def triangle01_poly_integral(d, p):
#*****************************************************************************80
#
## TRIANGLE01_POLY_INTEGRAL: polynomial integral over the unit triangle.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/triangle_integrals/triangle01_poly_integral.py
#
# Discussion:
#
# The unit triangle is T = ( (0,0), (1,0), (0,1) ).
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer D, the degree of the polynomial.
#
# Input, real P(M), the polynomial coefficients.
# M = ((D+1)*(D+2))/2.
#
# Output, real Q, the integral.
#
from i4_to_pascal import i4_to_pascal
from triangle01_monomial_integral import triangle01_monomial_integral
m = ((d + 1) * (d + 2)) / 2
q = 0.0
for k in range(1, m + 1):
km1 = k - 1
i, j = i4_to_pascal(k)
qk = triangle01_monomial_integral(i, j)
q = q + p[km1] * qk
return q
def triangle01_poly_integral ( d, p ):
#*****************************************************************************80
#
## TRIANGLE01_POLY_INTEGRAL: polynomial integral over the unit triangle.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/triangle_integrals/triangle01_poly_integral.py
#
# Discussion:
#
# The unit triangle is T = ( (0,0), (1,0), (0,1) ).
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer D, the degree of the polynomial.
#
# Input, real P(M), the polynomial coefficients.
# M = ((D+1)*(D+2))/2.
#
# Output, real Q, the integral.
#
from i4_to_pascal import i4_to_pascal
from triangle01_monomial_integral import triangle01_monomial_integral
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
q = 0.0
for k in range ( 1, m + 1 ):
km1 = k - 1
i, j = i4_to_pascal ( k )
qk = triangle01_monomial_integral ( i, j )
q = q + p[km1] * qk
return q
def triangle_xy_integral(x1, y1, x2, y2, x3, y3):
#*****************************************************************************80
#
## TRIANGLE_XY_INTEGRAL computes the integral of XY over a triangle.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/triangle_integrals/triangle_xy_integral.py
#
# Discussion:
#
# This function was written as a special test case for the general
# problem of integrating a monomial x^alpha * y^beta over a general
# triangle.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real X1, Y1, X2, Y2, X3, Y3, the coordinates of the
# triangle vertices.
#
# Output, real Q, the integral of X*Y over the triangle.
#
#
# x = x1 * ( 1 - xi - eta )
# + x2 * xi
# + x3 * eta
#
# y = y1 * ( 1 - xi - eta )
# + y2 * xi
# + y3 * eta
#
# Rewrite as linear polynomials in (xi,eta):
#
# x = x1 + ( x2 - x1 ) * xi + ( x3 - x1 ) * eta
# y = y1 + ( y2 - y1 ) * xi + ( y3 - y1 ) * eta
#
# Jacobian:
#
# J = [ ( x2 - x1 ) ( x3 - x1 ) ]
# [ ( y2 - y1 ) ( y3 - y1 ) ]
#
# det J = ( x2 - x1 ) * ( y3 - y1 ) - ( y2 - y1 ) * ( x3 - x1 )
#
# Integrand
#
# x * y = ( x1 + ( x2 - x1 ) * xi + ( x3 - x1 ) * eta )
# * ( y1 + ( y2 - y1 ) * xi + ( y3 - y1 ) * eta )
#
# Rewrite as linear combination of monomials:
#
# x * y = 1 * x1 * y1
# + eta * ( x1 * ( y3 - y1 ) + ( x3 - x1 ) * y1 )
# + xi * ( x1 * ( y2 - y1 ) + ( x2 - x1 ) * y1 )
# + eta^2 * ( x3 - x1 ) * ( y3 - y1 )
# + xi*eta * ( ( x2 - x1 ) * ( y3 - y1 ) + ( x3 - x1 ) * ( y2 - y1 ) )
# + xi^2 * ( x2 - x1 ) * ( y2 - y1 )
#
from triangle01_monomial_integral import triangle01_monomial_integral
det = (x2 - x1) * (y3 - y1) - (y2 - y1) * (x3 - x1)
p00 = x1 * y1
p01 = x1 * (y3 - y1) + (x3 - x1) * y1
p10 = x1 * (y2 - y1) + (x2 - x1) * y1
p02 = (x3 - x1) * (y3 - y1)
p11 = (x2 - x1) * (y3 - y1) + (x3 - x1) * (y2 - y1)
p20 = (x2 - x1) * (y2 - y1)
q = 0.0
q = q + p00 * triangle01_monomial_integral(0, 0)
q = q + p10 * triangle01_monomial_integral(1, 0)
q = q + p01 * triangle01_monomial_integral(0, 1)
q = q + p20 * triangle01_monomial_integral(2, 0)
q = q + p11 * triangle01_monomial_integral(1, 1)
q = q + p02 * triangle01_monomial_integral(0, 2)
q = q * det
return q
def triangle01_poly_integral_test():
#*****************************************************************************80
#
## TRIANGLE01_POLY_INTEGRAL_TEST: polynomial integrals over the unit triangle.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
from i4_to_pascal import i4_to_pascal
from poly_print import poly_print
from triangle01_monomial_integral import triangle01_monomial_integral
d_max = 6
k_max = ((d_max + 1) * (d_max + 2)) / 2
qm = np.zeros(k_max)
for k in range(1, k_max + 1):
km1 = k - 1
i, j = i4_to_pascal(k)
qm[km1] = triangle01_monomial_integral(i, j)
print ''
print 'TRIANGLE01_POLY_INTEGRAL_TEST'
print ' TRIANGLE01_POLY_INTEGRAL returns the integral Q of'
print ' a polynomial P(X,Y) over the interior of the unit triangle.'
d = 1
m = ((d + 1) * (d + 2)) / 2
p = np.array([1.0, 2.0, 3.0])
print ''
poly_print(d, p, ' p(x,y)')
q = triangle01_poly_integral(d, p)
print ''
print ' Q = %g' % (q)
q2 = np.dot(p, qm[0:m])
print ' Q (exact) = %g' % (q2)
d = 2
m = ((d + 1) * (d + 2)) / 2
p = np.array([0.0, 0.0, 0.0, 0.0, 1.0, 0.0])
print ''
poly_print(d, p, ' p(x,y)')
q = triangle01_poly_integral(d, p)
print ''
print ' Q = %g' % (q)
q2 = np.dot(p, qm[0:m])
print ' Q (exact) = %g' % (q2)
d = 2
m = ((d + 1) * (d + 2)) / 2
p = np.array([1.0, -2.0, 3.0, -4.0, 5.0, -6.0])
print ''
poly_print(d, p, ' p(x,y)')
q = triangle01_poly_integral(d, p)
print ''
print ' Q = %g' % (q)
q2 = np.dot(p, qm[0:m])
print ' Q (exact) = %g' % (q2)
#
# Terminate.
#
print ''
print 'TRIANGLE01_POLY_INTEGRAL_TEST'
print ' Normal end of execution.'
return
def triangle_xy_integral(x1, y1, x2, y2, x3, y3):
# *****************************************************************************80
#
## TRIANGLE_XY_INTEGRAL computes the integral of XY over a triangle.
#
# Location:
#
# http://people.sc.fsu.edu/~jburkardt/py_src/triangle_integrals/triangle_xy_integral.py
#
# Discussion:
#
# This function was written as a special test case for the general
# problem of integrating a monomial x^alpha * y^beta over a general
# triangle.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, real X1, Y1, X2, Y2, X3, Y3, the coordinates of the
# triangle vertices.
#
# Output, real Q, the integral of X*Y over the triangle.
#
#
# x = x1 * ( 1 - xi - eta )
# + x2 * xi
# + x3 * eta
#
# y = y1 * ( 1 - xi - eta )
# + y2 * xi
# + y3 * eta
#
# Rewrite as linear polynomials in (xi,eta):
#
# x = x1 + ( x2 - x1 ) * xi + ( x3 - x1 ) * eta
# y = y1 + ( y2 - y1 ) * xi + ( y3 - y1 ) * eta
#
# Jacobian:
#
# J = [ ( x2 - x1 ) ( x3 - x1 ) ]
# [ ( y2 - y1 ) ( y3 - y1 ) ]
#
# det J = ( x2 - x1 ) * ( y3 - y1 ) - ( y2 - y1 ) * ( x3 - x1 )
#
# Integrand
#
# x * y = ( x1 + ( x2 - x1 ) * xi + ( x3 - x1 ) * eta )
# * ( y1 + ( y2 - y1 ) * xi + ( y3 - y1 ) * eta )
#
# Rewrite as linear combination of monomials:
#
# x * y = 1 * x1 * y1
# + eta * ( x1 * ( y3 - y1 ) + ( x3 - x1 ) * y1 )
# + xi * ( x1 * ( y2 - y1 ) + ( x2 - x1 ) * y1 )
# + eta^2 * ( x3 - x1 ) * ( y3 - y1 )
# + xi*eta * ( ( x2 - x1 ) * ( y3 - y1 ) + ( x3 - x1 ) * ( y2 - y1 ) )
# + xi^2 * ( x2 - x1 ) * ( y2 - y1 )
#
from triangle01_monomial_integral import triangle01_monomial_integral
det = (x2 - x1) * (y3 - y1) - (y2 - y1) * (x3 - x1)
p00 = x1 * y1
p01 = x1 * (y3 - y1) + (x3 - x1) * y1
p10 = x1 * (y2 - y1) + (x2 - x1) * y1
p02 = (x3 - x1) * (y3 - y1)
p11 = (x2 - x1) * (y3 - y1) + (x3 - x1) * (y2 - y1)
p20 = (x2 - x1) * (y2 - y1)
q = 0.0
q = q + p00 * triangle01_monomial_integral(0, 0)
q = q + p10 * triangle01_monomial_integral(1, 0)
q = q + p01 * triangle01_monomial_integral(0, 1)
q = q + p20 * triangle01_monomial_integral(2, 0)
q = q + p11 * triangle01_monomial_integral(1, 1)
q = q + p02 * triangle01_monomial_integral(0, 2)
q = q * det
return q
def triangle01_poly_integral_test ( ):
#*****************************************************************************80
#
## TRIANGLE01_POLY_INTEGRAL_TEST: polynomial integrals over the unit triangle.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 23 April 2015
#
# Author:
#
# John Burkardt
#
import numpy as np
from i4_to_pascal import i4_to_pascal
from poly_print import poly_print
from triangle01_monomial_integral import triangle01_monomial_integral
d_max = 6
k_max = ( ( d_max + 1 ) * ( d_max + 2 ) ) / 2
qm = np.zeros ( k_max )
for k in range ( 1, k_max + 1 ):
km1 = k - 1
i, j = i4_to_pascal ( k )
qm[km1] = triangle01_monomial_integral ( i, j )
print ''
print 'TRIANGLE01_POLY_INTEGRAL_TEST'
print ' TRIANGLE01_POLY_INTEGRAL returns the integral Q of'
print ' a polynomial P(X,Y) over the interior of the unit triangle.'
d = 1
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
p = np.array ( [ 1.0, 2.0, 3.0 ] )
print ''
poly_print ( d, p, ' p(x,y)' )
q = triangle01_poly_integral ( d, p )
print ''
print ' Q = %g' % ( q )
q2 = np.dot ( p, qm[0:m] )
print ' Q (exact) = %g' % ( q2 )
d = 2
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
p = np.array ( [ 0.0, 0.0, 0.0, 0.0, 1.0, 0.0 ] )
print ''
poly_print ( d, p, ' p(x,y)' )
q = triangle01_poly_integral ( d, p )
print ''
print ' Q = %g' % ( q )
q2 = np.dot ( p, qm[0:m] )
print ' Q (exact) = %g' % ( q2 )
d = 2
m = ( ( d + 1 ) * ( d + 2 ) ) / 2
p = np.array ( [ 1.0, -2.0, 3.0, -4.0, 5.0, -6.0 ] )
print ''
poly_print ( d, p, ' p(x,y)' )
q = triangle01_poly_integral ( d, p )
print ''
print ' Q = %g' % ( q )
q2 = np.dot ( p, qm[0:m] )
print ' Q (exact) = %g' % ( q2 )
#
# Terminate.
#
print ''
print 'TRIANGLE01_POLY_INTEGRAL_TEST'
print ' Normal end of execution.'
return