    CH_TO_DIGIT returns the integer value of a base 10 digit.
//
//  Example:
//
//     CH  DIGIT
//    ---  -----
//    '0'    0
//    '1'    1
//    ...  ...
//    '9'    9
//    ' '    0
//    'X'   -1
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    13 June 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, char CH, the decimal digit, '0' through '9' or blank are legal.
//
//    Output, int CH_TO_DIGIT, the corresponding integer value.  If the
//    character was 'illegal', then DIGIT is -1.
//
{
  int digit;

  if ( '0' <= ch && ch <= '9' )
  {
    digit = ch - '0';
  }
  else if ( ch == ' ' )
  {
    digit = 0;
  }
  else
  {
    digit = -1;
  }

  return digit;
}
//****************************************************************************80

bool collinear ( double xa, double ya, double xb, double yb, double xc, 
  double yc )

//****************************************************************************80
//
//  Purpose:
//
//    COLLINEAR returns a measure of collinearity for three points.
//
//  Discussion:
//
//    In order to deal with collinear points whose coordinates are not
//    numerically exact, we compare the area of the largest square
//    that can be created by the line segment between two of the points
//    to (twice) the area of the triangle formed by the points.
//
//    If the points are collinear, their triangle has zero area.
//    If the points are close to collinear, then the area of this triangle
//    will be small relative to the square of the longest segment.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    10 September 2016
//
//  Author:
//
//    Original C version by Joseph ORourke.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Joseph ORourke,
//    Computational Geometry in C,
//    Cambridge, 1998,
//    ISBN: 0521649765,
//    LC: QA448.D38.
//
//  Parameters:
//
//    Input, double XA, YA, XB, YB, XC, YC, the vertex coordinates.
//
//    Output, bool COLLINEAR, is TRUE if the points are judged 
//    to be collinear.
//
{
  double area;
  const double r8_eps = 2.220446049250313E-016;
  double side_ab_sq;
  double side_bc_sq;
  double side_ca_sq;
  double side_max_sq;
  bool value;

  area = triangle_area ( xa, ya, xb, yb, xc, yc );

  side_ab_sq = pow ( xa - xb, 2 ) + pow ( ya - yb, 2 );
  side_bc_sq = pow ( xb - xc, 2 ) + pow ( yb - yc, 2 );
  side_ca_sq = pow ( xc - xa, 2 ) + pow ( yc - ya, 2 );

  side_max_sq = fmax ( side_ab_sq, fmax ( side_bc_sq, side_ca_sq ) );

  if ( side_max_sq <= r8_eps )
  {
    value = true;
  }
  else if ( 2.0 * fabs ( area ) <= r8_eps * side_max_sq )
  {
    value = true;
  }
  else
  {
    value = false;
  }

  return value;
}
//****************************************************************************80

bool diagonal ( int im1, int ip1, int n, int prev_node[], int next_node[], 
  double x[], double y[] )

//****************************************************************************80
//
//  Purpose:
//
//    DIAGONAL: VERTEX(IM1) to VERTEX(IP1) is a proper internal diagonal.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 May 2014
//
//  Author:
//
//    Original C version by Joseph ORourke.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Joseph ORourke,
//    Computational Geometry in C,
//    Cambridge, 1998,
//    ISBN: 0521649765,
//    LC: QA448.D38.
//
//  Parameters:
//
//    Input, int IM1, IP1, the indices of two vertices.
//
//    Input, int N, the number of vertices.
//
//    Input, int PREV_NODE[N], the previous neighbor of each vertex.
//
//    Input, int NEXT_NODE[N], the next neighbor of each vertex.
//
//    Input, double X[N], Y[N], the coordinates of each vertex.
//
//    Output, bool DIAGONAL, the value of the test.
//
{
  bool value;
  bool value1;
  bool value2;
  bool value3;

  value1 = in_cone ( im1, ip1, n, prev_node, next_node, x, y );
  value2 = in_cone ( ip1, im1, n, prev_node, next_node, x, y );
  value3 = diagonalie ( im1, ip1, n, next_node, x, y );

  value = ( value1 && value2 && value3 );

  return value;
}
//****************************************************************************80

bool diagonalie ( int im1, int ip1, int n, int next_node[], double x[], 
  double y[] )

//****************************************************************************80
//
//  Purpose:
//
//    DIAGONALIE is true if VERTEX(IM1):VERTEX(IP1) is a proper diagonal.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 May 2014
//
//  Author:
//
//    Original C version by Joseph ORourke.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Joseph ORourke,
//    Computational Geometry in C,
//    Cambridge, 1998,
//    ISBN: 0521649765,
//    LC: QA448.D38.
//
//  Parameters:
//
//    Input, int IM1, IP1, the indices of two vertices.
//
//    Input, int N, the number of vertices.
//
//    Input, int NEXT_NODE[N], the next neighbor of each vertex.
//
//    Input, double X[N], Y[N], the coordinates of each vertex.
//
//    Output, bool DIAGONALIE, the value of the test.
//
{
  int first;
  int j;
  int jp1;
  bool value;
  bool value2;

  first = im1;
  j = first;
  jp1 = next_node[first];

  value = true;
//
//  For each edge VERTEX(J):VERTEX(JP1) of the polygon:
//
  while ( 1 )
  {
//
//  Skip any edge that includes vertex IM1 or IP1.
//
    if ( j == im1 || j == ip1 || jp1 == im1 || jp1 == ip1 )
    {
    }
    else
    {
      value2 = intersect ( x[im1], y[im1], x[ip1], y[ip1], x[j], y[j], 
        x[jp1], y[jp1] );

      if ( value2 )
      {
        value = false;
        break;
      }
    }
    j = jp1;
    jp1 = next_node[j];

    if ( j == first )
    {
      break;
    }
  }

  return value;
}
//****************************************************************************80

int file_column_count ( string filename )

//****************************************************************************80
//
//  Purpose:
//
//    FILE_COLUMN_COUNT counts the columns in the first line of a file.
//
//  Discussion:
//
//    The file is assumed to be a simple text file.
//
//    Most lines of the file are presumed to consist of COLUMN_NUM words,
//    separated by spaces.  There may also be some blank lines, and some
//    comment lines, which have a "#" in column 1.
//
//    The routine tries to find the first non-comment non-blank line and
//    counts the number of words in that line.
//
//    If all lines are blanks or comments, it goes back and tries to analyze
//    a comment line.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string FILENAME, the name of the file.
//
//    Output, int FILE_COLUMN_COUNT, the number of columns assumed
//    to be in the file.
//
{
  int column_num;
  ifstream input;
  bool got_one;
  string text;
//
//  Open the file.
//
  input.open ( filename.c_str ( ) );

  if ( !input )
  {
    column_num = -1;
    cerr << "\n";
    cerr << "FILE_COLUMN_COUNT - Fatal error!\n";
    cerr << "  Could not open the file:\n";
    cerr << "  \"" << filename << "\"\n";
    exit ( 1 );
  }
//
//  Read one line, but skip blank lines and comment lines.
//
  got_one = false;

  for ( ; ; )
  {
    getline ( input, text );

    if ( input.eof ( ) )
    {
      break;
    }

    if ( s_len_trim ( text ) <= 0 )
    {
      continue;
    }

    if ( text[0] == '#' )
    {
      continue;
    }
    got_one = true;
    break;
  }

  if ( !got_one )
  {
    input.close ( );

    input.open ( filename.c_str ( ) );

    for ( ; ; )
    {
      input >> text;

      if ( input.eof ( ) )
      {
        break;
      }

      if ( s_len_trim ( text ) == 0 )
      {
        continue;
      }
      got_one = true;
      break;
    }
  }

  input.close ( );

  if ( !got_one )
  {
    cerr << "\n";
    cerr << "FILE_COLUMN_COUNT - Warning!\n";
    cerr << "  The file does not seem to contain any data.\n";
    return -1;
  }

  column_num = s_word_count ( text );

  return column_num;
}
//****************************************************************************80

int file_row_count ( string input_filename )

//****************************************************************************80
//
//  Purpose:
//
//    FILE_ROW_COUNT counts the number of row records in a file.
//
//  Discussion:
//
//    It does not count lines that are blank, or that begin with a
//    comment symbol '#'.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Output, int FILE_ROW_COUNT, the number of rows found.
//
{
  int comment_num;
  ifstream input;
  string line;
  int record_num;
  int row_num;

  row_num = 0;
  comment_num = 0;
  record_num = 0;

  input.open ( input_filename.c_str ( ) );

  if ( !input )
  {
    cerr << "\n";
    cerr << "FILE_ROW_COUNT - Fatal error!\n";
    cerr << "  Could not open the input file: \"" << input_filename << "\"\n";
    exit ( 1 );
  }

  for ( ; ; )
  {
    getline ( input, line );

    if ( input.eof ( ) )
    {
      break;
    }

    record_num = record_num + 1;

    if ( line[0] == '#' )
    {
      comment_num = comment_num + 1;
      continue;
    }

    if ( s_len_trim ( line ) == 0 )
    {
      comment_num = comment_num + 1;
      continue;
    }

    row_num = row_num + 1;

  }

  input.close ( );

  return row_num;
}
//****************************************************************************80

void i4mat_transpose_print ( int m, int n, int a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
//
//  Discussion:
//
//    An I4MAT is an MxN array of I4's, stored by (I,J) -> [I+J*M].
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    31 January 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows in A.
//
//    Input, int N, the number of columns in A.
//
//    Input, int A[M*N], the M by N matrix.
//
//    Input, string TITLE, a title.
//
{
  i4mat_transpose_print_some ( m, n, a, 1, 1, m, n, title );

  return;
}
//****************************************************************************80

void i4mat_transpose_print_some ( int m, int n, int a[], int ilo, int jlo,
  int ihi, int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
//
//  Discussion:
//
//    An I4MAT is an MxN array of I4's, stored by (I,J) -> [I+J*M].
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    15 October 2014
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows of the matrix.
//    M must be positive.
//
//    Input, int N, the number of columns of the matrix.
//    N must be positive.
//
//    Input, int A[M*N], the matrix.
//
//    Input, int ILO, JLO, IHI, JHI, designate the first row and
//    column, and the last row and column to be printed.
//
//    Input, string TITLE, a title.
//
{
# define INCX 10

  int i;
  int i2hi;
  int i2lo;
  int j;
  int j2hi;
  int j2lo;

  cout << "\n";
  cout << title << "\n";

  if ( m <= 0 || n <= 0 )
  {
    cout << "\n";
    cout << "  (None)\n";
    return;
  }
//
//  Print the columns of the matrix, in strips of INCX.
//
  for ( i2lo = ilo; i2lo <= ihi; i2lo = i2lo + INCX )
  {
    i2hi = i2lo + INCX - 1;
    if ( m < i2hi )
    {
      i2hi = m;
    }
    if ( ihi < i2hi )
    {
      i2hi = ihi;
    }
    cout << "\n";
//
//  For each row I in the current range...
//
//  Write the header.
//
    cout << "  Row: ";
    for ( i = i2lo; i <= i2hi; i++ )
    {
      cout << setw(6) << i - 1 << "  ";
    }
    cout << "\n";
    cout << "  Col\n";
    cout << "\n";
//
//  Determine the range of the rows in this strip.
//
    j2lo = jlo;
    if ( j2lo < 1 )
    {
      j2lo = 1;
    }
    j2hi = jhi;
    if ( n < j2hi )
    {
      j2hi = n;
    }
    for ( j = j2lo; j <= j2hi; j++ )
    {
//
//  Print out (up to INCX) entries in column J, that lie in the current strip.
//
      cout << setw(5) << j - 1 << ":";
      for ( i = i2lo; i <= i2hi; i++ )
      {
        cout << setw(6) << a[i-1+(j-1)*m] << "  ";
      }
      cout << "\n";
    }
  }

  return;
# undef INCX
}
//****************************************************************************80

void i4mat_write ( string output_filename, int m, int n, int table[] )

//****************************************************************************80
//
//  Purpose:
//
//    I4MAT_WRITE writes an I4MAT file with no header.
//
//  Discussion:
//
//    An I4MAT is an array of I4's.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    01 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string OUTPUT_FILENAME, the output filename.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input, int TABLE[M*N], the data.
//
{
  int i;
  int j;
  ofstream output;
//
//  Open the file.
//
  output.open ( output_filename.c_str ( ) );

  if ( !output )
  {
    cerr << "\n";
    cerr << "I4MAT_WRITE - Fatal error!\n";
    cerr << "  Could not open the output file.\n";
    exit ( 1 );
  }
//
//  Write the data.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      output << "  " << setw(10) << table[i+j*m];
    }
    output << "\n";
  }
//
//  Close the file.
//
  output.close ( );

  return;
}
//****************************************************************************80

bool in_cone ( int im1, int ip1, int n, int prev_node[], int next_node[], 
  double x[], double y[] )

//****************************************************************************80
//
//  Purpose:
//
//    IN_CONE is TRUE if the diagonal VERTEX(IM1):VERTEX(IP1) is strictly internal.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 May 2014
//
//  Author:
//
//    Original C version by Joseph ORourke.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Joseph ORourke,
//    Computational Geometry in C,
//    Cambridge, 1998,
//    ISBN: 0521649765,
//    LC: QA448.D38.
//
//  Parameters:
//
//    Input, int IM1, IP1, the indices of two vertices.
//
//    Input, int N, the number of vertices.
//
//    Input, int PREV_NODE[N], the previous neighbor of each vertex.
//
//    Input, int NEXT_NODE[N], the next neighbor of each vertex.
//
//    Input, double X[N], Y[N], the coordinates of each vertex.
//
//    Output, bool IN_CONE, the value of the test.
//
{
  int i;
  int im2;
  double t1;
  double t2;
  double t3;
  double t4;
  double t5;
  bool value;

  im2 = prev_node[im1];
  i = next_node[im1];

  t1 = triangle_area ( x[im1], y[im1], x[i], y[i], x[im2], y[im2] );

  if ( 0.0 <= t1 )
  {
    t2 = triangle_area ( x[im1], y[im1], x[ip1], y[ip1], x[im2], y[im2] );
    t3 = triangle_area ( x[ip1], y[ip1], x[im1], y[im1], x[i], y[i] );
    value = ( ( 0.0 < t2 ) && ( 0.0 < t3 ) );
  }
  else
  {
    t4 = triangle_area ( x[im1], y[im1], x[ip1], y[ip1], x[i], y[i] );
    t5 = triangle_area ( x[ip1], y[ip1], x[im1], y[im1], x[im2], y[im2] );
    value = ! ( ( 0.0 <= t4 ) && ( 0.0 <= t5 ) );
  }
  return value;
}
//****************************************************************************80

bool intersect ( double xa, double ya, double xb, double yb, double xc, 
  double yc, double xd, double yd )

//****************************************************************************80
//
//  Purpose:
//
//    INTERSECT is true if lines VA:VB and VC:VD intersect.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 May 2014
//
//  Author:
//
//    Original C version by Joseph ORourke.
//    C++ version by John Burkardt.
//
//  Reference:
//
//    Joseph ORourke,
//    Computational Geometry in C,
//    Cambridge, 1998,
//    ISBN: 0521649765,
//    LC: QA448.D38.
//
//  Parameters:
//
//    Input, double XA, YA, XB, YB, XC, YC, XD, YD, the X and Y 
//    coordinates of the four vertices.
//
//    Output, bool INTERSECT, the value of the test.
//
{
  bool value;

  if ( intersect_prop ( xa, ya, xb, yb, xc, yc, xd, yd ) )
  {
    value = true;
  }
  else if ( between ( xa, ya, xb, yb, xc, yc ) )
  {
    value = true;
  }
  else if ( between ( xa, ya, xb, yb, xd, yd ) )
  {
    value = true;
  }
  else if ( between ( xc, yc, xd, yd, xa, ya ) )
  {
    value = true;
  }
  else if ( between ( xc, yc, xd, yd, xb, yb ) )
  {
    value = true;
  }
  else
  {
    value = false;
  }
  return value;
}
//****************************************************************************80

bool intersect_prop ( double xa, double ya, double xb, double yb, double xc, 
  double yc, double xd, double yd )

//****************************************************************************80
//
//  Purpose:
//
//    INTERSECT_PROP is TRUE if lines VA:VB and VC:VD have a proper intersection.
//
//  Discussion:
//
//    Thanks to Gene Dial for correcting the call to intersect_prop(),
//    08 September 2016.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    08 September 2016
//
//  Author:
//
//    Original C version by Joseph ORourke.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Joseph ORourke,
//    Computational Geometry in C,
//    Cambridge, 1998,
//    ISBN: 0521649765,
//    LC: QA448.D38.
//
//  Parameters:
//
//    Input, double XA, YA, XB, YB, XC, YC, XD, YD, the X and Y 
//    coordinates of the four vertices.
//
//    Output, bool INTERSECT_PROP, the result of the test.
//
{
  double t1;
  double t2;
  double t3;
  double t4;
  bool value;
  bool value1;
  bool value2;
  bool value3;
  bool value4;

  if ( collinear ( xa, ya, xb, yb, xc, yc ) )
  {
    value = false;
  }
  else if ( collinear ( xa, ya, xb, yb, xd, yd ) )
  {
    value = false;
  }
  else if ( collinear ( xc, yc, xd, yd, xa, ya ) )
  {
    value = false;
  }
  else if ( collinear ( xc, yc, xd, yd, xb, yb ) )
  {
    value = false;
  }
  else
  {
    t1 = triangle_area ( xa, ya, xb, yb, xc, yc );
    t2 = triangle_area ( xa, ya, xb, yb, xd, yd );
    t3 = triangle_area ( xc, yc, xd, yd, xa, ya );
    t4 = triangle_area ( xc, yc, xd, yd, xb, yb );

    value1 = ( 0.0 < t1 );
    value2 = ( 0.0 < t2 );
    value3 = ( 0.0 < t3 );
    value4 = ( 0.0 < t4 );

    value = ( l4_xor ( value1, value2 ) ) && ( l4_xor ( value3, value4 ) );
  }
  return value;
}
//****************************************************************************80

bool l4_xor ( bool l1, bool l2 )

//****************************************************************************80
//
//  Purpose:
//
//    L4_XOR returns the exclusive OR of two L4's.
//
//  Discussion:
//
//    An L4 is a logical value.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    04 May 2014
//
//  Author:
//
//   John Burkardt
//
//  Parameters:
//
//    Input, bool L1, L2, two values whose exclusive OR is needed.
//
//    Output, bool L4_XOR, the exclusive OR of L1 and L2.
//
{
  bool value;
  bool value1;
  bool value2;

  value1 = (     l1   && ( ! l2 ) );
  value2 = ( ( ! l1 ) &&     l2   );

  value = ( value1 || value2 );

  return value;
}
//****************************************************************************80

void l4vec_print ( int n, bool a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    L4VEC_PRINT prints an L4VEC.
//
//  Licensing:
//
//    This code is distributed under the MIT license. 
//
//  Modified:
//
//    03 April 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of components of the vector.
//
//    Input, bool A[N], the vector to be printed.
//
//    Input, string TITLE, a title.
//
{
  int i;

  cout << "\n";
  cout << title << "\n";
  cout << "\n";
  for ( i = 0; i < n; i++ ) 
  {
    cout << "  " << setw(8) << i 
         << ": " << setw(1) << a[i]  << "\n";
  }

  return;
}
//****************************************************************************80

double polygon_area ( int n, double x[], double y[] )

//****************************************************************************80
//
//  Purpose:
//
//    POLYGON_AREA returns the area of a polygon.
//
//  Discussion:
//
//    The vertices should be listed in counter-clockwise order so that
//    the area will be positive.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    10 September 2016
//
//  Author:
//
//    John Burkardt.
//
//  Parameters:
//
//    Input, int N, the number of vertices.
//
//    Input, double X[N], Y[N], the vertex coordinates.
//
//    Output, double POLYGON_AREA, the area of the polygon.
//
{
  double area;
  int i;
  int im1;

  area = 0.0;
  im1 = n - 1;
  for ( i = 0; i < n; i++ )
  {
    area = area + x[im1] * y[i] - x[i] * y[im1];
    im1 = i;
  }
  area = 0.5 * area;

  return area;
}
//****************************************************************************80

int *polygon_triangulate ( int n, double x[], double y[] )

//****************************************************************************80
//
//  Purpose:
//
//    POLYGON_TRIANGULATE determines a triangulation of a polygon.
//
//  Discussion:
//
//    There are N-3 triangles in the triangulation.
//
//    For the first N-2 triangles, the first edge listed is always an
//    internal diagonal.
//
//    Thanks to Gene Dial for pointing out a mistake in the area calculation,
//    10 September 2016.
//
//    Gene Dial requested an angle tolerance of about 1 millionth radian or 
//    5.7E-05 degrees, 26 June 2018.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    26 June 2018
//
//  Author:
//
//    Original C version by Joseph ORourke.
//    This C++ version by John Burkardt.
//
//  Reference:
//
//    Joseph ORourke,
//    Computational Geometry in C,
//    Cambridge, 1998,
//    ISBN: 0521649765,
//    LC: QA448.D38.
//
//  Parameters:
//
//    Input, int N, the number of vertices.
//
//    Input, double X[N], Y[N], the coordinates of each vertex.
//
//    Output, int TRIANGLES[3*(N-2)], the triangles of the 
//    triangulation.
//
{
  double angle;
  const double angle_tol = 5.7E-05;
  double area;
  bool *ear;
  int i;
  int i0;
  int i1;
  int i2;
  int i3;
  int i4;
  int *next_node;
  int node;
  int node1;
  int node2;
  int node3;
  int node_m1;
  int *prev_node;
  int triangle_num;
  int *triangles;
//
//  We must have at least 3 vertices.
//
  if ( n < 3 )
  {
    cerr << "\n";
    cerr << "POLYGON_TRIANGULATE - Fatal error!\n";
    cerr << "  N < 3.\n";
    exit ( 1 );
  }
//
//  Consecutive vertices cannot be equal.
//
  node_m1 = n - 1;
  for ( node = 0; node < n; node++ )
  {
    if ( x[node_m1] == x[node] && y[node_m1] == y[node] )
    {
      cerr << "\n";
      cerr << "POLYGON_TRIANGULATE - Fatal error!\n";
      cerr << "  Two consecutive nodes are identical.\n";
      exit ( 1 );
    }
    node_m1 = node;
  }
//
//  No node can be the vertex of an angle less than 1 degree 
//  in absolute value.
//
  node1 = n - 1;

  for ( node2 = 0; node2 < n; node2++ )
  {
    node3 = ( ( node2 + 1 ) % n ); 

    angle = angle_degree ( 
      x[node1], y[node1], 
      x[node2], y[node2], 
      x[node3], y[node3] );

    if ( fabs ( angle ) <= angle_tol )
    {
      cerr << "\n";
      cerr << "POLYGON_TRIANGULATE - Fatal error!\n";
      cerr << "  Polygon has an angle smaller than " << angle_tol << "\n";
      cerr << "  occurring at node " << node2 << "\n";
      exit ( 1 );
    }
    node1 = node2;
  }
//
//  Area must be positive.
//
  area = polygon_area ( n, x, y );

  if ( area <= 0.0 )
  {
    cerr << "\n";
    cerr << "POLYGON_TRIANGULATE - Fatal error!\n";
    cerr << "  Polygon has zero or negative area.\n";
    exit ( 1 );
  }

  triangles = new int[3*(n-2)];
//
//  PREV_NODE and NEXT_NODE point to the previous and next nodes.
//
  prev_node = new int[n];
  next_node = new int[n];

  i = 0;
  prev_node[i] = n - 1;
  next_node[i] = i + 1;

  for ( i = 1; i < n - 1; i++ )
  {
    prev_node[i] = i - 1;
    next_node[i] = i + 1;
  }

  i = n - 1;
  prev_node[i] = i - 1;
  next_node[i] = 0;
//
//  EAR indicates whether the node and its immediate neighbors form an ear
//  that can be sliced off immediately.
//
  ear = new bool[n];
  for ( i = 0; i < n; i++ )
  {
    ear[i] = diagonal ( prev_node[i], next_node[i], n, prev_node, next_node, 
      x, y );
  }

  triangle_num = 0;

  i2 = 0;

  while ( triangle_num < n - 3 )
  {
//
//  If I2 is an ear, gather information necessary to carry out
//  the slicing operation and subsequent "healing".
//
    if ( ear[i2] )
    {
      i3 = next_node[i2];
      i4 = next_node[i3];
      i1 = prev_node[i2];
      i0 = prev_node[i1];
//
//  Make vertex I2 disappear.
//
      next_node[i1] = i3;
      prev_node[i3] = i1;
//
//  Update the earity of I1 and I3, because I2 disappeared.
//
      ear[i1] = diagonal ( i0, i3, n, prev_node, next_node, x, y );
      ear[i3] = diagonal ( i1, i4, n, prev_node, next_node, x, y );
//
//  Add the diagonal [I3, I1, I2] to the list.
//
      triangles[0+triangle_num*3] = i3;
      triangles[1+triangle_num*3] = i1;
      triangles[2+triangle_num*3] = i2;
      triangle_num = triangle_num + 1;
    }
//
//  Try the next vertex.
//
    i2 = next_node[i2];
  }
//
//  The last triangle is formed from the three remaining vertices.
//
  i3 = next_node[i2];
  i1 = prev_node[i2];

  triangles[0+triangle_num*3] = i3;
  triangles[1+triangle_num*3] = i1;
  triangles[2+triangle_num*3] = i2;
  triangle_num = triangle_num + 1;
//
//  Free memory.
//
  delete [] ear;
  delete [] next_node;
  delete [] prev_node;

  return triangles;
}
//****************************************************************************80

double *r8mat_data_read ( string input_filename, int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_DATA_READ reads the data from an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//    The file is assumed to contain one record per line.
//
//    Records beginning with '#' are comments, and are ignored.
//    Blank lines are also ignored.
//
//    Each line that is not ignored is assumed to contain exactly (or at least)
//    M real numbers, representing the coordinates of a point.
//
//    There are assumed to be exactly (or at least) N such records.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Input, int M, the number of spatial dimensions.
//
//    Input, int N, the number of points.  The program
//    will stop reading data once N values have been read.
//
//    Output, double R8MAT_DATA_READ[M*N], the data.
//
{
  bool error;
  ifstream input;
  int i;
  int j;
  string line;
  double *table;
  double *x;

  input.open ( input_filename.c_str ( ) );

  if ( !input )
  {
    cerr << "\n";
    cerr << "R8MAT_DATA_READ - Fatal error!\n";
    cerr << "  Could not open the input file: \"" << input_filename << "\"\n";
    exit ( 1 );
  }

  table = new double[m*n];

  x = new double[m];

  j = 0;

  while ( j < n )
  {
    getline ( input, line );

    if ( input.eof ( ) )
    {
      break;
    }

    if ( line[0] == '#' || s_len_trim ( line ) == 0 )
    {
      continue;
    }

    error = s_to_r8vec ( line, m, x );

    if ( error )
    {
      continue;
    }

    for ( i = 0; i < m; i++ )
    {
      table[i+j*m] = x[i];
    }
    j = j + 1;

  }

  input.close ( );

  delete [] x;

  return table;
}
//****************************************************************************80

void r8mat_header_read ( string input_filename, int &m, int &n )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_HEADER_READ reads the header from an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    23 February 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string INPUT_FILENAME, the name of the input file.
//
//    Output, int &M, the number of spatial dimensions.
//
//    Output, int &N, the number of points.
//
{
  m = file_column_count ( input_filename );

  if ( m <= 0 )
  {
    cerr << "\n";
    cerr << "R8MAT_HEADER_READ - Fatal error!\n";
    cerr << "  FILE_COLUMN_COUNT failed.\n";
    exit ( 1 );
  }

  n = file_row_count ( input_filename );

  if ( n <= 0 )
  {
    cerr << "\n";
    cerr << "R8MAT_HEADER_READ - Fatal error!\n";
    cerr << "  FILE_ROW_COUNT failed.\n";
    exit ( 1 );
  }

  return;
}
//****************************************************************************80

int s_len_trim ( string s )

//****************************************************************************80
//
//  Purpose:
//
//    S_LEN_TRIM returns the length of a string to the last nonblank.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, a string.
//
//    Output, int S_LEN_TRIM, the length of the string to the last nonblank.
//    If S_LEN_TRIM is 0, then the string is entirely blank.
//
{
  int n;

  n = s.length ( );

  while ( 0 < n )
  {
    if ( s[n-1] != ' ' )
    {
      return n;
    }
    n = n - 1;
  }

  return n;
}
//****************************************************************************80

double s_to_r8 ( string s, int *lchar, bool *error )

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_R8 reads an R8 from a string.
//
//  Discussion:
//
//    This routine will read as many characters as possible until it reaches
//    the end of the string, or encounters a character which cannot be
//    part of the real number.
//
//    Legal input is:
//
//       1 blanks,
//       2 '+' or '-' sign,
//       2.5 spaces
//       3 integer part,
//       4 decimal point,
//       5 fraction part,
//       6 'E' or 'e' or 'D' or 'd', exponent marker,
//       7 exponent sign,
//       8 exponent integer part,
//       9 exponent decimal point,
//      10 exponent fraction part,
//      11 blanks,
//      12 final comma or semicolon.
//
//    with most quantities optional.
//
//  Example:
//
//    S                 R
//
//    '1'               1.0
//    '     1   '       1.0
//    '1A'              1.0
//    '12,34,56'        12.0
//    '  34 7'          34.0
//    '-1E2ABCD'        -100.0
//    '-1X2ABCD'        -1.0
//    ' 2E-1'           0.2
//    '23.45'           23.45
//    '-4.2E+2'         -420.0
//    '17d2'            1700.0
//    '-14e-2'         -0.14
//    'e2'              100.0
//    '-12.73e-9.23'   -12.73 * 10.0^(-9.23)
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string containing the
//    data to be read.  Reading will begin at position 1 and
//    terminate at the end of the string, or when no more
//    characters can be read to form a legal real.  Blanks,
//    commas, or other nonnumeric data will, in particular,
//    cause the conversion to halt.
//
//    Output, int *LCHAR, the number of characters read from
//    the string to form the number, including any terminating
//    characters such as a trailing comma or blanks.
//
//    Output, bool *ERROR, is true if an error occurred.
//
//    Output, double S_TO_R8, the real value that was read from the string.
//
{
  char c;
  int ihave;
  int isgn;
  int iterm;
  int jbot;
  int jsgn;
  int jtop;
  int nchar;
  int ndig;
  double r;
  double rbot;
  double rexp;
  double rtop;
  char TAB = 9;

  nchar = s_len_trim ( s );
  *error = false;
  r = 0.0;
  *lchar = -1;
  isgn = 1;
  rtop = 0.0;
  rbot = 1.0;
  jsgn = 1;
  jtop = 0;
  jbot = 1;
  ihave = 1;
  iterm = 0;

  for ( ; ; )
  {
    c = s[*lchar+1];
    *lchar = *lchar + 1;
//
//  Blank or TAB character.
//
    if ( c == ' ' || c == TAB )
    {
      if ( ihave == 2 )
      {
      }
      else if ( ihave == 6 || ihave == 7 )
      {
        iterm = 1;
      }
      else if ( 1 < ihave )
      {
        ihave = 11;
      }
    }
//
//  Comma.
//
    else if ( c == ',' || c == ';' )
    {
      if ( ihave != 1 )
      {
        iterm = 1;
        ihave = 12;
        *lchar = *lchar + 1;
      }
    }
//
//  Minus sign.
//
    else if ( c == '-' )
    {
      if ( ihave == 1 )
      {
        ihave = 2;
        isgn = -1;
      }
      else if ( ihave == 6 )
      {
        ihave = 7;
        jsgn = -1;
      }
      else
      {
        iterm = 1;
      }
    }
//
//  Plus sign.
//
    else if ( c == '+' )
    {
      if ( ihave == 1 )
      {
        ihave = 2;
      }
      else if ( ihave == 6 )
      {
        ihave = 7;
      }
      else
      {
        iterm = 1;
      }
    }
//
//  Decimal point.
//
    else if ( c == '.' )
    {
      if ( ihave < 4 )
      {
        ihave = 4;
      }
      else if ( 6 <= ihave && ihave <= 8 )
      {
        ihave = 9;
      }
      else
      {
        iterm = 1;
      }
    }
//
//  Exponent marker.
//
    else if ( ch_eqi ( c, 'E' ) || ch_eqi ( c, 'D' ) )
    {
      if ( ihave < 6 )
      {
        ihave = 6;
      }
      else
      {
        iterm = 1;
      }
    }
//
//  Digit.
//
    else if ( ihave < 11 && '0' <= c && c <= '9' )
    {
      if ( ihave <= 2 )
      {
        ihave = 3;
      }
      else if ( ihave == 4 )
      {
        ihave = 5;
      }
      else if ( ihave == 6 || ihave == 7 )
      {
        ihave = 8;
      }
      else if ( ihave == 9 )
      {
        ihave = 10;
      }

      ndig = ch_to_digit ( c );

      if ( ihave == 3 )
      {
        rtop = 10.0 * rtop + ( double ) ndig;
      }
      else if ( ihave == 5 )
      {
        rtop = 10.0 * rtop + ( double ) ndig;
        rbot = 10.0 * rbot;
      }
      else if ( ihave == 8 )
      {
        jtop = 10 * jtop + ndig;
      }
      else if ( ihave == 10 )
      {
        jtop = 10 * jtop + ndig;
        jbot = 10 * jbot;
      }

    }
//
//  Anything else is regarded as a terminator.
//
    else
    {
      iterm = 1;
    }
//
//  If we haven't seen a terminator, and we haven't examined the
//  entire string, go get the next character.
//
    if ( iterm == 1 || nchar <= *lchar + 1 )
    {
      break;
    }

  }
//
//  If we haven't seen a terminator, and we have examined the
//  entire string, then we're done, and LCHAR is equal to NCHAR.
//
  if ( iterm != 1 && (*lchar) + 1 == nchar )
  {
    *lchar = nchar;
  }
//
//  Number seems to have terminated.  Have we got a legal number?
//  Not if we terminated in states 1, 2, 6 or 7!
//
  if ( ihave == 1 || ihave == 2 || ihave == 6 || ihave == 7 )
  {
    *error = true;
    return r;
  }
//
//  Number seems OK.  Form it.
//
  if ( jtop == 0 )
  {
    rexp = 1.0;
  }
  else
  {
    if ( jbot == 1 )
    {
      rexp = pow ( 10.0, jsgn * jtop );
    }
    else
    {
      rexp = jsgn * jtop;
      rexp = rexp / jbot;
      rexp = pow ( 10.0, rexp );
    }

  }

  r = isgn * rexp * rtop / rbot;

  return r;
}
//****************************************************************************80

bool s_to_r8vec ( string s, int n, double rvec[] )

//****************************************************************************80
//
//  Purpose:
//
//    S_TO_R8VEC reads an R8VEC from a string.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string to be read.
//
//    Input, int N, the number of values expected.
//
//    Output, double RVEC[N], the values read from the string.
//
//    Output, bool S_TO_R8VEC, is true if an error occurred.
//
{
  int begin;
  bool error;
  int i;
  int lchar;
  int length;

  begin = 0;
  length = s.length ( );
  error = 0;

  for ( i = 0; i < n; i++ )
  {
    rvec[i] = s_to_r8 ( s.substr(begin,length), &lchar, &error );

    if ( error )
    {
      return error;
    }
    begin = begin + lchar;
    length = length - lchar;
  }

  return error;
}
//****************************************************************************80

int s_word_count ( string s )

//****************************************************************************80
//
//  Purpose:
//
//    S_WORD_COUNT counts the number of "words" in a string.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    05 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string S, the string to be examined.
//
//    Output, int S_WORD_COUNT, the number of "words" in the string.
//    Words are presumed to be separated by one or more blanks.
//
{
  bool blank;
  int char_count;
  int i;
  int word_count;

  word_count = 0;
  blank = true;

  char_count = s.length ( );

  for ( i = 0; i < char_count; i++ )
  {
    if ( isspace ( s[i] ) )
    {
      blank = true;
    }
    else if ( blank )
    {
      word_count = word_count + 1;
      blank = false;
    }
  }

  return word_count;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}
//****************************************************************************80

double triangle_area ( double xa, double ya, double xb, double yb, double xc, 
  double yc )

//****************************************************************************80
//
//  Purpose:
//
//    TRIANGLE_AREA computes the signed area of a triangle.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    06 May 2014
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double XA, YA, XB, YB, XC, YC, the coordinates of
//    the vertices of the triangle, given in counterclockwise order.
//
//    Output, double TRIANGLE_AREA, the signed area of the triangle.
//
{
  double value;

  value = 0.5 * ( 
      ( xb - xa ) * ( yc - ya ) 
    - ( xc - xa ) * ( yb - ya ) );

  return value;
}