SPHERE_TRIANGLE_QUAD
Estimate Integrals over Spherical Triangles

SPHERE_TRIANGLE_QUAD, a MATLAB library which estimates the integral of a scalar function F(X,Y,Z) over a spherical triangle on the unit sphere.

Three methods of estimation are very crude and cannot be improved:

• the centroid rule, based on a single function value.
• the vertex rule, which averages the vertex values.
• the midside rule, which averages the midside values.

One method of estimation uses random sampling, the Monte Carlo method, whose accuracy can be improved by increasing the number of sample points.

Another method is based on the centroid rule, but allows the user to decompose the original spherical triangle into collection of smaller triangles. The accuracy of the estimate should improve as the size of these triangles decreases.

Licensing:

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

Languages:

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

Related Data and Programs:

RANDOM_DATA, a MATLAB library which generates sample points for various probability distributions, spatial dimensions, and geometries;

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_QUAD, a MATLAB library which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

SPHERE_TRIANGLE_MONTE_CARLO, a MATLAB library which estimates the integral of a function over a spherical triangle using the Monte Carlo method.

sphere_triangle_quad_test

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:

• 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:

• polyterm_exponent.m, gets or sets the exponents for the polynomial term.
• polyterm_value_3d.m, evaluates a single polynomial term in 3D.
• r8_acos.m, computes the arc cosine function, with argument truncation.
• r8_asin.m, computes the arc sine function, with argument truncation.
• r8_atan.m, computes the arc tangent function.
• r8_uniform_01.m, returns a unit pseudorandom R8.
• r8vec_norm.m, returns the L2 norm of an R8VEC.
• r8vec_normalize.m, normalizes an R8VEC.
• r8vec_polarize.m, polarizes an R8VEC.
• r8vec_transpose_print.m, prints an R8VEC, "transposed".
• sphere01_sample.m, picks random points on the unit sphere in 3D.
• sphere01_angles_to_area.m, area of a triangle on the unit sphere.
• sphere01_distance_xyz.m, computes great-circle distance on a sphere.
• sphere01_triangle_project.m, projects from plane to spherical triangle.
• sphere01_triangle_project2.m, projects from plane to spherical triangle.
• sphere01_triangle_quad_00.m, quadrature over a triangle on the unit sphere.
• sphere01_triangle_quad_01.m, quadrature over a triangle on the unit sphere.
• sphere01_triangle_quad_02.m, quadrature over a triangle on the unit sphere.
• sphere01_triangle_quad_03.m, quadrature over a triangle on the unit sphere.
• sphere01_triangle_quad_icos1c.m, centroid rule, subdivide then project.
• sphere01_triangle_quad_icos1m.m, midpoint rule, subdivide then project.
• sphere01_triangle_quad_icos1v.m, vertex rule, subdivide then project.
• sphere01_triangle_quad_icos2v.m, vertex rule, subdivide then project.
• sphere01_triangle_sample.m, sample spherical triangle on unit sphere.
• sphere01_triangle_quad_sides_to_angles.m, angles of triangle on unit sphere.
• sphere01_triangle_vertices_to_area.m, area of triangle on unit sphere.
• sphere01_triangle_vertices_to_centroid.m, centroid of triangle on unit sphere.
• sphere01_triangle_vertices_to_midpoint.m, midsides of triangle on unit sphere.
• sphere01_triangle_vertices_to_sides.m, sides of triangle on unit sphere.
• timestamp.m, prints the current YMDHMS date as a time stamp.