How to generate uniformly random points on n-spheres and balls?

How to generate uniformly random points on n-spheres and balls?

For many Monte Carlo methods, such as those in graphical computing, it is critical to uniformly sample from d -dimensional spheres and balls. This post describes over twenty different methods to uniformly random sample from the (surface of) a d -dimensional sphere or the (interior of) a d -dimensional ball.

How to generate points uniformly inside a 2-D circular region?

Last week I showed how to generate random points uniformly inside a 2-d circular region . That article showed that the distance of a point to the circle’s center cannot be distributed uniformly. Instead, you should use the square root of a uniform variate to generate 2-D distances to the origin. This result generalizes to higher dimensions.

How to generate random points uniformly inside a rectangle?

It is easy to generate random points that are uniformly distributed inside a rectangle. You simply generate independent random uniform values for each coordinate. However, nonrectangular regions are more complicated.

How are spheres and balls defined in D-dimensions?

Let us recap the standard definitions of spheres and balls in d -dimensions. A unit d − dimensional sphere is defined such that: And a unit d − dimensional ball is defined such that: So the perimeter of a circle is a 1-sphere, and the interior of the circle (a disk) is a 2-ball.

What kind of vector is uniformly distributed on sphere?

Normed gaussian 3d vector is uniformly distributed on sphere, see http://mathworld.wolfram.com/SpherePointPicking.html

How to sample from a uniform spherical shell?

There seem to be plenty of examples showing how to sample from a uniform spherical shell, but that seems to be easier an easier problem. The issue has to do with the scaling – there should be fewer particles at a radius of 0.1 than at a radius of 1.0 to generate a uniform sample from the volume of the sphere.

How to evenly distribute n points on a sphere?

Now the golden-ratio spiral trick spaces the points out in a nicely even pattern for θ so let’s integrate that out; for the unit disk we are left with F ( r) = r2. So the inverse function is F-1 ( u) = u1/2, and therefore we would generate random points on the disk in polar coordinates with r = sqrt (random ()); theta = 2 * pi * random ().

Which is the best algorithm for distributing points?

The Fibonacci sphere algorithm is great for this. It’s fast and gives results that at a glance will easily fool the human eye. You can see an example done with processing which will show the result over time as points are added.

How to find the latitude of a random point on a sphere?

The area of a surface patch between z and z+dz, phi and phi+dphi then depends only on dz and dphi, not at all on z or phi. To find the latitude (theta) of this point, note that z=Rsin (theta), so theta=sin -1 (z/R); its longitude is (surprise!) phi.

How to generate a random sample of points?

But then the points get distributed more densley on the poles of the sphere. In order for points to get uniformly distributed on the sphere surface, phi needs to be chosen as phi = acos (a) where -1 < a < 1 is chosen on an uniform distribution.

How to calculate the surface of a sphere?

Points on the surface of a sphere can be expressed using two spherical coordinates, theta and phi, with 0 < theta < 2pi and 0 < phi < pi. Conversion formula into cartesian x, y, z coordinates: x = r * cos (theta) * sin (phi) y = r * sin (theta) * sin (phi) z = r * cos (phi) where r is the radius of the sphere.

How to calculate surface area of the n-sphere?

We write Vn(R) = VnRn for the volume of the n -ball and Sn(R) = SnRn for the surface area of the n -sphere, both of radius R, where Vn = Vn(1) and Sn = Sn(1) are the values for the unit-radius case. In theory, one could compare the values of Sn(R) and Sm(R) for n ≠ m. However, this is not well-defined.

How to generate random points in spherical coordinates?

You can just generate random points in spherical coordinates (assuming that you are working in 3D): S(r, θ, φ ), where r ∈ [0, R), θ ∈ [0, π ], φ ∈ [0, 2π ), where R is the radius of your sphere.