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February 1st, 2012, 07:37 PM  #11  
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039  Re: Ellipse inside a circle  intersection Quote:
 
February 2nd, 2012, 03:33 AM  #12 
Newbie Joined: Feb 2012 Posts: 7 Thanks: 0  Re: Ellipse inside a circle  intersection
Greg: Thank you for your help. But I think your equation for the ellipse assumes it is centred on the origin. For my situation, we can assume the circle is centred on the origin, but the vertical centre of the ellipse in relation to the circle is not zero, so its equation is (x^2 / a^2) + ((y  ye)^2) / b^2 = 1 where ye is the vertical centre of the ellipse. Following your method, rearranging this for y and substituting into the equation of the circle, gives an equation which is somewhat more complex, and I then have the problem of solving it for x. That's where I get stuck. Denis: Well, that quartic may well be correct, but by itself it's really not much use to me  it's the solutions of it I need. 
February 2nd, 2012, 03:42 AM  #13 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,974 Thanks: 1156 Math Focus: Elementary mathematics and beyond  Re: Ellipse inside a circle  intersection
Yes, that was on oversight on my part. I will try to solve for x, being more careful. If I arrive at useful results I will post them.

February 2nd, 2012, 04:24 AM  #14 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,974 Thanks: 1156 Math Focus: Elementary mathematics and beyond  Re: Ellipse inside a circle  intersection
It gets pretty nasty. What language(s) are you writing your code in? My thinking is that you could use a module that handles these sorts of things. Though an algebraic solution would be ideal, a numerical solution, with suitable precision, should suffice. There is a program, PARI/GP, that has a 'solve' function. Here is the output from the associated help query: solve(X=a,b,expr): real root of expression expr (X between a and b), where expr(a)*expr(b)<=0. I know PARI/GP has a C library that probably includes this function, and it is freely distributed; you don't have to pay use it. Let us know what you think. Here is a simple example of solve: Code: input: solve(x=1,3,x^22*x+1) output: 1.000000000000000000000000000 
February 2nd, 2012, 05:21 AM  #15 
Newbie Joined: Feb 2012 Posts: 7 Thanks: 0  Re: Ellipse inside a circle  intersection
Thanks Greg  I tried putting the equations into an online solver (www.wolframalpha.com) and it did produce some solutions but, as you said, it does get pretty nasty, and I haven't had much luck so far integrating them successfully into my program. When I set out along this route I hadn't thought the solution would be so unwieldy. I'm probably going to have a rethink about the route I'm taking. I'm using ActionScript3 for Flash, btw. The whole purpose of this was to get my program to draw and fill some geometrical shapes, but given the complexity of doing it mathematically, I'll probably try prerendering those shapes in a graphics program instead. Thanks everyone for their help. 
February 2nd, 2012, 12:31 PM  #16 
Newbie Joined: Feb 2012 Posts: 7 Thanks: 0  Re: Ellipse inside a circle  intersection
Just a quick follow up  I finally got an equation (thanks to mathnerds.org) which works perfectly for this special case: y = w / (1  (b^2 / a^2)) where y is the vertical intersect point, w is the vertical height of the ellipse and a and b are the semi major/minor axis. The equation was derived by equating gradients  the gradient of the circle and the gradient of the ellipse are the same at the intersect point in this special case because they form a tangent. Thanks to everyone who helped. 
February 2nd, 2012, 01:45 PM  #17 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,974 Thanks: 1156 Math Focus: Elementary mathematics and beyond  Re: Ellipse inside a circle  intersection 
February 2nd, 2012, 02:17 PM  #18 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,974 Thanks: 1156 Math Focus: Elementary mathematics and beyond  Re: Ellipse inside a circle  intersection
I wonder what technique they used to derive that. Calculus, maybe? Is there a link to their solution? Did they show you how they worked it out?

February 3rd, 2012, 01:03 PM  #19  
Newbie Joined: Feb 2012 Posts: 7 Thanks: 0  Re: Ellipse inside a circle  intersection
Expressions for the gradients of the circle and tangent were found by differentiation, then equated to each other. This was the response from the volunteer at mathnerd.org: Quote:
 

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