A 'crumpled' sphere defined by means of three bidimensional fields [Une sphère 'froissée' définie à l'aide de trois champs bidimensionnels].

Many surfaces -bidimensional manifolds- in a tridimensional space can be defined using a set of three equations:
                    X = F (u,v)
                    Y = F (u,v)
                    Z = F (u,v)
                    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 |      :                           :
                         |      :                           :
                                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)
                    Y = M (i,j)
                    Z = M (i,j)
                    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...

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For the 'crumpled' sphere, the three {X,Y,Z} fields/matrices are as follows:

with 'fractal(u,v)' denoting a bidimensional periodical fractal generator (fractal(u,v) E [1-0.5,1+0.5]).
The one used for the 'Z' coordinate () differs from the one used for the 'X' and 'Y' coordinates () in order to avoid discontinuities at the two poles.

Only the left half part of each field is used for:
                    u E [0,pi]
                    v E [0,2.pi]

See the perfect sphere.
See an helix on this 'crumpled' sphere.
See its normal field.

(CMAP28 WWW site: this page was created on 11/13/2004 and last updated on 08/22/2020 11:13:54 -CEST-)

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