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MathGroup Archive 2006

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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


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