click to view the MPEG movie (cliquez pour voir le film MPEG)

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 = Fx(u,v)
                    Y = Fy(u,v)
                    Z = Fz(u,v)
with:
                    u  [Umin,Umax]
                    v  [Vmin,Vmax]
[Umin,Umax]*[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 = Mx(i,j)
                    Y = My(i,j)
                    Z = Mz(i,j)
with:
                    i = f(u,Umin,Umax,Nu)
                    j = g(v,Vmin,Vmax,Nv)
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


(CMAP28 WWW site: this page was created on 04/19/2006 and last updated on 04/16/2023 20:52:00 -CEST-)



[More information about that kind of picture and/or process [Plus d'informations sur ce type d'image et/ou de processus]]


[See the generator of this picture [Voir le générateur de cette image]]

[See all related pictures (including this one) [Voir toutes les images associées (incluant celle-ci)]]


[Go back to AVirtualMachineForExploringSpaceTimeAndBeyond [Retour à AVirtualMachineForExploringSpaceTimeAndBeyond]]

[The Y2K Bug [Le bug de l'an 2000]]

[Site Map, Help and Search [Plan du Site, Aide et Recherche]]
[Mail [Courrier]]
[About Pictures and Animations [A Propos des Images et des Animations]]


Copyright © Jean-François Colonna, 2006-2023.
Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2006-2023.