numpy - How to generate a randomly oriented, high dimension circle in python?

Question

I want to generate a randomly oriented circle in R^n. I have been able to successfully generate points on the surface of an n-sphere. I read that you can intersect it with a plane and get a circle, but I don't know how to do this in Python.

Or is there any other way to generate it in Python?

Thanks!

Answer 1

Since you already know how to generate a random point on the surface of an n-sphere, just generate two such points, call them P1 and P2. These will determine the plane in which the circle will lie.

I am assuming that both these points are a distance of 1 from the origin, (which will be true if you picked 1 as the radius of your n-sphere). If not, divide each point by its length.

Now we want to make P2 perpendicular to P1. This can be done by

P2 = P2 - dot(P1, P2) P1
P2 = P2 / || P2 ||

Then for each point, you can generate a random angle between 0 and 2pi. Convert this angle to a point on the circle by:

cos(angle) * P1 + sin(angle) * P2.