如何在Python中生成随机定向的高维圆？ [英] How to generate a randomly oriented, high dimension circle in python?

问题描述

我想在R ^ n中生成一个随机定向的圆。我已经能够成功地在n球表面上生成点。
我读到，你可以与飞机相交并得到一个圆，但我不知道如何在Python中做到这一点。

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.

或者是否有其他方法可以在Python中生成它？

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

谢谢！

推荐答案

由于您已经知道如何在n球面上生成随机点，只需生成两个这样的点，称他们为 P1 和 P2 。这些将决定圆圈所在的平面。

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.

我假设这两个点距离原点的距离为1（如果您选择1作为您的n球体的半径，则这是正确的）。如果没有，请将每个点除以其长度。

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.

现在我们要使 P2 垂直于 P1 。这可以通过

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

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

然后对于每个点，您可以生成0到2pi之间的随机角度。通过以下方式将此角度转换为圆上的点：$ b​​
$ b

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.

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