求两圆交点坐标

大概也可以说成是解二元二次方程组.

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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

其中有几个地方还是说明一下.......................................................................................

a的来源可以说是余弦定理   最后两个求x3 和y3 的式子可以看下面的图 标记粉颜色的角 这两个粉色的三角形相似..............