#! /usr/bin/env python
#
def simplex_coordinates1 ( m ):

#*****************************************************************************80
#
## SIMPLEX_COORDINATES1 computes the Cartesian coordinates of simplex vertices.
#
#  Discussion:
#
#    The simplex will have its centroid at 0
#
#    The sum of the vertices will be zero.
#
#    The distance of each vertex from the origin will be 1.
#
#    The length of each edge will be constant.
#
#    The dot product of the vectors defining any two vertices will be - 1 / M.
#    This also means the angle subtended by the vectors from the origin
#    to any two distinct vertices will be arccos ( - 1 / M ).
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 June 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer M, the spatial dimension.
#
#    Output, real X(M,M+1), the coordinates of the vertices
#    of a simplex in M dimensions.
#
  import numpy as np

  x = np.zeros ( [ m, m + 1 ] )

  for k in range ( 0, m ):
#
#  Set X(K,K) so that sum ( X(1:K,K)^2 ) = 1.
#
    s = 0.0
    for i in range ( 0, k ):
      s = s + x[i,k] ** 2

    x[k,k] = np.sqrt ( 1.0 - s )
#
#  Set X(K,J) for J = K+1 to M+1 by using the fact that XK dot XJ = - 1 / M.
#
    for j in range ( k + 1, m + 1 ):
      s = 0.0
      for i in range ( 0, k ):
        s = s + x[i,k] * x[i,j]

      x[k,j] = ( - 1.0 / float ( m ) - s ) / x[k,k]

  return x

def simplex_coordinates1_test ( m ):

#*****************************************************************************80
#
## SIMPLEX_COORDINATES1_TEST tests SIMPLEX_COORDINATES1.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    28 June 2015
#
#  Author:
#
#    John Burkardt
#
#  Parameters:
#
#    Input, integer M, the spatial dimension.
#
  import numpy as np
  import platform
  from r8_factorial import r8_factorial
  from r8mat_transpose_print import r8mat_transpose_print
  from simplex_volume import simplex_volume

  print ( '' )
  print ( 'SIMPLEX_COORDINATES1_TEST' )
  print ( '  Python version: %s' % ( platform.python_version ( ) ) )
  print ( '  Test SIMPLEX_COORDINATES1' )

  x = simplex_coordinates1 ( m )

  r8mat_transpose_print ( m, m + 1, x, '  Simplex vertex coordinates:' )

  s = 0.0
  for i in range ( 0, m ):
    s = s + ( x[i,0] - x[i,1] ) ** 2

  side = np.sqrt ( s )

  volume = simplex_volume ( m, x )

  volume2 = np.sqrt ( m + 1 ) / r8_factorial ( m ) \
    / np.sqrt ( 2.0 ** m ) * side ** m

  print ( '' )
  print ( '  Side length =     %g' % ( side ) )
  print ( '  Volume =          %g' % ( volume ) )
  print ( '  Expected volume = %g' % ( volume2 ) )

  xtx = np.dot ( np.transpose ( x ), x )

  r8mat_transpose_print ( m + 1, m + 1, xtx, '  Dot product matrix:' )
#
#  Terminate.
#
  print ( '' )
  print ( 'SIMPLEX_COORDINATES1_TEST' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( )
  simplex_coordinates1_test ( 3 )
  timestamp ( )