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Re: Ellipse equation simplification on Mathematica:
*To*: mathgroup at smc.vnet.net
*Subject*: [mg77125] Re: Ellipse equation simplification on Mathematica:
*From*: Narasimham <mathma18 at hotmail.com>
*Date*: Mon, 4 Jun 2007 03:52:48 -0400 (EDT)
*References*: <f2emof$35h$1@smc.vnet.net><200706010641.CAA25336@smc.vnet.net>
On Jun 2, 1:37 pm, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> This can't be true without some additional assumptions, since, as I
> showed in my reply:
>
> Expand[(-y^2)*(a - d)^2 + (a^2 - 2*d*a - c*(c + 2*cp*d))*(a - d)^2 +
> (-a^2 + 2*d*a + c^2 + (cp^2 - 1)*d^2 + 2*c*cp*d)*x^2 == 0 /.
> {a -> 181, c -> -(39/10), d -> 100, cp -> 1}]
>
> (267421*x^2)/100 - 6561*y^2 - 1754549181/100 == 0
>
> which is a hyperbola. However, in this case the original equations
> have no solutions.
a > c > 0, 2c is the inter focal distance same as for the standard
ellipse.
Narasimham
> Andrzej Kozlowski
>
> On 1 Jun 2007, at 15:41, Narasimham wrote:
>
> > On May 31, 12:19 pm, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> >> On 30 May 2007, at 18:29, Narasimham wrote:
>
> >> (-y^2)*(a - d)^2 + (a^2 - 2*d*a - c*(c + 2*cp*d))*(a - d)^2 +
> >> (-a^2 + 2*d*a + c^2 + (cp^2 - 1)*d^2 + 2*c*cp*d)*x^2
>
> > The above can be simplified, and they are shown capable of being cast
> > into canonical form of ellipses :
>
> > (x/aEff)^2 + y^2/(aEff^2 - cEff^2) = 1
>
> > For case x-axis parallel tube cp = 1, forming trapeziums at each
> > stage:
>
> > aEff = (a - d), cEff = (c + d).
>
> > For case x-axis perpendicular to tube cp = 0:
>
> > aEff = (a - d) V, cEff = c V
>
> > where V = Sqrt((a^2 -2 a d -c^2)/(a^2 -2 a d -c^2 + d^2) )
>
> > Narasimham
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