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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: [mg72099] Re: [mg72005] Re: Finding the periphery of a region
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 11 Dec 2006 04:55:40 -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>

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





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