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MathGroup Archive 2002

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Intersection Ellipse & Circle

  • To: mathgroup at smc.vnet.net
  • Subject: [mg32503] Intersection Ellipse & Circle
  • From: Philipp Schramek <philipp at physics.usyd.edu.au>
  • Date: Thu, 24 Jan 2002 05:20:56 -0500 (EST)
  • Organization: The University of Sydney, Australia
  • Sender: owner-wri-mathgroup at wolfram.com

Hi
I want to calculate the intersection of an Ellips ((x-c)^2/b^2 +
(y-d)^2/a^2 ==1)which centre is at the point (c,d) and a Circle (x^2 +
y^2 == 1) which centre is (0,0).
I thought I might be able to solve this problem for any a!=0 && b!=0
with Mathmatica 3. There should be 4 solutions.
Therefore I did following calcuatlion:
In[15]:= Eliminate[{y==Sqrt[1-x^2],((x-c)^2)/(b^2) + ((y-d)^2)/(a^2)
==1},y]
                                 2
   Out[15]= a != 0 && b != 0 && c  - 2 c x ==

              2    2  2         2  2      2             2
         2   b    b  d     2   b  x    2 b  d Sqrt[1 - x ]
  >     b  - -- - ----- - x  + ----- + -------------------
              2     2            2              2
             a     a            a              a

   In[16]:= Solve[%,x]

   Out[16]:= .....................
My problem is that the solutions I got from Mathmatica are very very
long and therefore far to long for me to use it as a analytical
approach. In fact I was very suprised that there was not an easier
solution. Even if I assume the special case b==1 the results were too
long.I even tried to solve the problem defining the circle and the
ellips by two angles:
Eliminate[{Cos[x]==a*Cos[y]+c, Sin[x]==b*Sin[y]+d},y]
But solving this result was even worse.

Does anyone has a suggestion how I could get a shorter result for my
problem?

Thanks for you help
Philipp



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