A Bidimensional Hilbert-like Curve defined with {X1(...),Y1(...)} -iteration 1- [Une courbe bidimensionnelle du type Hilbert définie avec {X1(...),Y1(...)} -itération 1-].

# The bidimensional Hilbert Curves:

Let's C1(T) being a parametric curve defined by means of 2 real functions of T (T [0,1]) X1(T) [0,1] and Y1(T) [0,1] such as :
```                    X1(T=0)=0 Y1(T=0)=0 (lower left corner)
```
```                    X1(T=1)=1 Y1(T=1)=0 (lower right corner)
```

Then one defines a sequence of curves Ci(T) (i >= 1) as follows :
```                    Ci(T) = {Xi(T),Yi(T)} ∈ [0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T)} ∈ [0,1]x[0,1]
```

```                    if T ∈ [0,1/4[:
Xi+1(T) =   Yi(4T-0)
Yi+1(T) =   Xi(4T-0)
Transformation 1
```
```                    if T ∈ [1/4,2/4[:
Xi+1(T) =   Xi(4T-1)
Yi+1(T) = 1+Yi(4T-1)
Transformation 2
```
```                    if T ∈ [2/4,3/4[:
Xi+1(T) = 1+Xi(4T-2)
Yi+1(T) = 1+Yi(4T-2)
Transformation 3
```
```                    if T ∈ [3/4,1]:
Xi+1(T) = 2-Yi(4T-3)
Yi+1(T) = 1-Xi(4T-3)
Transformation 4
```

Please note that 4=2d where d=2 is the space dimension.

Here are the five first bidimensional Hilbert curves with an increasing number of iterations :

[See the used color set to display the parameter T]

Here are some examples of Hilbert-like bidimensional curves using different generating curves :

(CMAP28 WWW site: this page was created on 04/29/2022 and last updated on 07/04/2023 12:13:42 -CEST-)

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