TETRAHEDRON_INTEGRALS
Integrals Inside the Unit Tetrahedron in 3D

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

The interior of the unit tetrahedron in 3D is defined by

        0 <= x
        0 <= y
        0 <= z
             x + y + z <= 1
      

The integrands are all of the form

        f(x,y,z) = x^e1 * y^e2 * z^e3
      

Licensing:

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

Languages:

TETRAHEDRON_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 Python library which returns the exact value of the integral of any monomial over the interior of the unit ball in 3D.

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

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

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

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

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

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

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

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

TETRAHEDRON_MONTE_CARLO, a Python library which uses the Monte Carlo method to estimate integrals over the interior of the unit tetrahedron in 3D.

TRIANGLE_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the interior of the unit triangle in 2D.

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

Reference:

• Jean Lasserre, Konstantin Avrachenkov,
  The multidimensional version of the integral from A to B of X to the P,
  American Mathematics Monthly,
  Volume 108, Number 2, 2001, pages 151-154.

Source Code:

• i4vec_print.py, prints an I4VEC.
• i4vec_transpose_print.py, prints an I4VEC "transposed", that is, 5 entries to a line.
• i4vec_uniform_ab.py, returns a uniform scaled pseudorandom I4VEC.
• monomial_value.py, evaluates a monomial.
• r8mat_print.py, prints an R8MAT.
• r8mat_print_some.py, prints some of an R8MAT.
• r8mat_transpose_print.py, prints an R8MAT, transposed.
• r8mat_transpose_print_some.py, prints some of an R8MAT, transposed.
• r8mat_uniform_ab.py, returns a scaled pseudorandom R8MAT.
• r8vec_print.py, prints an R8VEC.
• r8vec_uniform_01.py returns a unit pseudorandom R8VEC.
• tetrahedron01_monomial_integral.py, monomial integrals in the interior of the unit tetrahedron in 3D.
• tetrahedron01_sample.py samples points from the interior of the unit tetrahedron in 3D.
• tetrahedron01_area.py computes the volume of the unit tetrahedron in 3D.
• timestamp.py prints the current YMDHMS date as a time stamp.

Examples and Tests:

• tetrahedron_integrals_test.py, calls all the tests.
• tetrahedron_integrals_test.sh, runs all the tests.
• tetrahedron_integrals_test.txt, the output file.

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