# sphere_lebedev_rule

sphere_lebedev_rule, a FORTRAN90 code which computes a Lebedev quadrature rule for approximating integrals over the surface of the unit sphere in 3D.

Vyacheslav Lebedev determined a family of 65 quadrature rules for the unit sphere, increasing in precision from 3 to 131, by 2 each time. This software library computes any one of a subset of 32 of these rules.

Each rule is defined as a list of N values of theta, phi, and w. Here:

• theta is a longitudinal angle, measured in degrees, and ranging from -180 to +180.
• phi is a latitudinal angle, measured in degrees, and ranging from 0 to 180.
• w is a weight.

Of course, each pair of values (thetai, phii) has a corresponding Cartesian representation:

xi = cos ( thetai ) * sin ( phii )
yi = sin ( thetai ) * sin ( phii )
zi = cos ( phii )
which may be more useful when evaluating integrands.

The integral of a function f(x,y,z) over the surface of the unit sphere can be approximated by

integral f(x,y,z) = 4 * pi * sum ( 1 <= i <= N ) f(xi,yi,zi)

### Languages:

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

### Related Programs:

CUBE_FELIPPA_RULE, a FORTRAN90 code which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

PYRAMID_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

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

SPHERE_DESIGN_RULE, a FORTRAN90 code 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_GRID, a FORTRAN90 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 surface of the unit sphere in 3D.

SPHERE_INTEGRALS, a FORTRAN90 code which defines test functions for integration over the surface of the unit sphere in 3D.

SPHERE_LEBEDEV_RULE, a dataset directory which contains sets of points on a sphere which can be used for quadrature rules of a known precision;

SPHERE_LEBEDEV_RULE_DISPLAY, a MATLAB program which reads a file defining a Lebedev quadrature rule for the sphere and displays the point locations.

SPHERE_QUAD, a FORTRAN90 code which estimates the integral of a function defined on the sphere.

SPHERE_STEREOGRAPH, a FORTRAN90 code which computes the stereographic mapping between points on the unit sphere and points on the plane Z = 1; a generalized mapping is also available.

SPHERE_TRIANGLE_QUAD, a FORTRAN90 code which estimates the integral of a function over a spherical triangle.

SPHERE_VORONOI, a FORTRAN90 code which computes and plots the Voronoi diagram of points over the surface of the unit sphere in 3D.

SPHERE_XYZ_DISPLAY_OPENGL, a C++ program which reads XYZ information defining points in 3D, and displays a unit sphere and the points, using OpenGL.

SPHERE_XYZF_DISPLAY, a MATLAB program which reads XYZF information defining points and faces, and displays a unit sphere, the points, and the faces, in the MATLAB 3D graphics window. This can be used, for instance, to display Voronoi diagrams or Delaunay triangulations on the unit sphere.

SQUARE_FELIPPA_RULE, a FORTRAN90 code which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

WEDGE_FELIPPA_RULE, a FORTRAN90 code which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

### Reference:

1. Axel Becke,
A multicenter numerical integration scheme for polyatomic molecules,
Journal of Chemical Physics,
Volume 88, Number 4, 15 February 1988, pages 2547-2553.
2. Vyacheslav Lebedev, Dmitri Laikov,
A quadrature formula for the sphere of the 131st algebraic order of accuracy,
Volume 59, Number 3, 1999, pages 477-481.
3. Vyacheslav Lebedev,
A quadrature formula for the sphere of 59th algebraic order of accuracy,
Volume 50, 1995, pages 283-286.
4. Vyacheslav Lebedev, A.L. Skorokhodov,
Quadrature formulas of orders 41, 47, and 53 for the sphere,
Volume 45, 1992, pages 587-592.
5. Vyacheslav Lebedev,
Spherical quadrature formulas exact to orders 25-29,
Siberian Mathematical Journal,
Volume 18, 1977, pages 99-107.
6. Vyacheslav Lebedev,