TRIANGLE_INTEGRALS
Integrals of Polynomials over Triangles

TRIANGLE_INTEGRALS is a FORTRAN90 library which returns the exact value of the integral of a polynomial over the interior of an arbitrary triangle in 2D.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

TRIANGLE_INTEGRALS is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BALL_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

CIRCLE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit circle in 2D.

CUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit cube in 3D.

DISK01_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit disk in 2D.

HYPERBALL_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit hyperball in M dimensions.

HYPERCUBE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit hypercube in M dimensions.

HYPERSPHERE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.

LINE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the length of the unit line in 1D.

POLYGON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

PYRAMID_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit pyramid in 3D.

SIMPLEX_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit simplex in M dimensions.

SPHERE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the surface of the unit sphere in 3D.

SQUARE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit square in 2D.

TETRAHEDRON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TRIANGLE_DUNAVANT_RULE, a FORTRAN90 library which sets up a Dunavant quadrature rule over the interior of a triangle in 2D.

TRIANGLE_FEKETE_RULE, a FORTRAN90 library which defines Fekete rules for quadrature or interpolation over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_LYNESS_RULE, a FORTRAN90 library which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.

TRIANGLE_MONTE_CARLO, a FORTRAN90 library which uses the Monte Carlo method to estimate the integral of a function over the interior of the unit triangle in 2D.

TRIANGLE_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules over the interior of a triangle in 2D.

TRIANGLE_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a triangle.

TRIANGLE_SYMQ_RULE, a FORTRAN90 library which returns efficient symmetric quadrature rules, with exactness up to total degree 50, over the interior of an arbitrary triangle in 2D, by Hong Xiao and Zydrunas Gimbutas.

TRIANGLE_WANDZURA_RULE, a FORTRAN90 library which sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30 over the interior of a triangle in 2D.

TRIANGLE01_INTEGRALS, a FORTRAN90 library which determines the exact integral of a monomial over the unit triangle in 2D.

WEDGE_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit wedge in 3D.

Source Code:

• triangle_integrals.f90, the source code.

Examples and Tests:

• triangle_integrals_test.f90, a sample calling program.
• triangle_integrals_test.txt, the output file.

List of Routines:

• I4_TO_PASCAL converts a linear index to Pascal triangle coordinates.
• I4_TO_PASCAL_DEGREE converts a linear index to a Pascal triangle degree.
• PASCAL_TO_I4 converts Pacal triangle coordinates to a linear index.
• POLY_POWER computes a power of a polynomial.
• POLY_POWER_LINEAR computes the polynomial ( a + b*x + c*y ) ^ n.
• POLY_PRINT prints an XY polynomial.
• POLY_PRODUCT computes P3(x,y) = P1(x,y) * P2(x,y) for polynomials.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• RS_TO_XY_MAP returns the linear map from reference to physical triangle.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE01_MONOMIAL_INTEGRAL: monomial integrals in the unit triangle.
• TRIANGLE01_POLY_INTEGRAL: polynomial integral over the unit triangle.
• TRIANGLE_AREA returns the area of a triangle.
• TRIANGLE_MONOMIAL_INTEGRAL integrates a monomial over an arbitrary triangle.
• TRIANGLE_POLY_INTEGRAL: polynomial integral over a triangle.
• TRIANGLE_XY_INTEGRAL computes the integral of XY over a triangle.
• TRINOMIAL computes a trinomial coefficient.
• XY_TO_RS_MAP returns the linear map from physical to reference triangle.

You can go up one level to the FORTRAN90 source codes.