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

SPHERE_INTEGRALS is a C 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 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_EXACTNESS, a C program which tests the monomial exactness of a quadrature rule on the surface of the unit sphere in 3D.

SPHERE_MONTE_CARLO, a C library which uses the Monte Carlo method to estimate the integral of a function 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_INTEGRALS, a C library which returns the exact value of the integral of any monomial over the interior of a 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.

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:

sphere_integrals.c, the source code.
sphere_integrals.h, the include file.

Examples and Tests:

sphere_integrals_test.c a sample calling program.
sphere_integrals_test.txt, the output file.

List of Routines:

I4VEC_UNIFORM_AB_NEW returns a scaled pseudorandom I4VEC.
MONOMIAL_VALUE evaluates a monomial.
R8_GAMMA evaluates Gamma(X) for a real argument.
R8MAT_NORMAL_01_NEW returns a unit pseudonormal R8MAT.
R8VEC_NORMAL_01_NEW returns a unit pseudonormal R8VEC.
R8VEC_SUM returns the sum of an R8VEC.
R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC.
SPHERE01_AREA returns the surface area of the unit sphere.
SPHERE01_MONOMIAL_INTEGRAL: monomial integrals on the surface of the unit sphere.
SPHERE01_SAMPLE samples points uniformly from the surface of the unit sphere.
TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 06 January 2014.

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