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An interpolation between the Klein bottle and the double Jeener bottle defined by means of three sets of bidimensional fields [Une interpolation entre la bouteille de Klein et la double bouteille de Jeener définie à l'aide de trois ensembles de champs bidimensionnels].




Many surfaces -bidimensional manifolds- in a tridimensional space can be defined using a set of three equations:
                    X = F (u,v)
                         x
                    Y = F (u,v)
                         y
                    Z = F (u,v)
                         z
with:
                    u E [U   ,U   ]
                          min  max
                    v E [V   ,V   ]
                          min  max
[Umin,Umax]x[Vmin,Vmax] then defined a bidimensional rectangular domain D.
                       v ^
                         |
                    V    |...... ---------------------------
                     max |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                         |      |+++++++++++++++++++++++++++|
                    V    |...... ---------------------------
                     min |      :                           :
                         |      :                           :
                         O------------------------------------------------->
                                U                           U              u
                                 min                         max

If D is sampled by means of a bidimensional rectangular grid (made of Nu.Nv points), the three {X,Y,Z} coordinates can be defined by means of three rectangular matrices:
                    X = M (i,j)
                         x
                    Y = M (i,j)
                         y
                    Z = M (i,j)
                         z
with:
                    i = f(u,U   ,U   ,N )
                             min  max  u
                    j = g(v,V   ,V   ,N )
                             min  max  v
where 'f' and 'g' denote two obvious linear functions...


[for more information about this process]
[Plus d'informations sur ce processus]


For the interpolation between the Klein bottle and the double Jeener bottle, the three sets of {X,Y,Z} fields/matrices are as follows:



See some of the interpolated surfaces:

The Klein bottle defined by means of three bidimensional fields A surface between the Klein bottle and the double Jeener bottle defined by means of three bidimensional fields A surface between the Klein bottle and the double Jeener bottle defined by means of three bidimensional fields A surface between the Klein bottle and the double Jeener bottle defined by means of three bidimensional fields A double Jeener bottle defined by means of three bidimensional fields


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