sphere_quad, a C code which estimates the integral of a scalar function F(X,Y,Z) over the surface of the unit sphere in 3D.

The library includes one function, SPHERE01_QUAD_MC(), which estimates the integral using a Monte Carlo approach. It randomly samples points on the surface, and estimates the integral as the average of these values times the area of the surface.

The library includes three functions based on the idea of a latitudinal/longitudinal grid: SPHERE01_QUAD_LLC(), SPHERE01_QUAD_LLV() and SPHERE01_QUAD_LLM(). The surface of the sphere is divided into rectangles whose sides are always lines of latitude or longitude. Each rectangle is then split diagonally into a pair of triangles (except for the degenerate rectangles that include the north or south pole as a vertex.)

The user controls the accuracy of the integral estimate by specifying a maximum side length H. The functions determine angular increments that guarantee the size restriction. Of course, this means that the restriction on latitude, enforced at the equator, will result in excessively small triangles away from the equator. That is a penalty of using this simple subdivision scheme.

The library includes three functions based on the idea of first subdividing the surface into 20 congruent spherical triangles, based on the projection of a regular icosahedron. The functions SPHERE01_QUAD_ICOS1C, SPHERE01_QUAD_ICOS1V and SPHERE01_QUAD_ICOS1M use this idea, along with subdivision.

The function SPHERE01_QUAD_ICOS2V is similar to SPHERE01_QUAD_ICOS1V() but uses a more sophisticated algorithm to project points from the planar triangle to the unit sphere. However, the modifications seem to make little difference to the resulting integral estimate.

### Languages:

sphere_quad is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

SPHERE_EXACTNESS, a C code which tests the monomial exactness of a quadrature rule on the surface of the unit sphere in 3D.

SPHERE_GRID, a C code which provides a number of ways of generating grids of points, or of points and lines, or of points and lines and faces, over the unit sphere.

SPHERE_LEBEDEV_RULE, a C code which computes Lebedev quadrature rules for the unit sphere;

STROUD, a C code which approximates the integral of a function on the surface or in the interior of a variety of geometric shapes.

### Reference:

• James Arvo,
Stratified sampling of spherical triangles,
Computer Graphics Proceedings, Annual Conference Series,
ACM SIGGRAPH '95, pages 437-438, 1995.
• Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
• Jacob Goodman, Joseph ORourke, editors,
Handbook of Discrete and Computational Geometry,
Second Edition,
CRC/Chapman and Hall, 2004,
ISBN: 1-58488-301-4,
LC: QA167.H36.

### Source Code:

Last revised on 08 August 2019.