重叠的圆圈联合区 [英] Combined area of overlapping circles

问题描述

我最近碰到一个问题，我有四个圈（中点和半径），不得不计算这些圈的结合区域。

I recently came across a problem where I had four circles (midpoints and radius) and had to calculate the area of the union of these circles.

示例图片：

有关两界是很容易的，

我可以只计算每个界区不是三角形内的分数，然后计算此三角形的面积。

I can just calculate the fraction of the each circles area that is not within the triangles and then calculate the area of the triangles.

但有一个聪明的算法，我可以用的时候有两个以上的圈子？

But is there a clever algorithm I can use when there is more than two circles?

推荐答案

查找的外周圈的所有路口（如B，D，F，H下面的图上）。与相应的圆的中心将其连接在一起，以形成一个多边形。圆的工会的面积是多边形+通过连续的交叉点，并在它们之间的圆心确定的圆形切片的区域的面积。你还需要考虑任何漏洞。

Find all circle intersections on the outer perimeter (e.g. B,D,F,H on the following diagram). Connect them together with the centres of the corresponding circles to form a polygon. The area of the union of the circles is the area of the polygon + the area of the circle slices defined by consecutive intersection points and the circle center in between them. You'll need to also account for any holes.

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