单位球面上的均匀随机 (Monte-Carlo) 分布 [英] Uniform random (Monte-Carlo) distribution on unit sphere

问题描述

我需要澄清一下为我的宠物光线追踪器生成随机值的算法.
我从一点发出光线.我有这些光线分布的问题:我需要分布均匀，但它不是......

I need a clarification with algorithm generating random values for my pet ray-tracer.
I emit rays from one point. And I have the problem with distribution of these rays: I need the distribution to be uniform, but it isn't...

我现在面临的问题是，在我对结果空间进行扭曲之后，最初均匀的分布并不均匀.

The problem I face now is that the distribution being uniform initially is not uniform after my distortions of the space of results.

例如，如果是极坐标系，我会生成 r 和 t 角.分布是不均匀的，也不能是均匀的:靠近每个极点的空间比靠近赤道的空间具有更多的结果密度.原因很明显:我将均匀分布的点从圆柱空间转换为球面.我扭曲了结果.同样的问题是，如果我对立方体中随机生成的点进行归一化.

So for example, I generate r and t angles if the polar coordinate system. The distribution is not uniform and it cannot be uniform: space close to each pole has much more density of results than, say, close to equator. The reason is pretty clear: I convert uniformly distributed points from cylindrical space to the spherical. And I distort results. The same problem is if I normalize points generated randomly in the cube.

我现在的想法是:我想创建一个四面体，标准化它的顶点，用中间的点分割每个面(三角形)，标准化它并递归重复，直到我有足够的点.然后我稍微扭曲"这些点.然后我再次将它们标准化.就是这样.

My idea now is this: I want to create a tetrahedron, normalize its vertexes, split each face (triangle) with the point in the middle, normalize it and repeat recursively until I have enough points. Then I "distort" these points a little bit. Then I normalize them again. That's it.

我知道这个方法本身并不是纯数学的蒙特卡罗方法，因为除了最后一步之外，我没有在任何步骤中使用随机分布.我不喜欢这种复杂性的解决方案.

I understand that this method is not pure mathematical Monte-Carlo method itself, because I do not use random distribution in any step except for the last one. And I do not like this solution for this complexity.

谁能建议更简单的东西

• 随机
• 统一
• 快
• 简单

谢谢！

我需要一种快速的方法，而不仅仅是正确的方法.这就是为什么我要问蒙特卡洛.提供的答案是正确的，但速度不快.四面体的方法很快，但不是很随机"=> 不正确.
我真的需要更合适的东西.

I need a fast method, not just the correct one. That's why I'm asking about Monte-Carlo. Answers provided are correct, but not fast. The method with tetrahedron is fast, but not very "random" => incorrect.
I really need something more suitable.

推荐答案

这是一个算法允许您生成随机分布在单位球面上的点.

Here's an algorithm that allows you to generate points randomly distributed on the unit sphere.

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