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Re: Re: Ellipse equation simplification on Mathematica:
*To*: mathgroup at smc.vnet.net
*Subject*: [mg76291] Re: [mg76222] Re: Ellipse equation simplification on Mathematica:
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Sat, 19 May 2007 04:35:17 -0400 (EDT)
*References*: <f2emof$35h$1@smc.vnet.net> <200705181006.GAA12812@smc.vnet.net>
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. Consdier 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 broght to
standard form by hand. But now conider 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 "stadard 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
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