SPHERE_TRIANGLE_QUAD
Estimate Integrals over Spherical Triangles

SPHERE_TRIANGLE_QUAD is a FORTRAN90 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 FORTRAN90 version.

Related Data and Programs:

RANDOM_DATA, a FORTRAN90 library which generates sample points for various probability distributions, spatial dimensions, and geometries;

SPHERE_CVT, a FORTRAN90 library which creates a mesh of well-separated points on a unit sphere using Centroidal Voronoi Tessellations.

SPHERE_DELAUNAY, a FORTRAN90 program which computes and plots the Delaunay triangulation of points on the unit sphere.

SPHERE_DESIGN_RULE, a FORTRAN90 library which returns point sets on the surface of the unit sphere, known as "designs", which can be useful for estimating integrals on the surface, among other uses.

SPHERE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule for the unit sphere;

SPHERE_GRID, a FORTRAN90 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 FORTRAN90 library which computes Lebedev quadrature rules for the unit sphere;

SPHERE_MONTE_CARLO, a FORTRAN90 library which applies a Monte Carlo method to estimate the integral of a function over the surface of the sphere in 3D;

SPHERE_QUAD, a FORTRAN90 library which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

SPHERE_TRIANGLE_MONTE_CARLO, a FORTRAN90 library which estimates the integral of a function over a spherical triangle using the Monte Carlo method.

SPHERE_VORONOI, a FORTRAN90 program which computes the Voronoi diagram of points on a sphere.

STRIPACK, a FORTRAN90 library which computes the Voronoi diagram or Delaunay triangulation of pointsets on a sphere.

STROUD, a FORTRAN90 library which approximates the integral of a function on the surface or in the interior of a variety of geometric shapes.

XYZ_DISPLAY, a MATLAB program which reads XYZ information defining points in 3D, and displays an image in the MATLAB graphics window.

XYZ_DISPLAY_OPENGL, a C++ program which reads XYZ information defining points in 3D, and displays an image using OpenGL.

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:

• sphere_triangle_quad.f90, the source code.
• sphere_triangle_quad.sh, commands to compile the source code.

Examples and Tests:

• sphere_triangle_quad_prb.f90, a sample problem.
• sphere_triangle_quad_prb.sh, commands to compile and run the problem.
• sphere_triangle_quad_prb_output.txt, the output file.

List of Routines:

• ARC_COSINE computes the arc cosine function, with argument truncation.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8VEC_NORM returns the L2 norm of an R8VEC.
• S_CAT concatenates two strings to make a third string.
• SPHERE01_SAMPLE picks random points on the unit sphere in 3D.
• SPHERE01_TRIANGLE_ANGLES_TO_AREA: area of a triangle on the unit sphere.
• SPHERE01_TRIANGLE_PROJECT projects from plane to spherical triangle.
• SPHERE01_TRIANGLE_QUAD_00: quadrature over a triangle on the unit sphere.
• SPHERE01_TRIANGLE_QUAD_01: quadrature over a triangle on the unit sphere.
• SPHERE01_TRIANGLE_QUAD_02: quadrature over a triangle on the unit sphere.
• SPHERE01_TRIANGLE_QUAD_03: quadrature over a triangle on the unit sphere.
• SPHERE01_TRIANGLE_QUAD_ICOS1C: centroid rule, subdivide then project.
• SPHERE01_TRIANGLE_SAMPLE: sample spherical triangle on unit sphere.
• SPHERE01_TRIANGLE_SIDES_TO_ANGLES: angles of triangle on unit sphere.
• SPHERE01_TRIANGLE_VERTICES_TO_AREA: area of triangle on unit sphere.
• SPHERE01_TRIANGLE_VERTICES_TO_CENTROID: centroid of triangle on unit sphere.
• SPHERE01_TRIANGLE_VERTICES_TO_MIDPOINTS: midsides of triangle on unit sphere.
• SPHERE01_TRIANGLE_VERTICES_TO_SIDES_3D: sides of triangle on unit sphere.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.