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

**Follow-Ups**:**Re: Ellipse equation simplification on Mathematica:***From:*Narasimham <mathma18@hotmail.com>

**Re: Re: Ellipse equation simplification on Mathematica:***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**References**:**Re: Ellipse equation simplification on Mathematica:***From:*Narasimham <mathma18@hotmail.com>