Linked Questions

17
votes
5answers
2k views

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'...
52
votes
5answers
38k views

Picking random points in the volume of sphere with uniform probability

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 ...
45
votes
5answers
15k views

How to find a random axis or unit vector in 3D?

I would like to generate a random axis or unit vector in 3D. In 2D it would be easy, I could just pick an angle between 0 and 2*Pi and use the unit vector pointing in that direction. But in 3D I don'...
16
votes
2answers
7k views

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. ...
2
votes
1answer
445 views

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 ...
0
votes
1answer
366 views

Standard Deviation of random unit vector [closed]

I have a question i ve been thinking about for a few days: I want to better understand the statistics of random 3 dimensional unit vectors. For example, i pick two angles $\theta$ and $\phi$ from $U(...
3
votes
1answer
107 views

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 ...
2
votes
1answer
81 views

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 ...
0
votes
1answer
77 views

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 ...
1
vote
2answers
87 views

Given a vector $v$ and an angle $\theta$, find a vector $w$ such that the angle between $v$ and $w$ is exactly $\theta$

Given: an $n$-dimensional vector $v = (v_1,\ldots,v_n) \in \mathbb{R}^n$ an angle $\theta \in (0, \frac{\pi}{2})$ I am looking for a technique to construct a vector $w = (w_1,\ldots, w_n) \in \...