Let's restate the originally intended problem: we have a uniform sample of D-1 sphere. How to convert this into uniform sampling of D-2 sphere? For instance, if we have a sample on a surface of a sphere, how to convert it to a uniform sample on a *circle*?

The idea is to define that given circle on a sphere with the axis that goes from the center of the sphere through a center of a circle. This axis would be z-axis of spherical coordinate system, and make the given circle one of latitudes:

[![enter image description here][1]][1].

Now, if you observe only the angle $\varphi$ (azimuth angle or longitude) of the uniformly random points on a sphere, this will give you a uniform sample of points on a circle, because a point on a circle can easily be located by that angle.

Now, for a general D-1 sphere, we have the same approach. We define a circle on the sphere by the z-axis. After that azimuth angle can be calculated easily by projecting the points on sphere to the hyperplane that is orthogonal to this z-axis. Then you calculate the cosine (azimuth angle) to any fixed vector on that hyperplane that gives through the center of the D-sphere. 

A disk can be sampled from a ball in a similar way. The difference is that for a disk we need two spherical coordinates: an azimuth angle and the  distance to the point from the center of **a disk**.  We then re-scale the distance so that it corresponds to the distance on the given disk.

# Examples for a circle


Here's an example. I sampled 100 points on a sphere, shown in blue, using [Marsaglia's algorithm][2]. Next, I converted the cartesian coordinates into [spherical coordinates][3]. Assuming that $x_1$ is my z-axis, I proceeded replacing $\varphi_1$ with a given angle $\pi/6$ that defines the circle. The resulting points are shown in black, and lie on a circle, as expected.

[![enter image description here][4]][4]

Here's an example in 4-dimensions. Since I can't plot 4 dimensional objects, only the projected points will be shown. So, I sampled from 3-sphere in 4 dimensions. Then proceeded applying the same algorithm to project the points onto a 2-sphere in 3 dimensions. The result is shown below. As you can see the sphere is, of course, smaller than the unit sphere, but the points are uniformly distributed on it.

[![enter image description here][5]][5]


  [1]: https://i.sstatic.net/jvWte.png
  [2]: https://en.wikipedia.org/wiki/N-sphere#Uniformly_at_random_on_the_(n_%E2%88%92_1)-sphere
  [3]: https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates
  [4]: https://i.sstatic.net/einSH.png
  [5]: https://i.sstatic.net/qEhyo.png