![](image.jpg)
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 = 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 'crumpled' sphere, the three {X,Y,Z} fields/matrices are as follows:
with a bidimensional periodical fractal generator (
) being applied only on the radius coordinate.
Only the left half part of each field is used for:
u ∈ [0,pi]
when:
v ∈ [0,2.pi]
See the perfect sphere.
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