Let's state 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:

.

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.

Examples

Here's an example. I sampled 100 points on a sphere, shown in blue, using Marsaglia's algorithm. Next, I converted the cartesian coordinates into spherical coordinates. 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.

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.

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:

.

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. Next, I converted the cartesian coordinates into spherical coordinates. 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.

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.

Let's state 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 trough a center of a circle. This axis would be z-axis of spherical coordinate system:  .

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.

Examples

Here's an example. I sampled 100 points on a sphere, shown in blue, using Marsaglia's algorithm. Next, I converted the cartesian coordinates into spherical coordinates. 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.

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 that unit sphere, but the points are uniformly distributed on it.

Let's state 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: 

.

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.

Examples

Here's an example. I sampled 100 points on a sphere, shown in blue, using Marsaglia's algorithm. Next, I converted the cartesian coordinates into spherical coordinates. 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.

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.

Let's state 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 trough a center of a circle. This axis would be z-axis of spherical coordinate system: .

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.

Here's an example. I samples 100 points on a sphere, shown in blue., using Marsaglia's algorithm. Next, I converted the cartesian coordinates into spherical coordinates. 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.

Here's an example in 4-dimensions. I can't plot 4 dimensional objects. 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 that unit sphere, but the points are uniformly distributed on it.

Let's state 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 trough a center of a circle. This axis would be z-axis of spherical coordinate system: .

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.

Examples

Here's an example. I sampled 100 points on a sphere, shown in blue, using Marsaglia's algorithm. Next, I converted the cartesian coordinates into spherical coordinates. 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.

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 that unit sphere, but the points are uniformly distributed on it.