hypersphere

hypersphere, a MATLAB code which carries out various operations for an M-dimensional hypersphere, including converting between Cartesian and spherical coordinates, stereographic projection, sampling the surface of the sphere, and computing the surface area and volume.

Licensing:

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

Languages:

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

Related Data and Programs:

hypersphere_test

geometry, a MATLAB code which performs geometric calculations in 2, 3 and M dimensional space.

hypersphere_angle, a MATLAB code which considers the problem of describing the typical value of the angle between a pair of points randomly selected on the unit hypersphere in M dimensions. Since by symmetry, this will be zero, we instead look at the average of the absolute value of the dot product, and the corresponding angle. This starts out at 1 for dimension 1, and rapidly decreases as the spatial dimension increases.

hypersphere_distance, a MATLAB code which computes the expected value of the distance between a pair of points randomly selected on the unit hypersphere in M dimensions.

hypersphere_integrals, a MATLAB code which returns the exact value of the integral of any monomial over the surface of the unit hypersphere in M dimensions.

hypersphere_monte_carlo, a MATLAB code which applies a Monte Carlo method to estimate the integral of a function on the surface of the unit sphere in M dimensions;

hypersphere_surface, a MATLAB code which illustrates a procedure for estimating the location of a hypersurface defined by a characteristic function or a signed function.

random_data, a MATLAB code which generates sample points for various probability distributions, spatial dimensions, and geometries, including the M-dimensional cube, ellipsoid, simplex and sphere.

sphere_stereograph, a MATLAB 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_stereograph_display, a MATLAB code which computes and displays the results of several stereographic projections between a sphere and a plane.

Reference:

George Marsaglia,
Choosing a point from the surface of a sphere,
Annals of Mathematical Statistics,
Volume 43, Number 2, April 1972, pages 645-646.

Source Code:

• cartesian_to_hypersphere.m converts from a cartesian to a hypersphere coordinate system.
• hypersphere_01_area.m computes the area of the unit hypersphere.
• hypersphere_01_area_values.m returns selected values of the area of the unit hypersphere.
• hypersphere_01_surface_uniform.m samples points uniformly from the surface of the unit hypersphere.
• hypersphere_01_interior_uniform.m samples points uniformly inside the unit hypersphere.
• hypersphere_01_volume.m computes the volume of the unit hypersphere.
• hypersphere_01_volume_values.m returns selected values of the volume of the unit hypersphere.
• hypersphere_area.m computes the area of a hypersphere.
• hypersphere_to_cartesian.m converts from a hypersphere to a cartesian coordinate system.
• hypersphere_surface_uniform.m samples points uniformly from the surface of a hypersphere.
• hypersphere_volume.m computes the volume of a hypersphere.
• r8vec_transpose_print.m prints an R8VEC, transposed.
• hypersphere_stereograph.m stereographically maps points on the hypersphere to the plane.
• hypersphere_stereograph_inverse.m inverse stereographically maps points in the plane to the hypersphere.