Re: Intersection Ellipse & Circle

• To: mathgroup at smc.vnet.net
• Subject: [mg32525] Re: [mg32503] Intersection Ellipse & Circle
• From: Tomas Garza <tgarza01 at prodigy.net.mx>
• Date: Fri, 25 Jan 2002 02:57:57 -0500 (EST)
• References: <200201241020.FAA06041@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```I'm afraid the answer to your query is that you get very long solutions
because they are in fact very long, and very little can be done about it.
The straightforward approach would be

In[1]:=
elipse = (x - c)^2/b^2 + (y - d)^2/a^2 == 1;
circ = x^2 + y^2 == 1;

In[2]:=
sols=Solve[{elipse,circ},{x,y}];

And of course you get your four solutions, which turn out to be exceedingly
long. It depends, though, on what you want to do with it. You can look at a
simplified version if, for example, you set one of the four parameters equal
to zero, say c:

In[3]:=
sols2 = Solve[{elipse, circ} /. c -> 0, {x, y}]

(I omit the output). Here you have much shorter expressions for each of the
four solutions, and you may convince yourself  that it is almost impossible
to do anything in order to obtain anything "shorter" therefrom.

Tomas Garza
Mexico City

----- Original Message -----
From: "Philipp Schramek" <philipp at physics.usyd.edu.au>
To: mathgroup at smc.vnet.net
Subject: [mg32525] [mg32503] Intersection Ellipse & Circle

> 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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