圆圆交点 [英] Circle-circle intersection points

问题描述

如何计算两个圆的交点.我希望在所有情况下都有两个、一个或没有交点.

我有中心点的 x 和 y 坐标，以及每个圆的半径.

python 中的答案将是首选，但任何工作算法都是可以接受的.

解决方案

How do I calculate the intersection points of two circles. I would expect there to be either two, one or no intersection points in all cases.

I have the x and y coordinates of the centre-point, and the radius for each circle.

An answer in python would be preferred, but any working algorithm would be acceptable.

解决方案

Intersection of two circles

Written by Paul Bourke

The following note describes how to find the intersection point(s) between two circles on a plane, the following notation is used. The aim is to find the two points P₃ = (x₃, y₃) if they exist.

First calculate the distance d between the center of the circles. d = ||P₁ - P₀||.

• If d > r₀ + r₁ then there are no solutions, the circles are separate.

• If d < |r₀ - r₁| then there are no solutions because one circle is contained within the other.

• If d = 0 and r₀ = r₁ then the circles are coincident and there are an infinite number of solutions.

Considering the two triangles P₀P₂P₃ and P₁P₂P₃ we can write

a² + h² = r₀² and b² + h² = r₁²

Using d = a + b we can solve for a,

a = (r₀² - r₁² + d² ) / (2 d)

It can be readily shown that this reduces to r₀ when the two circles touch at one point, ie: d = r₀ + r₁

Solve for h by substituting a into the first equation, h² = r₀² - a²

So

P₂ = P₀ + a ( P₁ - P₀ ) / d

And finally, P₃ = (x₃,y₃) in terms of P₀ = (x₀,y₀), P₁ = (x₁,y₁) and P₂ = (x₂,y₂), is

x₃ = x₂ +- h ( y₁ - y₀ ) / d

y₃ = y₂ -+ h ( x₁ - x₀ ) / d

Source: http://paulbourke.net/geometry/circlesphere/

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