Re: Re: Finding the periphery of a region
- To: mathgroup at smc.vnet.net
- Subject: [mg72119] Re: [mg72005] Re: Finding the periphery of a region
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Wed, 13 Dec 2006 06:38:52 -0500 (EST)
- References: <el8ufm$st3$1@smc.vnet.net> <200612081117.GAA20172@smc.vnet.net> <6DA7D258-EA36-456E-A8F7-B4CBE82001B8@mimuw.edu.pl> <457D0248.7020004@metrohm.ch> <78BE6846-13C6-4E0F-B440-B211BBB62250@mimuw.edu.pl> <457D0BAF.9040000@metrohm.ch> <8197B414-9148-43A4-8BBE-EB019F2CCD02@mimuw.edu.pl>
On 11 Dec 2006, at 17:55, Andrzej Kozlowski wrote:
> On 11 Dec 2006, at 16:41, Daniel Huber wrote:
>> Hi Andrzej ,
>> I think the whole problem arises because we are loking at
>> solutions to (in)equations in R that actually lay in C. In C there
>> wouldn't be any "singular" points.
>> On the other hand, if somebodey has an applied problem involving
>> area, then I think he can most of the time simply ignore isolated
>> points.
>> Anyway, thank's a lot that you pointed me to this interesting fact.
>> Daniel
>
>
> Well, form the point of view of measure theory an isolated point is
> nothing but from the point of view of topology it is enormously
> important. (And there are real life applications, e.g. in robotics,
> where the existence of isolated points can have quite dramatic
> effect).
> Partly because of such things, from the point of view of topology,
> real algebraic sets are much more interesting than complex ones.
> Among well known examples that can be plotted in Mathematica are:
>
> The "Cartan umbrella":
>
> z*(x^2 + y^2) - x^3==0
>
> another "umbrella":
>
> x^3+ z*x - y^2 ==0
>
> a surface with curious properties:
>
> x^2 (1-z^2) == x^4 + y^2.
>
> The Nash-Tognoli theorem says that every compact smooth manifold is
> diffeomorphic to some non singular real algebraic vaiety. For
> example the Klein bottle here is an algebraic equation of the Klein
> bottle:
>
> 768*x^4 - 1024*x^5 - 128*x^6 +
> 512*x^7 - 80*x^8 - 64*x^9 + 16*x^10 + 144*x^2*y^2 -
> 768*x^3*y^2 - 136*x^4*y^2 + 896*x^5*y^2 -
> 183*x^6*y^2 - 176*x^7*y^2 + 52*x^8*y^2 + 400*y^4 +
> 256*x*y^4 - 912*x^2*y^4 + 256*x^3*y^4 + 315*x^4*y^4 -
> 144*x^5*y^4 - 16*x^6*y^4 + 4*x^8*y^4 - 904*y^6 -
> 128*x*y^6 + 859*x^2*y^6 - 16*x^3*y^6 - 200*x^4*y^6 +
> 16*x^6*y^6 + 441*y^8 + 16*x*y^8 - 224*x^2*y^8 +
> 24*x^4*y^8 - 76*y^10 + 16*x^2*y^10 + 4*y^12 -
> 2784*x^3*y*z + 4112*x^4*y*z - 968*x^5*y*z -
> 836*x^6*y*z + 416*x^7*y*z - 48*x^8*y*z +
> 1312*x*y^3*z + 2976*x^2*y^3*z - 5008*x^3*y^3*z -
> 12*x^4*y^3*z + 2016*x^5*y^3*z - 616*x^6*y^3*z -
> 64*x^7*y^3*z + 32*x^8*y^3*z - 1136*y^5*z -
> 4040*x*y^5*z + 2484*x^2*y^5*z + 2784*x^3*y^5*z -
> 1560*x^4*y^5*z - 192*x^5*y^5*z + 128*x^6*y^5*z +
> 1660*y^7*z + 1184*x*y^7*z - 1464*x^2*y^7*z -
> 192*x^3*y^7*z + 192*x^4*y^7*z - 472*y^9*z -
> 64*x*y^9*z + 128*x^2*y^9*z + 32*y^11*z - 752*x^4*z^2 +
> 1808*x^5*z^2 - 1468*x^6*z^2 + 512*x^7*z^2 -
> 64*x^8*z^2 + 6280*x^2*y^2*z^2 - 5728*x^3*y^2*z^2 -
> 4066*x^4*y^2*z^2 + 5088*x^5*y^2*z^2 -
> 820*x^6*y^2*z^2 - 384*x^7*y^2*z^2 + 96*x^8*y^2*z^2 -
> 136*y^4*z^2 - 7536*x*y^4*z^2 + 112*x^2*y^4*z^2 +
> 8640*x^3*y^4*z^2 - 2652*x^4*y^4*z^2 -
> 1152*x^5*y^4*z^2 + 400*x^6*y^4*z^2 + 2710*y^6*z^2 +
> 4064*x*y^6*z^2 - 3100*x^2*y^6*z^2 - 1152*x^3*y^6*z^2 +
> 624*x^4*y^6*z^2 - 1204*y^8*z^2 - 384*x*y^8*z^2 +
> 432*x^2*y^8*z^2 + 112*y^10*z^2 + 3896*x^3*y*z^3 -
> 7108*x^4*y*z^3 + 3072*x^5*y*z^3 + 768*x^6*y*z^3 -
> 768*x^7*y*z^3 + 128*x^8*y*z^3 - 3272*x*y^3*z^3 -
> 4936*x^2*y^3*z^3 + 8704*x^3*y^3*z^3 - 80*x^4*y^3*z^3 -
> 2496*x^5*y^3*z^3 + 608*x^6*y^3*z^3 + 2172*y^5*z^3 +
> 5632*x*y^5*z^3 - 2464*x^2*y^5*z^3 - 2688*x^3*y^5*z^3 +
> 1056*x^4*y^5*z^3 - 1616*y^7*z^3 - 960*x*y^7*z^3 +
> 800*x^2*y^7*z^3 + 224*y^9*z^3 + 752*x^4*z^4 -
> 1792*x^5*z^4 + 1472*x^6*z^4 - 512*x^7*z^4 +
> 64*x^8*z^4 - 3031*x^2*y^2*z^4 + 1936*x^3*y^2*z^4 +
> 2700*x^4*y^2*z^4 - 2304*x^5*y^2*z^4 +
> 448*x^6*y^2*z^4 + 697*y^4*z^4 + 3728*x*y^4*z^4 +
> 24*x^2*y^4*z^4 - 3072*x^3*y^4*z^4 + 984*x^4*y^4*z^4 -
> 1204*y^6*z^4 - 1280*x*y^6*z^4 + 880*x^2*y^6*z^4 +
> 280*y^8*z^4 - 800*x^3*y*z^5 + 1488*x^4*y*z^5 -
> 768*x^5*y*z^5 + 128*x^6*y*z^5 + 992*x*y^3*z^5 +
> 1016*x^2*y^3*z^5 - 1728*x^3*y^3*z^5 +
> 480*x^4*y^3*z^5 - 472*y^5*z^5 - 960*x*y^5*z^5 +
> 576*x^2*y^5*z^5 + 224*y^7*z^5 + 16*x^4*z^6 +
> 388*x^2*y^2*z^6 - 384*x^3*y^2*z^6 + 96*x^4*y^2*z^6 -
> 76*y^4*z^6 - 384*x*y^4*z^6 + 208*x^2*y^4*z^6 +
> 112*y^6*z^6 - 64*x*y^3*z^7 + 32*x^2*y^3*z^7 +
> 32*y^5*z^7 + 4*y^4*z^8==0
>
> One can plot this using Mathematica's Graphics`ContourPlot3D
> package, but it won't come out very nice. It is much better to use
> Jens Kuska's excellent MVContourPlot3D funciton, which is a part of
> his MathGL3d (one does not need the commercial version to be able
> to use this function). Of course one can get the same picture much
> more easily with ParametricPlot3D using well known parametric
> description of the Klein bottle:
>
> {x,y,z}={(Cos[Ï?/2]*Sin[θ] - Sin[Ï?/2]*Sin[2*θ] + 2)*
> Cos[Ï?], (Cos[Ï?/2]*Sin[θ] -
> Sin[Ï?/2]*Sin[2*θ] + 2)*Sin[Ï?],
> Sin[Ï?/2]*Sin[θ] + Cos[Ï?/2]*Sin[2*θ]}
>
> Actually, the above algebraic equation was obtained form the
> parametric one by using GroebnerBasis. It is a nice illustration of
> the relation (and difference) between non-constructive mathematics,
> like the Nash-Tognoli theorem and computational one. Getting the
> algebraic equation from the parametric one without using
> Mathematica or a similar program would be quite challenging.
>
> Andrej Kozlowski
>
>
>
>
I just realized that this Klein bottle is not a good illustration of
the Nash-Tognoli theorem because the algebraic surface in R^3 that we
get is obviously singular. (It is not really diffeomorphic to the
Klein bottle but the image of an immersion of the Klein bottle in
R^3). I should have used a torus or a Klein bottle in R^4 but that
would have been less impressive (both the formulas and the pictures).
Andrzej
- References:
- Re: Finding the periphery of a region
- From: dh <dh@metrohm.ch>
- Re: Finding the periphery of a region