Génération et Animation d'Entrelacs,

Effets Larsen et plus

Jean-François COLONNA
[Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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[in english/en anglais]

Résumé : Comment construire et animer des entrelacs ? Comment générer des effets Larsen ?

Mots-Clefs : Picture Synthesis, Entrelacs, Intertwinings.

Plan de ce document :

1-UNE LIBRAIRIE DE MOTIFS ELEMENTAIRES :
2-DEFINITION DE TRANSFORMATIONS :
3-ENTRELACS "BIDIMENSIONNELS" :
4-ENTRELACS "PSEUDO-TRIDIMENSIONNELS" :
5-ENTRELACS "BIDIMENSIONNELS" DYNAMIQUES :
6-ENTRELACS "PSEUDO-TRIDIMENSIONNELS" DYNAMIQUES :
7-ENTRELACS "TRIDIMENSIONNELS" :
8-EFFETS LARSEN "BIDIMENSIONNELS" :
9-EFFETS DIVERS :

1-UNE LIBRAIRIE DE MOTIFS ELEMENTAIRES :

2-DEFINITION DE TRANSFORMATIONS :

PR[x,y] = PA[FX[x,y],FY[x,y]]

bidimensionnel

```
FX[x,y] = A11.PX[x,y] + A12.PY[x,y]
```
```
FY[x,y] = A21.PX[x,y] + A22.PY[x,y]
```

pseudo-tridimensionnel

FX[x,y] = A11.PX[x,y] + A12.PY[x,y] + A13.PZ[x,y]

FY[x,y] = A21.PX[x,y] + A22.PY[x,y] + A23.PZ[x,y]

```
PX[PX[x,y],PY[x,y]]
```
```
PY[PX[x,y],PY[x,y]]
```

tridimensionnel

PR[x,y,z] = PA[FX[x,y,z],FY[x,y,z],FZ[x,y,z]]

FX[x,y,z] = A11.PX[x,y,z] + A12.PY[x,y,z] + A13.PZ[x,y,z]

FY[x,y,z] = A21.PX[x,y,z] + A22.PY[x,y,z] + A23.PZ[x,y,z]

FZ[x,y,z] = A31.PX[x,y,z] + A32.PY[x,y,z] + A33.PZ[x,y,z]

3-ENTRELACS "BIDIMENSIONNELS" :

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

(PR=) = (PX=,PY=)(PA=) and then a transposition.

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

(PR=) = (PX=,PY=)(PA=TapestryEffect(CenterOf()))

(PR=) = (PX=,PY=)(PA=TapestryEffect(CenterOf()))

(PR=) = (PX=,PY=)(PA=TapestryEffect(CenterOf()))

(PR=) = (PX=,PY=)(PA=TapestryEffect(CenterOf()))

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = MountainEffect((PX=,PY=)(PA=))
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

4-ENTRELACS "PSEUDO-TRIDIMENSIONNELS" :

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

(PR=) =TapestryEffect((PX=,PY=,PZ=)(PA=))

(PR=) =MountainEffect(TapestryEffect((PX=,PY=,PZ=)(PA=)))

Un plan -courbure nulle-.

Une sphère -courbure positive-.

La triple bouteille de Bonan-Jeener-Klein.

La surface de Boy.

5-ENTRELACS "BIDIMENSIONNELS" DYNAMIQUES :

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

6-ENTRELACS "PSEUDO-TRIDIMENSIONNELS" DYNAMIQUES :

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

7-ENTRELACS "TRIDIMENSIONNELS" :

(PR=) = (PX=,PY=,PZ=)(PA=) avec un double effet miroir.

```
(PR=) = (PX=,PY=,PZ=)(PA=).
```
```
(PR=) = (PX=,PY=,PZ=)(PA=).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```
```
(PR=) = (PX=,PY=,PZ=)(PA==).
```

Une sphère -courbure positive-.

La triple bouteille de Bonan-Jeener-Klein.

8-EFFETS LARSEN "BIDIMENSIONNELS" :

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

9-EFFETS DIVERS :

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=,PZ=)(PA=)
```
```
(PR=) = (PX=,PY=,PZ=)(PA=)
```

```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```
```
(PR=) = (PX=,PY=)(PA=)
```

Génération et Animation d'Entrelacs, Effets Larsen et plus

Jean-François COLONNA [Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

1-UNE LIBRAIRIE DE MOTIFS ELEMENTAIRES :

2-DEFINITION DE TRANSFORMATIONS :

3-ENTRELACS "BIDIMENSIONNELS" :

4-ENTRELACS "PSEUDO-TRIDIMENSIONNELS" :

5-ENTRELACS "BIDIMENSIONNELS" DYNAMIQUES :

6-ENTRELACS "PSEUDO-TRIDIMENSIONNELS" DYNAMIQUES :

7-ENTRELACS "TRIDIMENSIONNELS" :

8-EFFETS LARSEN "BIDIMENSIONNELS" :

9-EFFETS DIVERS :

Copyright © Jean-François COLONNA, 2008-2024. Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2008-2024.

Génération et Animation d'Entrelacs,

Effets Larsen et plus

Jean-François COLONNA
[Contact me]

Copyright © Jean-François COLONNA, 2008-2024.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2008-2024.