
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)
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]
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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 E [0,pi]
when:
v E [0,2.pi]
See the perfect sphere.
(CMAP28 WWW site: this page was created on 11/17/2019 and last updated on 08/22/2020 11:15:27 -CEST-)
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