TRIANGLE_INTEGRALS
Integrals of Polynomials over Triangles

TRIANGLE_INTEGRALS is a C++ 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 C++ library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

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

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

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

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

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

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

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

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

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

TRIANGLE_ANALYZE, a C++ program which reads a triangle defined in a file, and uses the triangle_properties() library to compute angles, area, centroid, circumcircle, edge lengths, incircle, orientation, orthocenter, and quality.

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

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

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

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

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

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

TRIANGLE_SYMQ_RULE, a C++ 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 C++ 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 C++ library which determines the exact integral of a monomial over the unit triangle in 2D.

WEDGE_INTEGRALS, a C++ 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.cpp, the source code.
• triangle_integrals.hpp, the include file.

Examples and Tests:

• triangle_integrals_test.cpp, 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 C++ source codes.