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August 9th, 2010, 02:12 PM   #1
nom
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Angular Circle Intersection

I am looking for a method to solve the problem in the below image, generally.



Ideally I would like to find a formula for both the start and extent angles of the arc. The following formulas produce the xy coordinates of the intersection points:

x = (-[x1-x2][r12-r22-x12+x22]+[x1+x2][y1-y2]2+[y1-y2]sqrt[Delta])/(2[x1-x2]2+2[y1-y2]2),
y = (-[y1-y2][r12-r22-y12+y22]+[y1+y2][x1-x2]2-[x1-x2]sqrt[Delta])/(2[x1-x2]2+2[y1-y2]2),

and

x = (-[x1-x2][r12-r22-x12+x22]+[x1+x2][y1-y2]2-[y1-y2]sqrt[Delta])/(2[x1-x2]2+2[y1-y2]2),
y = (-[y1-y2][r12-r22-y12+y22]+[y1+y2][x1-x2]2+[x1-x2]sqrt[Delta])/(2[x1-x2]2+2[y1-y2]2),

where

Delta = -([x1-x2]2+[y1-y2]2-[r1-r2]2)([x1-x2]2+[y1-y2]2-[r1+r2]2).

I used the law of cosines to find the angular starting point of the arc, but the rounding was unacceptable for acute angles.

While the solution currently eludes me, I feel there must be a fundamental relationship here that is easily computed by formula to find the angular circle intersection of the arc.
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August 15th, 2010, 03:08 AM   #2
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From: Osijek

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Re: Angular Circle Intersection

I found this problem interesting, so I worked on it for a while. And this is what I concluded:





Now, I forgot that you need a starting point too, so I'll try and find an optimal solution soon.
Oh and sorry for posting this in .jpg images, I failed to see the "latex" button on forum.
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