Generation and Animation of Intertwinings,

Larsen Effects and more

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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[en français/in french]

Abstract: How to build and animate intertwinings? How to generate Larsen effects?

Keywords: Picture Synthesis, Entrelacs, Intertwinings.

Contents of this page:

1-A LIBRARY OF ELEMENTARY SYMBOLS:
2-DEFINITION OF TRANSFORMATIONS:
3-"BIDIMENSIONAL" INTERTWININGS:
4-"PSEUDO-TRIDIMENSIONAL" INTERTWININGS:
5-"BIDIMENSIONAL" DYNAMICAL INTERTWININGS:
6-"PSEUDO-TRIDIMENSIONAL" DYNAMICAL INTERTWININGS:
7-"TRIDIMENSIONAL" INTERTWININGS:
8-"BIDIMENSIONAL" LARSEN EFFECTS:
9-MISCELLANEOUS EFFECTS:

1-A LIBRARY OF ELEMENTARY SYMBOLS:

2-DEFINITION OF TRANSFORMATIONS:

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

bidimensional

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

pseudo-tridimensional

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]]
```

tridimensional

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-"BIDIMENSIONAL" INTERTWININGS:

```
(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-"PSEUDO-TRIDIMENSIONAL" INTERTWININGS:

```
(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=)))

A plane -zero curvature-.

A sphere -positive curvature-.

The Bonan-Jeener-Klein triple bottle.

The Boy surface.

5-"BIDIMENSIONAL" DYNAMICAL INTERTWININGS:

```
(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-"PSEUDO-TRIDIMENSIONAL" DYNAMICAL INTERTWININGS:

```
(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-"TRIDIMENSIONAL" INTERTWININGS:

(PR=) = (PX=,PY=,PZ=)(PA=) with a double mirror effect.

```
(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==).
```

A sphere -positive curvature-.

The Bonan-Jeener-Klein triple bottle.

8-"BIDIMENSIONAL" LARSEN EFFECTS:

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

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

9-MISCELLANEOUS EFFECTS:

```
(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=)
```

Generation and Animation of Intertwinings, Larsen Effects and more

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-A LIBRARY OF ELEMENTARY SYMBOLS:

2-DEFINITION OF TRANSFORMATIONS:

3-"BIDIMENSIONAL" INTERTWININGS:

4-"PSEUDO-TRIDIMENSIONAL" INTERTWININGS:

5-"BIDIMENSIONAL" DYNAMICAL INTERTWININGS:

6-"PSEUDO-TRIDIMENSIONAL" DYNAMICAL INTERTWININGS:

7-"TRIDIMENSIONAL" INTERTWININGS:

8-"BIDIMENSIONAL" LARSEN EFFECTS:

9-MISCELLANEOUS EFFECTS:

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

Generation and Animation of Intertwinings,

Larsen Effects and more

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.