10 questions linked to/from How to generate random points on a sphere?
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### If $U$ is uniformly distributed on $S^{d-1} \subset \mathbb{R}^d$, what's the distribution of its orthogonal projection onto any vector?

Let $U \in S^{d-1} \subset \mathbb{R}^d$ follow a uniform distribution on a sphere. Let $v \in \mathbb{R}^d.$ Then is the orthogonal projection $U^{T}v=\langle U,v \rangle$ uniformly distributed, and ...
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### Choose Points at Random With a Uniform Distribution on a Sphere

I want to choose a point at random so that it is located on the unit sphere in $N$ dimensions. How is this done? According to this post, one can choose each component of an $N$ dimensional vector ...
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### Is there a way to generate individual uniformly distributed points on a sphere from a fixed amount of random real numbers per point? [duplicate]

The obvious solution of Lattitude & Longitude doesn't work because it generates points more densely near the poles, and the other thing I came up with (Pick a random point in the unit cube, if it'...
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### Proof of generating random rotations using quaternions

I’m trying to understand the proof of why uniformly distributed pseudo-random rotations on a sphere can be generated using quaternions and still keep the uniform distribution, but I don’t really ...
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### Generate Random Sparse Vectors at a Given Distance on the Unit Hypersphere

I'm trying to write a Python function that would take 3 parameters: n: Number of dimensions s: Sparsity factor (example 0.9) d: Distance The function would return two vectors in n dimensions that are ...
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### How can I pick a random point on the surface of a sphere with equal distribution?

I've got a random number generator that yields values between 0 and 1, and I'd like to use it to select a random point on the surface of a sphere where all points on the sphere are equally likely. ...
I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of ...