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MathGroup Archive 2006

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Re: Intersection of Conic Sections

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66288] Re: [mg66277] Intersection of Conic Sections
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 8 May 2006 00:46:05 -0400 (EDT)
  • References: <200605070350.XAA08497@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On 7 May 2006, at 12:50, irchans wrote:

> This seems like a simple question, but I am finding it diffucult to
> find a simple solution.
>
> What are the points of intersection of
>
> 0 = A1 + A2*x + A3*y + A4*x^2 + A5*x*y + A6*y^2
>
> and
>
> 0 = B1 + B2*x + B3*y + B4*x^2 + B5*x*y + B6*y^2
>
>
> where Ai and Bi are constants.
>
> When I solve this with mathematica, I get a rather complex expression.
> Is there a simple algorithm that finds all the roots?
>
> Cheers,
> Irchans
>

It depends on what you mean by a simple. By using the GroebnerBasis  
algorithm you can reduce the problem to solving a quartic (fourth  
degree) equation in either y or x. There is a complicated formula for  
the roots of a quartic due to Ferrari, a student of Cardano, and that  
is essentially all that you need. Whether this algorithm is "simple'  
or "complicated' algorithm depends on  your viewpoint.

It certainly does not mean that there is a "simple" general formula,;  
on the  contrary the general formula is horribly complicated and is  
exactly the one Mathematica returns.

Andrzej Kozlowski

Tokyo, Japan


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