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Re: the ellipse and the circle

• To: mathgroup at smc.vnet.net
• Subject: [mg69209] Re: [mg69198] the ellipse and the circle
• From: "Carl K. Woll" <carlw at wolfram.com>
• Date: Fri, 1 Sep 2006 18:41:16 -0400 (EDT)
• References: <200609011041.GAA25668@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Jack Kennedy wrote:
> 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
> 

Your equations are equivalent to finding the roots of a 4th order 
polynomial. By default, Solve will use radicals to express the root of a 
  quartic polynomial, and hence yields an enormous and useless result in 
this case. We can control this behavior by changing the options of Roots:

SetOptions[Roots, Cubics -> False, Quartics -> False]

Now using Solve will produce much smaller results at the cost of 
containing explicit Root objects.

Carl Woll
Wolfram Research

> 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]

• References:
  • the ellipse and the circle
    • From: "Jack Kennedy" <jack@realmode.com>