the ellipse and the circle

*To*: mathgroup at smc.vnet.net*Subject*: [mg69198] the ellipse and the circle*From*: "Jack Kennedy" <jack at realmode.com>*Date*: Fri, 1 Sep 2006 06:41:52 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Greetings Math People, I'm trying to solve the following problem. Please see the figure at http://oldnews.org/ellipse2.gif. A circle of radius r sits on an ellipse with semi-major axis a and semi-minor axis b as shown in the figure. The center of the circle is theta radians around the ellipse. The radius r is small compared to a and b. In terms of theta, a, b, and r, what are the coordinates of the two points of intersection (shown in red)? What follows is a transcript of my attempt. I did not include the results of the final Solve command because it is page upon pages. While I have to hand it to Mathematica for being able to produce such a fantastical answer, my gut tells me the "real" answer is nowhere near this complicated. I tried applying Simplify to the result, but I ran out of memory before it could finish (I have 1GB physical, 2GB virtual). Can anyone provide advice on how to coax Mathematica into a simpler answer? (Or confirm that the answer really is this complicated.) Thanks, J. Kennedy In[1]:= $Version Out[1]= "5.1 for Microsoft Windows (October 25, 2004)" In[3]:= e = x^2/a^2 + y^2/b^2 == 1 Out[3]= x^2/a^2 + y^2/b^2 == 1 In[4]:= x0 = a*Cos[\[Theta]] Out[4]= a*Cos[\[Theta]] In[5]:= y0 = b*Sin[\[Theta]] Out[5]= b*Sin[\[Theta]] In[6]:= c = (x - x0)^2 + (y - y0)^2 == r^2 Out[6]= (x - a*Cos[\[Theta]])^2 + (y - b*Sin[\[Theta]])^2 == r^2 In[7]:= Solve[{e, c}, {x, y}] [thousand of lines deleted]

**Follow-Ups**:**Re: the ellipse and the circle***From:*"Carl K. Woll" <carlw@wolfram.com>