SPHERE_INTEGRALS
Integrals on the Surface of the Unit Sphere in 3D

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

The surface of the unit sphere in 3D is defined by

        x^2 + y^2 + z^2 = 1

The integrands are all of the form

        f(x,y,z) = x^a y^b z^c

where the exponents are nonnegative integers. If any exponent is an odd integer, the integral will be zero. Thus, the "interesting" results occur when all exponents are even.

Licensing:

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

Languages:

SPHERE_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_MONTE_CARLO, a Python library which applies a Monte Carlo method to estimate the integral of a function on 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_INTEGRALS, a Python library which returns the exact value of the integral of any monomial 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:

Gerald Folland,
How to Integrate a Polynomial Over a Sphere,
American Mathematical Monthly,
Volume 108, Number 5, May 2001, pages 446-448.

Source Code:

gamma_values.py, returns selected values of the gamma function;
i4vec_print.py, prints an I4VEC.
i4vec_transpose_print.py, prints an I4VEC "transposed", that is, 5 entries to a line.
i4vec_uniform_ab.m returns a scaled pseudorandom I4VEC.
monomial_value.m evaluates a monomial.
r8_gamma.py, returns the gamma function of an R8.
r8_normal_01.py, returns a unit pseudonormal R8;
r8_uniform_01.py, returns a unit pseudorandom R8.
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_norm.py, returns the L2 norm of an R8VEC.
r8vec_normal_01.py, returns a uniform pseudonormal R8VEC.
r8vec_print.py, prints an R8VEC.
r8vec_uniform_ab.py, returns a scaled uniform pseudorandom R8VEC.
sphere01_area.m returns the surface area of the unit sphere in 3D.
sphere01_monomial_integral.m monomial integrals on the surface of the unit sphere in 3D.
sphere01_sample.m samples points uniformly from the surface of the unit sphere in 3D.
timestamp.m prints the current YMDHMS date as a time stamp.

Examples and Tests:

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

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

Last revised on 23 June 2015.