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Re: intersection of circles

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Krishna Myneni wrote:

> Surprisingly, the following problem isn't easily handled by a couple
> of scientific computing packages:
> 
> If solution(s) exist, find the intersection points of two circles in
> the x-y plane. This requires solving a system of two qudratic
> equations in two variables:
> 
> 1)  (x - a)^2 + (y - b)^2 = r1^2
> 
> 2)  (x - c)^2 + (y - d)^2 = r2^2
> 
> where the points (a, b) and (c, d) are the centers of two circles of
> radii r1 and r2.

I had that problem a bit over a month ago, when I solved the Triceps 2 
geometry.  There, intersection of two circles is one of the steps to be 
done.  Fortunately, I didn't know that this was a "hard" problem, which 
required a quadratic solver, and just decided to use the law of cosines.

You know:  The distance of the two points r3²=(a-c)²+(b-d)².  You know 
r1 and r2.  These are the three sides of a triangle, which could be on 
either side (for Triceps 2, only one side is possible, as gravity goes 
downwards).

cos alpha=(r1²+r2²-r3²)/(2*r1*r2)

You need to add the angle of the rectangular triangle where r3 is the 
hypothenuse, and the a and b sides are parallel to x and y, but this is 
left as an exersice to the reader.

-- 
Bernd Paysan
"If you want it done right, you have to do it yourself"
http://bernd-paysan.de/

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intersection of circles Krishna Myneni <krishna.myneni@ccreweb.org> - 2012-06-22 17:31 -0700
  Re: intersection of circles mhx@iae.nl (Marcel Hendrix) - 2012-06-23 12:26 +0200
    Re: intersection of circles Krishna Myneni <krishna.myneni@ccreweb.org> - 2012-06-23 04:24 -0700
  Re: intersection of circles Krishna Myneni <krishna.myneni@ccreweb.org> - 2012-06-23 04:28 -0700
  Re: intersection of circles Bernd Paysan <bernd.paysan@gmx.de> - 2012-06-23 22:12 +0200
    Re: intersection of circles Krishna Myneni <krishna.myneni@ccreweb.org> - 2012-06-23 16:06 -0700
      Re: intersection of circles Bernd Paysan <bernd.paysan@gmx.de> - 2012-06-24 02:11 +0200

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