A bidimensional Peano-like curve defined with {X4(...),Y4(...)} -iteration 4- [Une courbe bidimensionnelle du type Peano définie avec {X4(...),Y4(...)} -itération 4-].

## The bidimensional Peano-like Curves:

Let's C1(T) being a parametric curve defined by means of 2 real functions of T (T E [0,1]) X1(T) E [0,1] and Y1(T) E [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)} E [0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T)} E [0,1]x[0,1]

if T E [0,1/4[:
Xi+1(T) =   Yi(4T-0)
Yi+1(T) =   Xi(4T-0)
Transformation 1
if T E [1/4,2/4[:
Xi+1(T) =   Xi(4T-1)
Yi+1(T) = 1+Yi(4T-1)
Transformation 2
if T E [2/4,3/4[:
Xi+1(T) = 1+Xi(4T-2)
Yi+1(T) = 1+Yi(4T-2)
Transformation 3
if T E [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.

See a special C1(T) curve in order to understand the geometrical meaning of the 4 transformations and of their order .

See the 6 first Ci(T) curves (i E {1,2,3,4,5,6}):

See the used color set to display the parameter T.

## The tridimensional Peano-like Curves:

Let's C1(T) being a parametric curve defined by means of 3 real functions of T (T E [0,1]) X1(T) E [0,1], Y1(T) E [0,1] and Z1(T) E [0,1 such as :
X1(T=0)=0 Y1(T=0)=0 Z1(T=0)=0 (lower left foreground corner)
X1(T=1)=0 Y1(T=1)=0 Z1(T=1)=1 (lower right foreground corner)

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

if T E [0,1/8[:
Xi+1(T) =   Xi(8T-0)
Yi+1(T) =   Zi(8T-0)
Zi+1(T) =   Yi(8T-0)
Transformation 1
if T E [1/8,2/8[:
Xi+1(T) =   Zi(8T-1)
Yi+1(T) = 1+Yi(8T-1)
Zi+1(T) =   Xi(8T-1)
Transformation 2
if T E [2/8,3/8[:
Xi+1(T) = 1+Xi(8T-2)
Yi+1(T) = 1+Yi(8T-2)
Zi+1(T) =   Zi(8T-2)
Transformation 3
if T E [3/8,4/8[:
Xi+1(T) = 1+Zi(8T-3)
Yi+1(T) = 1-Xi(8T-3)
Zi+1(T) = 1-Yi(8T-3)
Transformation 4
if T E [4/8,5/8[:
Xi+1(T) = 2-Zi(8T-4)
Yi+1(T) = 1-Xi(8T-4)
Zi+1(T) = 1+Yi(8T-4)
Transformation 5
if T E [5/8,6/8[:
Xi+1(T) = 1+Xi(8T-5)
Yi+1(T) = 1+Yi(8T-5)
Zi+1(T) = 1+Zi(8T-5)
Transformation 6
if T E [6/8,7/8[:
Xi+1(T) = 1-Zi(8T-6)
Yi+1(T) = 1+Yi(8T-6)
Zi+1(T) = 2-Xi(8T-6)
Transformation 7
if T E [7/8,1]:
Xi+1(T) =   Xi(8T-7)
Yi+1(T) = 1-Zi(8T-7)
Zi+1(T) = 2-Yi(8T-7)
Transformation 8

Please note that 8=2d where d=3 is the space dimension.

See a special C1(T) curve in order to understand the geometrical meaning of the 8 transformations and of their order .

See the 5 first Ci(T) curves (i E {1,2,3,4,5}):

See the used color set to display the parameter T.

See various bidimensional Hilbert and Peano curves (possibly including this one):

See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.

See various tridimensional Hilbert and Peano curves (possibly including this one):

See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.
See the used color set to display the parameter T.

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