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An interpolation between the Möbius strip and the Klein bottle defined by means of three sets of bidimensional fields [Une interpolation entre le ruban de Möbius et la bouteille de Klein 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 Möbius strip and the Klein bottle, the three sets of {X,Y,Z} fields/matrices are as follows:



See some of the interpolated surfaces:

The Möbius strip defined by means of three bidimensional fields A surface between the Möbius strip and the Klein bottle defined by means of three bidimensional fields A surface between the Möbius strip and the Klein bottle defined by means of three bidimensional fields A surface between the Möbius strip and the Klein bottle defined by means of three bidimensional fields The Klein bottle defined by means of three bidimensional fields

See some artistic views of this dynamics:

From the Möbius strip to the Klein bottle From the Klein bottle to the Möbius strip


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