SPHERE_QUAD
Quadrature on the Surface of the Unit Sphere

SPHERE_QUAD is a MATLAB library which estimates the integral of a scalar function F(X,Y,Z) over the surface of the unit sphere centered at the origin.

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.

Licensing:

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

Languages:

SPHERE_QUAD is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

SPHERE_EXACTNESS, a MATLAB program which tests the polynomial exactness of a quadrature rule for the unit sphere;

SPHERE_GRID, a MATLAB library 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 MATLAB library which computes Lebedev quadrature rules for the unit sphere;

SPHERE_MONTE_CARLO, a MATLAB library which applies a Monte Carlo method to estimate the integral of a function over the surface of the sphere in 3D;

STROUD, a MATLAB library 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:

• arc_cosine.m computes the arc cosine function, with argument truncation.
• arc_sine.m computes the arc sine function, with argument truncation.
• atan4.m computes the inverse tangent of the ratio Y / X.
• ch_cap.m capitalizes a character.
• icos_shape.m describes an icosahedron.
• icos_size.m gives "sizes" for an icosahedron in 3D.
• r8vec_polarize.m decomposes an R8VEC into normal and parallel components.
• s_eqi.m compares two strings for equality, ignoring case.
• sphere01_distance_xyz.m computes great circle distances on a unit sphere.
• sphere01_monomial_integral.m returns monomial integrals on the unit sphere.
• sphere01_quad_icos1c.m centroid rule, subdivide then project.
• sphere01_quad_icos1m.m midside rule, subdivide then project.
• sphere01_quad_icos1v.m vertex rule, subdivide then project.
• sphere01_quad_icos2v.m vertex rule, project, then subdivide.
• sphere01_quad_llc.m Longitude/Latitude grid with centroid rule.
• sphere01_quad_llm.m longitude/latitude grid plus midside rule.
• sphere01_quad_llv.m longitude/latitude grid with vertex rule.
• sphere01_quad_mc.m uses the Monte Carlo rule for sphere quadrature.
• sphere01_quad_mc_size.m sizes a Monte Carlo rule for sphere quadrature.
• sphere01_sample_3d.m picks random points on a sphere in 3D.
• sphere01_triangle_angles_to_area.m computes the area of a spherical triangle.
• sphere01_triangle_project.m projects from a plane triangle to a spherical triangle.
• sphere01_triangle_project2.m projects from a plane triangle to a spherical triangle using a more sophisticated algorithm.
• sphere01_triangle_sample.m samples a spherical triangle.
• sphere01_triangle_sides_to_angles.m computes spherical triangle angles.
• sphere01_triangle_vertices_to_areas.m computes the area of a spherical triangle.
• sphere01_triangle_vertices_to_centroid.m gets a spherical triangle "centroid".
• sphere01_triangle_vertices_to_midpoints.m gets the midsides of a spherical triangle.
• sphere01_triangle_vertices_to_sides.m computes spherical triangle sides.
• timestamp.m prints the current YMDHMS date as a time stamp.
• tp_to_xyz.m converts spherical TP coordinates to XYZ coordinates.

Examples and Tests:

• sphere_quad_test.m, runs all the tests.
• polyterm_exponent.m, allows the user to set or get the exponents defining a monomial.
• polyterm_value_3d.m, evaluates a monomial.
• sphere_quad_test01.m, tests the Monte Carlo rule and the latitude/longitude grids.
• sphere_quad_test02.m, tests the icosahedral centroid subdivision rule.
• sphere_quad_test03.m, tests the icosahedral midpoint subdivision rule.
• sphere_quad_test04.m, tests the icosahedral vertex subdivision rule.
• sphere_quad_test05.m, tests the icosahedral vertex subdivision rule, using a more sophisticated projection.
• sphere_quad_test_output.txt, the output file.

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