by Means of Pseudo-Projections,

Picture Self-Transformations

Jean-François COLONNA

jean-francois.colonna@polytechnique.edu

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France

france telecom, France Telecom R&D

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(CMAP28 WWW site: this page was created on 12/24/2004 and last updated on 04/13/2016 12:27:47 -CEST-)

- 1-PICTURES:
- 2-TRIDIMENSIONAL SURFACES (BIDIMENSIONAL MANIFOLDS):
- 3-ANIMATION OF TRIDIMENSIONAL SURFACES (BIDIMENSIONAL MANIFOLDS):
- 4-TRIDIMENSIONAL MANIFOLDS:
- 5-PICTURE SELF-TRANSFORMATIONS:

X = F (u,v) x

Y = F (u,v) y

Z = F (u,v) zwith:

u E [U ,U ] min max

v E [V ,V ] min maxFor example:

F (u,v) = R.sin(u).cos(v) x

F (u,v) = R.sin(u).sin(v) y

F (u,v) = R.cos(u) zwith:

u E [0,pi]

v E [0,2.pi]defines a sphere with R as the radius and the origin of the coordinates as the center.

[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) zwith:

i = f(u,U ,U ,N ) min max u

j = g(v,V ,V ,N ) min max vwhere 'f' and 'g' denote two obvious linear functions...

Then it is possible

Let's give a deeper understanding of this process using the case of the sphere. Here are the three matrices Mx, My and Mz:

- Mx(i,j) = = Fx(u,v) = sin(u).cos(v)
- My(i,j) = = Fy(u,v) = sin(u).sin(v)
- Mz(i,j) = = Fz(u,v) = cos(u)

The "additive" superposition (red+green+blue) of these three matrices (black and white pictures) gives the following color picture:

+ + =

that is the color picture associated to the sphere (its

The surface colors are obviously arbitrary. But for all examples here provided, the gradients of the three functions Fx, Fy and Fz are locally defining the three Red, Green and Blue components.

Here are some numerous examples:

- == a plane.
- == a distorted plane.
- == a "pseudo-gaussian plane".
- == a "pseudo-bi-gaussian plane".
- == a "spiraling plane".
- == a Peano "fractal plane".
- == a "fractal plane" with overhangings.
- == a "fractal plane" with overhangings.
- == the relief -modulus- of the function exp(1/z).
- == the relief -modulus- of the function sin(z).
- == a crumpled cylinder (some fractal "noise" added).
- == a torus.
- == a crumpled torus (some fractal "noise" added).
- == the second 'power' of a torus.
- == the third 'power' of a torus.
- == the modulus of the complex sine function.
- == the Möbius strip.
- == the Klein bottle.
- ==
the Klein bottle (the coloring is made using the
*pseudo-projection*itself). - == a crumpled Klein bottle (some fractal "noise" added).
- == the double Jeener bottle.
- == the Jeener's triple Klein bottle.
- == the Jeener's quadruple bilateral bottle.
- == the Jeener's quintuple Klein bottle.
- == the Bonan-Jeener's triple Klein bottle.
- == a shell (Jeener surface 1).
- == the Boy surface.
- == a fractal surface (two iterations).
- == a fractal surface (three iterations).
- == a fractal surface (four iterations).
- == a fractal surface (five iterations).
- == a fractal surface (six iterations).
- == a fractal surface (numerous iterations...).
- == a linear mixing of a sphere and of a torus.
- == a linear mixing of a torus and of a sphere.
- == a gaussian mixing of a sphere and of the Möbius strip.
- == a Fourier interpolation between a "double sphere" and the Bonan-Jeener's triple Klein bottle.
- == the addition of a sphere and of the Bonan-Jeener's triple Klein bottle.
- == a fractal interpolation between a "double sphere" and the Bonan-Jeener's triple Klein bottle.

Here are some examples:

- == a dynamics of a 'pseudo-bi-gaussian plane'.
- => => == a dynamic crumpled Klein bottle(some fractal "noise" added).
- == a dynamic fractal surface (four iterations).
- == a dynamical gaussian smoothing of the Bonan-Jeener's triple Klein bottle.
- == an interpolation between a rectangle and a rounded cube.
- == an interpolation between a 'double sphere' and a cylinder.
- == an interpolation between a rectangle and a cylinder.
- == an interpolation between a cylinder and a torus.
- == an interpolation between a rectangle and a torus.
- == an interpolation between a rectangle and the Möbius strip.
- == an interpolation between the Möbius strip and a "double sphere".
- == an interpolation between the Möbius strip and the Klein bottle.
- == an interpolation between a "double sphere" and a torus.
- == an interpolation between a shell (Jeener surface 1) and a "double sphere".
- == an interpolation between the Klein bottle and a "double sphere".
- == an interpolation between the Klein bottle and the Klein bottle -the two Klein bottles being defined by means of two different sets of equations-.
- == an interpolation between the Klein bottle and the double Jeener bottle.
- == an interpolation between the quadruple Jeener bottle and the octuple Jeener bottle.
- == an interpolation between the Bonan-Jeener's triple Klein bottle and a "double sphere", that can be seen too as a tridimensional manifold.
- == the evolution of a sphere using the Lorenz attractor.
- == the evolution of the Klein bottle using the Lorenz attractor.

At last, it is obvious that a

Here are some examples:

- == the Olympic pseudo-Rings.
- == the Borromean Rings.
- == four interlaced torus.
- == a dynamics of four interlaced torus.
- == a dynamics of four interlaced torus.
- == an interpolation between four interlaced torus and the quadruple Jeener bottle.
- == the extended Borromean Rings.
- == sixteen interlaced torus.
- == an interpolation between two sets of interlaced torus (four and sixteen).
- == an interpolation between sixteen interlaced torus and the quadruple Jeener bottle.
- == sixty-four interlaced torus.

Holes and distorsions of the 'u' and 'v' coordinates can be added as on these examples:

At last, this process facilitates the interpolation between surfaces as on this example:

Many tridimensional manifolds in a tridimensional space can be defined using a set of three equations:

X = F (u,v,w) x

Y = F (u,v,w) y

Z = F (u,v,w) zwith:

u E [U ,U ] min max

v E [V ,V ] min max

w E [W ,W ] min max[Umin,Umax]x[Vmin,Vmax]x[Wmin,Wmax] then defined a tridimensional rectangular domain D. If D is sampled by means of a tridimensional rectangular grid (made of Nu.Nv.Nw points), the three {X,Y,Z} coordinates can be defined by means of a set of Nw rectangular matrice triplets:

k X = {M (i,j)} x

k Y = {M (i,j)} y

k Z = {M (i,j)} zwith:

i = f(u,U ,U ,N ) min max u

j = g(v,V ,V ,N ) min max v

k = g(w,W ,W ,N ) min max wwhere 'f', 'g' and 'h' denote two obvious linear functions...

Then it is possible

Here are some examples:

- == a tridimensional twisted torus-like manifold.
- == a tridimensional Möbius-like manifold.
- == a Jeener-Möbius tridimensional manifold.
- == a Jeener-Möbius tridimensional manifold.
- == a tridimensional mesh.
- == a tridimensional distorted mesh.
- == a tridimensional fractal manifold.
- == a tridimensional manifold defined by means of an interpolation between the Bonan-Jeener's triple Klein bottle and a "double sphere".

Here is the used process:

X = F (u,v) x

Y = F (u,v) y

Z = F (u,v) zThe tridimensional coordinates {X,Y,Z} are then projected:

X = ProjectionX(X,Y,Z) p

Y = ProjectionY(X,Y,Z) pThe distorsion is the defined as:

DISTORSION(u,v) = TEXTURE(X ,Y ) p p'TEXTURE' being the picture "texture" to be self-transformed.

Here are some examples of picture self-transformations without displaying the associated surfaces:

- =>
- =>
- =>
- =>
- =>
- =>
- => (in this example, a Fourier filtering was applied to the initial picture in order to smooth it ).

This process can be applied to color animations thus giving birth to

Here are some examples:

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