重叠圆的组合面积 [英] Combined area of overlapping circles

问题描述

我最近遇到了一个问题，我有四个圆(中点和半径)并且必须计算这些圆的并集面积.

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:

对于两个圆圈来说，这很容易，

For two circles it's quite easy,

我可以只计算每个圆面积不在三角形内的分数，然后计算三角形的面积.

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.

但是当有两个以上的圆时，我可以使用一种聪明的算法吗?

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

推荐答案

找到外周上的所有圆交点(例如下图中的 B、D、F、H).将它们与相应圆的中心连接在一起形成一个多边形.圆的并集面积是多边形的面积+由连续交点和它们之间的圆心定义的圆切片的面积.您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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