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Intersection of two disks
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kamuran...@gmail.com
Oct 5, 2012, 4:47:55 PM10/5/12
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I have two disks:
C1: (x-a1)^2+(y-b1)^2<=r1^2
C2: (x-a2)^2+(y-b2)^2<=r2^2
These disks have non-empty intersection.
I define the third circle as:
C3: (x-a3)^2+(y-b3)^2<=r3^2 where
a3=a1*(1-t)+a2*t
b3=b1*(1-t)+b2*t
r3=sqrt(a3^2+b3^2-(a1^2+b1^2-r1^2)*(1-t)-(a2^2+b2^2-r2^2)*t)
where 0<=t<=1.
Claim: C3 contains the intersection of C1 and C3 for all values of t such that 0<=t<=1
Numerically when i substitute t values and check it the claim works. However i could not prove it. Any suggestions?
C1: (x-a1)^2+(y-b1)^2<=r1^2
C2: (x-a2)^2+(y-b2)^2<=r2^2
These disks have non-empty intersection.
I define the third circle as:
C3: (x-a3)^2+(y-b3)^2<=r3^2 where
a3=a1*(1-t)+a2*t
b3=b1*(1-t)+b2*t
r3=sqrt(a3^2+b3^2-(a1^2+b1^2-r1^2)*(1-t)-(a2^2+b2^2-r2^2)*t)
where 0<=t<=1.
Claim: C3 contains the intersection of C1 and C3 for all values of t such that 0<=t<=1
Numerically when i substitute t values and check it the claim works. However i could not prove it. Any suggestions?
William Elliot
Oct 7, 2012, 4:34:27 AM10/7/12
to
On Fri, 5 Oct 2012, kamuran...@gmail.com wrote:
> I have two disks:
> C1: (x-a1)^2+(y-b1)^2<=r1^2
> C2: (x-a2)^2+(y-b2)^2<=r2^2
>
> These disks have non-empty intersection.
>
> I define the third circle as:
> C3: (x-a3)^2+(y-b3)^2<=r3^2 where
>
> a3=a1*(1-t)+a2*t
> b3=b1*(1-t)+b2*t
> r3=sqrt(a3^2+b3^2-(a1^2+b1^2-r1^2)*(1-t)-(a2^2+b2^2-r2^2)*t)
> where 0<=t<=1.
>
> Claim: C3 contains the intersection of C1 and C3 for all values of t such that 0<=t<=1
>
Of course C3 contains the intersection of C1 and C3.
> I have two disks:
> C1: (x-a1)^2+(y-b1)^2<=r1^2
> C2: (x-a2)^2+(y-b2)^2<=r2^2
>
> These disks have non-empty intersection.
>
> I define the third circle as:
> C3: (x-a3)^2+(y-b3)^2<=r3^2 where
>
> a3=a1*(1-t)+a2*t
> b3=b1*(1-t)+b2*t
> r3=sqrt(a3^2+b3^2-(a1^2+b1^2-r1^2)*(1-t)-(a2^2+b2^2-r2^2)*t)
> where 0<=t<=1.
>
> Claim: C3 contains the intersection of C1 and C3 for all values of t such that 0<=t<=1
>
Did you mean C3 contains the intersection C1 and C2?
TherestIignoreforit'stoohardtoreadwithmathwrittenlikethis.
Thomas Nordhaus
Oct 7, 2012, 5:06:39 AM10/7/12
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C1: x^2 + y^2 <= r1^2
C2: (x-D)^2 + y^2 <= r2^2
C3: (x-t)^2 + y^2 <= r3^2
where D>0 and 0 <= t <= D. That should make the analysis simpler.
--
Thomas Nordhaus
Narasimham
Oct 7, 2012, 12:16:37 PM10/7/12
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On Saturday, October 6, 2012 2:17:55 AM UTC+5:30, kamuran turksoy wrote:
> I have two disks:
> C1: (x-a1)^2+(y-b1)^2<= r1^2
> C2: (x-a2)^2+(y-b2)^2<= r2^2
> These disks have non-empty intersection.
> I define the third circle as:
> C3: (x-a3)^2+(y-b3)^2<=r3^2 where
> a3=a1*(1-t)+a2*t
> b3=b1*(1-t)+b2*t
> r3=sqrt(a3^2+b3^2-(a1^2+b1^2-r1^2)*(1-t)-(a2^2+b2^2-r2^2)*t)
> where 0<=t<=1.
> Claim: C3 contains the intersection of C1 and C2 for all values of t such that 0 <= t <= 1.
> I have two disks:
> C1: (x-a1)^2+(y-b1)^2<= r1^2
> C2: (x-a2)^2+(y-b2)^2<= r2^2
> These disks have non-empty intersection.
> I define the third circle as:
> C3: (x-a3)^2+(y-b3)^2<=r3^2 where
> a3=a1*(1-t)+a2*t
> b3=b1*(1-t)+b2*t
> r3=sqrt(a3^2+b3^2-(a1^2+b1^2-r1^2)*(1-t)-(a2^2+b2^2-r2^2)*t)
> where 0<=t<=1.
> Numerically when I substitute t values and check it, the claim works. However I could not prove it. Any suggestions?
Hint : The Power (product of tangent lengths) is negative originating from a point contained inside the circle (and positive if outside).
Narasimham
kamuran turksoy
Oct 9, 2012, 5:20:14 PM10/9/12
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