tetrahedron_integrals

tetrahedron_integrals, an Octave code 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 information on this web page is distributed under the MIT license.

Languages:

tetrahedron_integrals is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

tetrahedron_integrals_test

octave_integrals, an Octave code which returns the exact value of the integral of any monomial over the surface or interior of some geometric object, including a line, quadrilateral, box, circle, disk, sphere, ball and others.

tetrahedron_exactness, an Octave code which computes the monomial exactness of a quadrature rule over the interior of a tetrahedron 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:

• monomial_value.m, evaluates a monomial.
• tetrahedron01_monomial_integral.m, monomial integrals in the interior of the unit tetrahedron in 3D.
• tetrahedron01_sample.m samples points from the interior of the unit tetrahedron in 3D.
• tetrahedron01_volume.m computes the volume of the unit tetrahedron in 3D.