Solving a Circle Problem: Max Arc Length

[Moni]

Hello! I am back! :)...little busy for few days...exams and projects :(...but at last... :)

A math problem can you help? :

Given a circle: C1 centre at (a,b) and radius r1.

Now there is a point outside the circle: (m,n) say this is the centre of the 2nd circle C2.

Now you have to measure the radius r2 of the C2 that gives maximum arc length of part of the C2, inside C1.

 

[NateTG]

Do you have any starting ideas?

 

[Moni]

Originally posted by NateTG
Do you have any starting ideas?

Sorry! I can't get you!
please make it clear!

 

[Monique]

Hi Moni, thanks for the PM, you blocked other members from sending you PMs so I will just leave you a note here :) I hope your exams went well. Monique

 

[NateTG]

Well, I can see the answer, but it wouldn't help you any if I just told you what it was.

So, I was wondering if you had any ideas about how to sove the problem to start out with.

Perhaps you used a compass to make drawings or something?

P.S. It should be pretty obvious that r1 is unimportant

 

[Moni]

Oh! I was just solving a geometry puzzle :)

But this is not that easy as I thought...so posted...

Can you tell me the solution?

 

[NateTG]

So, we've got circle C1 with it's center at (a,b) and circle C2 which does not contain (a,b), that has radius r2 center at (c,d).

Then the raidus of the circle around (a,b) that has maximal arc length inside C2 is at most
sqrt((a-c)²+(b-d)²+r2²)

This is the radius of the circle centered at (a,b) that intersects with C2 in two places that form a diameter.

It's a little tricky to explain without drawings, but it's easy to show that any longer radius would lead to a smaller section of arc.

Proving the other direction is a bit trickier.

 

[Moni]

Thanks Nate!

I have shown to another this problem...

And he says something like:

r1 is Radius of C2 (variable)
r2 is Radius of C1
d is distance of both centers
r1*r1-x*x = r2*r2-(d-x)*(d-x)
r1*r1-x*x = r2*r2-d*d+2*d*x-x*x
r1*r1-r2*r2+d*d = 2*d*x
phi = acos(x/r1)
arclen(r1) = r1*acos((d*d-r2*r2+r1*r1)/2.0/d/r1)


And then he suggested to use Bisection Method!

 

[NateTG]

Well, I'm pretty positive that the optimal radius is
sqrt(d²+r₂²)

 

[Moni]

Originally posted by NateTG
Well, I'm pretty positive that the optimal radius is
sqrt(d²+r₂²)

How you've come onto this formula ?

 

[NateTG]

I may be misunderstanding your problem, but it's the radius that makes a diameter of the small circle a sector of the large circle.

I am positive that the optimal large radius is between
sqrt(d²+r²)
and
sqrt(d²-r²)
for geometric reasons.

 

[Moni]

I still didn't find any clue