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Subject: Re: analytical solution?
Date: Wed, 18 Mar 2009 06:24:02 +0000 (UTC)
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"Peter Schreiber" <schreiber.peter15@gmail.com> wrote in message <gpq19u$c4d$1@fred.mathworks.com>...
> Hi guys,
> Does somebody know if there exists an analytical solution for a line - conic intersection, e.g.
> z1=z2, with
> 
> z1=c.*r.^2./(1+sqrt(1-(1+k).*c.^2.*r.^2))
> z2=a.*r+b?
> 
> Any hints would be highly appreciated.
> 
> Best Regards,
> Peter
> 
> 
> %code showing the line and conic
> clear all
> clc
> close all
> c=-1/100;
> k=-1;
> a=1;
> b=2;
> r=linspace(-10,10,50);
> 
> z1=c.*r.^2./(1+sqrt(1-(1+k).*c.^2.*r.^2))
> z2=a.*r+b
> plot(z1,r)
> hold on
> plot(z2,r)
> axis equal

  Certainly!  The first equation can be reduced to a quadratic expression in z and r.  Using the second equation to substitute a*r+b for z yields a quadratic equation in the single variable r, which has an analytic solution.  However, I am going to leave all that algebraic manipulation for you to do, Peter, since I am feeling lazy at the moment.

Roger Stafford