均匀的生成圆和三角形内的随机点 - tenos - 博客园



均匀的生成圆和三角形内的随机点 - tenos - 博客园

一、均匀生成圆内的随机点

我们知道生成矩形内的随机点比较容易,只要分别随机生成相应的横坐标和纵坐标,比如随机生成范围[-10,10]内横坐标x,随机生成范围[-20,20]内的纵坐标y,那么(x,y)就是生成的随机点。由此,我们很容易的想到了算法1

算法1(正确的):

每个圆对应一个外切矩形,我们随机生成矩形内的点,如果该点在圆内,就返回改点,否则重新生成直到生成的点在圆内。

该方法的缺点是有可能连续几次都生成不了符合要求的点。(可以求得:每次生成点时,该点有 image 的概率在圆内)

image

 

算法2(错误的):

可能有的人想到该方法,根据圆的解析式image (假设圆心在原点)我们可以先随机生成[-R, R]范围内横坐标x,然后生成image 范围内的随机数y,(x,y)就是需要的点。

我们写程序模拟了该过程,从下图可以看出,我们可以看到当x靠近圆的边缘使,y的范围减小,因此两边边缘的点较密集,靠近圆心的点较稀疏。

image

 

算法3(错误的):

还可以根据极坐标,圆内的点可以如下描述

x = r*sin(theta)

y = r*cos(theta)

其中0 <= r <= R, 0 <= theta < 360

先随机生成[0, 360)内的theta,然后随机生成[0, R]内的r。

theta固定后,当r越靠近R,即点越靠近圆的边缘,因此如果r是[0,R]等概率生成的,那么圆的边缘的点会比靠近圆心处要稀疏。


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