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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 nonrotated 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: Then the height you want should be: ... 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 xaxis?

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 xaxis 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: 
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, is the width of the ellipse and 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*w4*a^4*w)*y^3 + (2*b^2*r^2*a^22*b^2*a^2*w^2+2*b^2*z^2*a^2+2*ab^2*b^2+6*a^4*w^22*b^4*z^22*b^4*r^22*ab^2*a^2+4*z)*y^2 + (4*ab^2*a^2*w4*b^2*r^2*a^2*w4*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^22*ab^2*a^2*w^22*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^44*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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