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MathGroup Archive 2006

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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]


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