在圆柱体中生成随机点 [英] Generating random point in a cylinder

问题描述

如果圆柱体的半径r和高度h给定，那么在圆柱体的体积内产生一个随机3d点[x，y，z]的最佳方法或算法是什么？

p $ p $ s = random.uniform（0，1）
theta = random.uniform（0，2 * pi）
z = random.uniform（0，H）
r = sqrt（s）* R
x = r * cos（theta）
y = r * sin（theta）
z = z＃.. for symmetry :-)

简单地取 x = r * cos（角度）和 y = r * sin（angle）是那么当r很小时，即在圆的中心，r的微小变化不会改变x和y的位置非常。 IOW，它导致笛卡尔坐标中的非均匀分布，并且这些点集中在圆的中心。以平方根来纠正这一点，至少如果我已经正确地做了我的算术。

[啊，它看起来像sqrt 是正确的。]

（请注意，我假设没有考虑到圆柱体与z轴对齐，圆柱体中心位于（0,0，H / 2）。在圆柱体中心设置（0,0,0）不太随意，在这种情况下z应该选择在-H / 2和H / 2之间，而不是0，H。）

What is best way or an algorithm for generating a random 3d point [x,y,z] inside the volume of the circular cylinder if radius r and height h of the cylinder are given?

解决方案

How about -- in Python pseudocode, letting R be the radius and H be the height:

s = random.uniform(0, 1)
theta = random.uniform(0, 2*pi)
z = random.uniform(0, H)
r = sqrt(s)*R
x = r * cos(theta)
y = r * sin(theta)
z = z # .. for symmetry :-)

The problem with simply taking x = r * cos(angle) and y = r * sin(angle) is that then when r is small, i.e. at the centre of the circle, a tiny change in r doesn't change the x and y positions very much. IOW, it leads to a nonuniform distribution in Cartesian coordinates, and the points get concentrated toward the centre of the circle. Taking the square root corrects this, at least if I've done my arithmetic correctly.

[Ah, it looks like the sqrt was right.]

(Note that I assumed without thinking about it that the cylinder is aligned with the z-axis and the cylinder centre is located at (0,0,H/2). It'd be less arbitrary to set (0,0,0) at the cylinder centre, in which case z should be chosen to be between -H/2 and H/2, not 0,H.)

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