![]() Hence, a larger and larger fraction of the uniformly generated numbers in the cube will be rejected using Alternative method 2, and so this algorithm will be inefficient. On the other hand, as the number of dimensions grows, the ratio of the volume of the edges of a cube to the volume of the ball inside of it grows (see Wikipedia article). As each vector within the region has a random direction, these points will be uniformly distributed on a sphere of radius 1.Īlternative method 1 can be used to efficiently generate uniformly distributed numbers on a hypersphere– i.e. Next, normalize each random vector to have unit norm so that the vector retains its direction but is extended to the sphere of unit radius. This will ensure a uniform distribution in the region. Alternative method 2Ĭredit to FX Coudert for pointing this out in a comment, another method is to generate uniformly distributed numbers in the cube and ignore any points that are further than a unit distance from the origin. By finding where the ray determined by this vector intersects the sphere, we have a sample from a uniform distribution on the sphere. The vectors are thus indeed pointing in uniformly random directions. This shows that the probability distribution of only depends on its magnitude and not any direction and. To prove this, note that the standard normal distribution is:Īs, , and each follow the standard normal distribution and are generated independently: V = v / norm ( v ) # normalize to unit ( f_normal, "%f,%f,%f \n ", v, v, v ) V = # initialize so we go into the while loop We have to be careful in the case that the vector has a norm close to zero, in which we must worry about floating point precision by dividing by a very small number. The following Julia code implements this. If we normalize the vector, it will then lie on the sphere. Intuitively, this vector will have a uniformly random orientation in space, but will not lie on the sphere. Alternative method 1Īn alternative method to generate uniformly disributed points on a unit sphere is to generate three standard normally distributed numbers, , and to form a vector. There are more points around (the equator) than around the poles (), as we had hoped for.Īnd, finally, a uniform distribution of points on the sphere. We then get the following distribution of points in the plane. Incorrect = fopen ( "incorrect.csv", "w" ) įprintf ( incorrect, "Theta,Phi,x,y,z \n " ) įprintf ( correct, "Theta,Phi,x,y,z \n " ) ĭouble theta = 2 * M_PI * uniform01 ( generator ) ĭouble phi = acos ( 1 - 2 * uniform01 ( generator )) įprintf ( correct, "%f,%f,%f,%f,%f \n ", theta, phi, x, y, z ) Std :: uniform_real_distribution uniform01 ( 0. Unsigned seed = std :: chrono :: system_clock :: now (). In spherical coordinates, is the radius, is the azimuthal angle, and is the polar angle.Ī tempting way to generate uniformly distributed numbers in a sphere is to generate a uniform distribution of and, then apply the above transformation to yield points in Cartesian space, as with the following C code. Spherical coordinates give us a nice way to ensure that a point is on the sphere for any : This came up today in writing a code for molecular simulations. So, we want to generate uniformly distributed random numbers on a unit sphere. Please read Section Separable grid refinement in spherical coordinates. : Mars global surveyor/Mars orbiter laser altimeter (MOLA) data. In Cartesian coordinates RADMC-3D enforces perfectly cubic grid cells (i.e.: GEBCO gridded bathymetry data, one minute grid.Coxeter H., Regular Polytopes, Methuen
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