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   Circles in a corner
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   Author  Topic: Circles in a corner  (Read 517 times)
Noke Lieu
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Circles in a corner   two_circles_corner.gif
« on: Jan 21st, 2008, 10:10pm »
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Whilst wrestling with somethingelse, this presented itself to me.
 
Take two circles of different radii and back them into a corner (as per the diagram)
 
What's the question though? That's something that's eluding me. I guess I'll settle for
The ratio of AB:BC
 
(because I haven't made it look particularly pretty, and hope someone around here can Kiss )
 
 
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Obob
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Re: Circles in a corner  
« Reply #1 on: Jan 22nd, 2008, 12:46am »
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It is not too difficult to find the coordinates of the points of intersection of the two circles, from which a formula for the ratio follows.
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towr
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Re: Circles in a corner  
« Reply #2 on: Jan 22nd, 2008, 1:12am »
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If the ratio is constant, it seems 3+2sqrt(2)
« Last Edit: Jan 22nd, 2008, 11:31am by towr » IP Logged

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Obob
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Re: Circles in a corner  
« Reply #3 on: Jan 22nd, 2008, 10:05am »
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The way I interpreted the problem, the radi of the two circles are arbitrary.  The circles are not necessarily orthogonal.  And thus the ratio is not constant.
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Re: Circles in a corner  
« Reply #4 on: Jan 22nd, 2008, 10:23am »
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Actually, I'm not assuming the ratio would be constant because supposedly the circles are orthogonal. If the ratio isn't constant regardless, I doubt the value I found is the ratio in the case of orthogonal circles (because that's not the case I used to calculate it).
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Obob
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Re: Circles in a corner  
« Reply #5 on: Jan 22nd, 2008, 10:29am »
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Can we get some clarification as to where the point B is?  Is it on the y-axis or is it the point where the two circles meet?
 
My calculations were based on B being on the y-axis.
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Noke Lieu
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Re: Circles in a corner  
« Reply #6 on: Jan 22nd, 2008, 3:22pm »
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on Jan 22nd, 2008, 10:29am, Obob wrote:
Can we get some clarification as to where the point B is?  Is it on the y-axis or is it the point where the two circles meet?
 
My calculations were based on B being on the y-axis.

 
 
I intended B to be on the y axis.
 
But your question makes me think of the lengths of the arcs.  
 
 
I am curious about how you guys went about determining where the circles intercept. I went about it in a seemingly obvious fashion, then I noticed something cuter.
(Doesn't quite work when there is no intercept though)
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Re: Circles in a corner  
« Reply #7 on: Jan 22nd, 2008, 4:04pm »
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I just took the equations (x-R)^2+(y-R)^2 = R^2 and (x-r)^2+(y-r)^2=r^2.  Subtract the second equation from the first to get a linear equation relating x and y; solve for one of them and put it into the first equation to get a quadratic in x, say.  Then use the quadratic formula to solve for x, and solving for y in the second equation gives the four solutions (x,y).
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Noke Lieu
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Re: Circles in a corner  
« Reply #8 on: Jan 22nd, 2008, 7:48pm »
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Exactly what I did.  
Then I noticed that the result of
 (x-R)^2+(y-R)^2 = R^2  minus
(x-r)^2+(y-r)^2=r^2
 
could be re-arranged to y= -x + (R+r)/2
 
(so on the line that is  perpendicular to the line joining centres; intercepting the x and y axes at the average of the two radii)  
 
obvious when one thinks about it.
« Last Edit: Jan 22nd, 2008, 7:49pm by Noke Lieu » IP Logged

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