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An interpolation between four interlaced torus and the quadruple Jeener bottle defined by means of three sets of bidimensional fields [Une interpolation entre quatre tores entrelacés et la quadruple 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 = 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...


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For the interpolation between four interlaced torus and the quadruple Jeener bottle, the three sets of {X,Y,Z} fields/matrices are as follows:



See some of the interpolated surfaces:

Four interlaced torus defined by means of three bidimensional fields A surface between four interlaced torus and the quadruple Jeener bottle defined by means of three bidimensional fields A surface between four interlaced torus and the quadruple Jeener bottle defined by means of three bidimensional fields A surface between four interlaced torus and the quadruple Jeener bottle defined by means of three bidimensional fields The quadruple Jeener bottle defined by means of three bidimensional fields


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