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Re: Ellipse equation simplification on Mathematica:
*To*: mathgroup at smc.vnet.net
*Subject*: [mg76830] Re: Ellipse equation simplification on Mathematica:
*From*: Narasimham <mathma18 at hotmail.com>
*Date*: Mon, 28 May 2007 01:00:43 -0400 (EDT)
*References*: <f2emof$35h$1@smc.vnet.net><200705181006.GAA12812@smc.vnet.net>
On May 19, 1:54 pm, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> On 18 May 2007, at 19:06, Narasimham wrote:
>
> > Reference is made to:
>
> >http://groups.google.co.in/group/geometry.puzzles/browse_thread/
> > threa...
>
> > Constant[c,d th,ph] ;
>
> > (* th, ph are spherical cords of tip of tube *) ;
>
> > cp = Cos[ph] ; sp = Sin[ph] ; cth = Cos[th] ; sth = Sin[th] ;
>
> > (* earlier typo corrected *)
>
> > d1 = Sqrt[(x + d cp cth + c )^2 + ( y + d cp sth )^2 + (d sp)^2 ]
>
> > d2 =Sqrt[(x - d cp cth - c )^2 + ( y - d cp sth )^2 + (d sp)^2 ]
>
> > FullSimplify[ d1 + d2 + 2 d - 2 a == 0] ;
>
> > When d = 0, algebraic/trigonometric simplification brings about
> > common ellipse form:
>
> > (x/a)^2 + y^2/(a^2-c^2) = 1
>
> > Request help for bringing to standard form involving constants a,c
> > and the new tube length constant d.
>
> > Regards,
> > Narasimham
>
> I don't think such a form exists. Consider the following.
>
> id1 = {d1^2 - ((x + d*cp*cth + c)^2 + (y + d*cp*sth)^2 + (d*sp)^2),
> d2^2 - ((x - d*cp*cth - c)^2 + (y - d*cp*sth)^2 + (d*sp)^2),
> sp^2 + cp^2 - 1, sth^2 + cth^2 - 1};
>
> id = Prepend[id1, d1 + d2 + 2 d - 2 a];
>
> Now consdier first the case of the ellipse:
>
> d = 0;
>
> gr = GroebnerBasis[id, {x, y, a, c}, {cp, sp, cth, sth, d1, d2},
> MonomialOrder -> EliminationOrder]
> {-a^4 + c^2 a^2 + x^2 a^2 + y^2 a^2 - c^2 x^2}
>
> This tells us that
>
> First[%] == 0
> -a^4 + c^2*a^2 + x^2*a^2 + y^2*a^2 - c^2*x^2 == 0
>
> is the equation of the ellipse, and this can be easily brought to
> standard form by hand. But now consider your "general" case:
>
> Clear[d]
> gr = GroebnerBasis[id, {x, y, a, c, d}, {cp, sp, cth, sth, d1, d2},
> MonomialOrder -> EliminationOrder]
> {}
>
> This means that elimination cannot be performed and no "standard form"
> of the kind you had in mind exists. Unless of course there is a bug
> in GroebnerBasis (v. unlikely) or I have misunderstood what you had
> in mind.
>
> Andrzej Kozlowski
I checked for case of tube parallel to x- or y-axis produces ellipses
and suspected validity even in 3-D general case.
Narasimham
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