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February 1st, 2012, 06:55 AM   #1
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Ellipse inside a circle - intersection

I have a problem I'm trying to figure out for a piece of computer software I'm writing. Can anybody please help?

I have a non-rotated ellipse inside a circle, as in this diagram. The ellipse just touches the inside of the circle forming a tangent at two points.



What I know: The x and y coordinates for the centres of both the circle and the ellipse, the radius of the circle and the width and height of the ellipse

What I am trying to find out: The y coordinate of P, i.e. the vertical height above the x axis of the point at which the ellipse just touches the circle.

Any help is much appreciated - thanks!
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February 1st, 2012, 07:13 AM   #2
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Re: Ellipse inside a circle - intersection

This might work:

Let a = half the width of your ellipse
Let b = half the height of your ellipse
Let r = the radius of the circle.

First you must calculate this:



Then the height you want should be:



... assuming my algebra is correct.
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February 1st, 2012, 08:04 AM   #3
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Re: Ellipse inside a circle - intersection

Your diagram, quite nice(!), begs the question: is ellipse always tangent to x-axis?
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February 1st, 2012, 09:03 AM   #4
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Re: Ellipse inside a circle - intersection

mrtwhs: - Thank you, I will give that a try and report back.

denis: - The ellipse can actually have different y values. It was just a coincidence that it was a tangent to the x-axis in my diagram .... it could be the case, but not necessarily.
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February 1st, 2012, 10:39 AM   #5
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Re: Ellipse inside a circle - intersection

Well then:
is the center of circle always the origin...or somewhere on x axis?
is the center of ellipse always above x axis...or such that P is always above x axis?
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February 1st, 2012, 01:37 PM   #6
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Re: Ellipse inside a circle - intersection

mrtwhs: I've tried your equation and unfortunately it doesn't seem to work. That doesn't necessarily mean the equation is wrong - it could be my implementation of it within my program. But for the moment I'm still hoping for someone else to either verify or alter that equation.

denis: The circle is centred on the origin. For the ellipse, you can assume that it's centre and p is always above the x axis if that makes things easier. In reality, the ellipse may move below the x axis at some point as well, but if I can get the equation for above it it's no problem to find the values for below it.
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February 1st, 2012, 02:31 PM   #7
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Re: Ellipse inside a circle - intersection

I think I've got another way to do it:







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February 1st, 2012, 02:47 PM   #8
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Re: Ellipse inside a circle - intersection

By the way, is the width of the ellipse and is the height.
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February 1st, 2012, 06:01 PM   #9
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Re: Ellipse inside a circle - intersection

What did you think of this nightmare:

(-2*b^2*a^2+b^4+a^4)*y^4
+ (4*b^2*a^2*w-4*a^4*w)*y^3
+ (2*b^2*r^2*a^2-2*b^2*a^2*w^2+2*b^2*z^2*a^2+2*ab^2*b^2+6*a^4*w^2-2*b^4*z^2-2*b^4*r^2-2*ab^2*a^2+4*z)*y^2
+ (4*ab^2*a^2*w-4*b^2*r^2*a^2*w-4*b^2*z^2*a^2*w^4*a^4*w^3)*y
+ 2*b^2*r^2*a^2*w^2+2*b^2*z^2*a^2*w^2-2*ab^2*a^2*w^2-2*ab^2*b^2*r^2
-2*ab^2*b^2*z^2+2*b^4*r^2*z^2+ab^4+b^4*r^4+b^4*z^4+ a^4*w^4-4*z*r^2

...that you got here:
http://www.physicsforums.com/showthread ... ost3737507

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February 1st, 2012, 06:20 PM   #10
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Re: Ellipse inside a circle - intersection

Me? No.

This problem seemed more difficult than it actually is.
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