A 'crumpled' torus defined by means of three bidimensional fields [Un tore 'froissé' défini 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...

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

• Fx = fractal(u,v).(R+r.cos(u)).cos(v)
• Fy = fractal(u,v).(R+r.cos(u)).sin(v)
• Fz = fractal(u,v).r.sin(u)
with 'fractal(u,v)' denoting a bidimensional periodical fractal generator (fractal(u,v) E [1-0.5,1+0.5]).
The same one was used () for the 'X', 'Y' and 'Z' coordinates.

See a related picture:

See the perfect torus.

(CMAP28 WWW site: this page was created on 04/01/2021 and last updated on 04/02/2021 10:20:38 -CEST-)

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