Re: Re: Ellipse equation simplification on Mathematica:

*To*: mathgroup at smc.vnet.net*Subject*: [mg76898] Re: [mg76830] Re: Ellipse equation simplification on Mathematica:*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Wed, 30 May 2007 05:04:33 -0400 (EDT)*References*: <f2emof$35h$1@smc.vnet.net><200705181006.GAA12812@smc.vnet.net> <200705280500.BAA15766@smc.vnet.net> <A2AFF4CF-D65F-424C-A963-97043E71D6E7@mimuw.edu.pl> <7D2D10D0-B53E-4A9F-A03D-6D1755AA2505@mimuw.edu.pl> <6EB71EC4-37CE-48D1-842E-7DF0B82D4DB9@mimuw.edu.pl>

On 29 May 2007, at 12:20, Andrzej Kozlowski wrote: > > > On 29 May 2007, at 12:15, Andrzej Kozlowski wrote: > >> >> On 29 May 2007, at 10:53, Andrzej Kozlowski wrote: >> >>> >>> On 28 May 2007, at 14:00, Narasimham wrote: >>> >>>> 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 >>>> >>> >>> >>> OK., now I see that I misundertood you and you wrote that >>> cd,th, ph (and presumably a) are supposed to be constants, so you >>> do not wish to eliminate them. But now one can easily prove that >>> what you get is not, in general, an ellipse. In this situation >>> Groebner basis works and you can obtain a rather horrible >>> quartic equation of your surface: >>> >>> 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]; >>> >>> >>> v = GroebnerBasis[id, {x, y, a, c, cp, cth}, {sth, sp, d1, d2}, >>> MonomialOrder -> EliminationOrder][[1]]; >>> >>> First[v] == 0 >>> >>> is the equation (I prefer not to include the output here). >>> >>> Looking at v see that the non zero coefficients are only the free >>> coefficient, the coefficients of x^2, y^2, x^2 y^2, x^4 and y^4. >>> So only in some cases you will get a quadratic (for example when >>> the quartic happens to be a perfect square as in the case d=0, >>> or when the free coefficient vanishes, as in the trivial case >>> a=d, or when the coefficients of 4-degree terms vanish). One can >>> work out all the cases when gets a quadratic but it is also easy >>> to find those when one does not. For example, taking both th and >>> ph to be 60 degrees (so that cth and cph are both 1/2) we get: >>> >>> v /. {cp -> 1/2, cth -> 1/2, d -> 4, a -> 2, c -> 1} >>> >>> y^4 - 48 x^2 y^2 + 120 y^2 + 3600 >>> >>> This is certianly is not the equation of an ellipse. >>> >>> Andrzej Kozlowski >>> >>> >> >> >> Sorry; that last example was a bad one, because with these >> parameters (d>a) the original equations do not have solutions. >> The point is that the equations that we get after elimination will >> have more solutions than the ones we start with, but all the >> solutions of the original ones will satisfy the new ones. So to >> see that we do not normally get an ellipse we need a different >> choice of parameters, with a>d. So take >> instead >> >> w = First[v ]/. {cp -> 1/2, cth -> 1/2, sp -> Sqrt[3]/2, sth -> >> Sqrt[3]/2, d -> 1, >> a -> 3, c -> 1} >> >> (1521*x^4)/256 + (2229*y^2*x^2)/128 - (117*x^2)/4 + (3721*y^4)/256 >> - (183*y^2)/4 + 36 >> >> >> Now, this is clearly not the equation of an ellipse. We can see >> now the relationship between this and your original equation. With >> the above values of the parameters your equation takes the form: >> >> eq = d1 + d2 + 2*d - 2*a == 0 /. {cp -> 1/2, cth -> 1/2, sp -> Sqrt >> [3]/2, >> sth -> Sqrt[3]/2, d -> 1, a -> 3, >> c -> 1} >> Sqrt[(x - 5/4)^2 + (y - Sqrt[3]/4)^2 + 3/4] + >> Sqrt[(x + 5/4)^2 + (y + Sqrt[3]/4)^2 + 3/4] - 4 == 0 >> >> So now look at the graph: >> >> gr1=ContourPlot[ >> Sqrt[(x - 5/4)^2 + (y - Sqrt[3]/4)^2 + 3/4] + >> Sqrt[(x + 5/4)^2 + (y + Sqrt[3]/4)^2 + 3/4] - 4 == 0, {x, -5, >> 5}, {y, -5, >> 5}] >> >> >> Looks like a nice ellipse, right? Unfortunately it is only a part >> of the graph of >> >> w =First[v] >> >> gr2 = ContourPlot[Evaluate[w == 0], {x, -5, 5}, {y, -5, 5}] >> >> The picture is not quite convincing, because the latter contour >> plot is not very accurate but when you see them together: >> >> Show[gr1, gr2] >> >> the relationship becomes obvious. In any case, what you get a >> quartic curve that looks of course like an ellipse but isn't. >> >> Andrzej Kozlowski > > > Sorry, a small correction is needed. I wrote: > >> w=First[v] >> >> gr2 = ContourPlot[Evaluate[w == 0], {x, -5, 5}, {y, -5, 5}] > > Ignre the first line (w=First[v]) in the above since it will > prevent the plot from working. Just evaluate the second line with > the value of w defined earlier. > > Andrzej Kozlowski > > > One more comment: if you want a really good plot of gr2 you need to use a lot more plot points. With; gr2 = ContourPlot[Evaluate[w == 0], {x, -5, 5}, {y, -5, 5}, PlotPoints -> 300] one can see clearly how gr1 forms a part of gr2 (a quartic and not a quadratic). Andrzej Kozlowski

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

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

**Tracking point on a plot (follow-up)**

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**Re: Re: Ellipse equation simplification on Mathematica:**

**Re: Ellipse equation simplification on Mathematica:**