My Math Forum Ellipse inside a circle - intersection

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 February 1st, 2012, 06:55 AM #1 Newbie   Joined: Feb 2012 Posts: 7 Thanks: 0 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!
 February 1st, 2012, 07:13 AM #2 Senior Member     Joined: Feb 2010 Posts: 706 Thanks: 141 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: $k= \sqrt{(a^2-b^2)(\dfrac{r^2-a^2}{a^2})}$ Then the height you want should be: $y= \dfrac{2ka^2}{a^2-b^2}$ ... assuming my algebra is correct.
 February 1st, 2012, 08:04 AM #3 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,587 Thanks: 1038 Re: Ellipse inside a circle - intersection Your diagram, quite nice(!), begs the question: is ellipse always tangent to x-axis?
 February 1st, 2012, 09:03 AM #4 Newbie   Joined: Feb 2012 Posts: 7 Thanks: 0 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.
 February 1st, 2012, 10:39 AM #5 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,587 Thanks: 1038 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?
 February 1st, 2012, 01:37 PM #6 Newbie   Joined: Feb 2012 Posts: 7 Thanks: 0 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.
 February 1st, 2012, 02:31 PM #7 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,944 Thanks: 1135 Math Focus: Elementary mathematics and beyond Re: Ellipse inside a circle - intersection I think I've got another way to do it: $\text{Equation of ellipse: }\frac{x^2}{a^2}\,+\,\frac{y^2}{b^2}\,=\,1\,\Right arrow\,y\,=\,\frac{b}{a}\sqrt{a^2\,-\,x^2}\,$$\text{Eq. 1}$$$ $\text{Equation of circle (centered on origin): }x^2\,+\,y^2\,=\,r^2,\,\text{where }r\text{ is the radius.}$ $\text{Substitute the RHS of Eq 1 for }y\text{ into the equation of the circle: }x^2\,+\,$$\frac{b}{a}\sqrt{a^2\,-\,x^2}$$^2\,=\,r^2$ $\text{Plug in your value for the radius and solve for }x\text{, then substitute that into }$$\text{Eq 1}$$\text{ and there is }y.$
 February 1st, 2012, 02:47 PM #8 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,944 Thanks: 1135 Math Focus: Elementary mathematics and beyond Re: Ellipse inside a circle - intersection By the way, $a$ is the width of the ellipse and $b$ is the height.
 February 1st, 2012, 06:01 PM #9 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,587 Thanks: 1038 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
 February 1st, 2012, 06:20 PM #10 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,944 Thanks: 1135 Math Focus: Elementary mathematics and beyond Re: Ellipse inside a circle - intersection Me? No. This problem seemed more difficult than it actually is.

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# a circle touching the ellipse from inside

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