
Bidimensional Hilbert Curve -iterations 1 to 5- [Courbe de Hilbert bidimensionnelle -itérations 1 à 5-].
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.
The bidimensional Peano Surjection:
Peano defined the following surjection:
[0,1] --> [0,1]x[0,1]
Let's T being a real number defined using the base 3:
T = 0.A1A2A3... E [0,1] with Ai E {0,1,2}
Let's X(T) and Y(T) being 2 real functions of T defined as:
X(T) = 0.B1B2B3... E [0,1] with Bi E {0,1,2}
Y(T) = 0.C1C2C3... E [0,1] with Ci E {0,1,2}
with:
Bn = A2n-1 if A2+A4+...+A2n-0 is even
Bn = 2-A2n-1 otherwise
Cn = A2n if A1+A3+...+A2n-1 is even
Cn = 2-A2n otherwise
These 2 functions X(T) and Y(T) are the coordinates of a point P(T) inside the [0,1]x[0,1] square.
The displayed "curve" is the trajectory of P(T) -displayed as little spheres- when T varies from 0 (lower left corner) to 1-epsilon (upper right corner).
See some related pictures (possibly including this one):
See the used color set to display the parameter T.
The tridimensional Peano Surjection:
A tridimensional surjection can be defined:
[0,1] --> [0,1]x[0,1]x[0,1]
as a generalization of the bidimensional one.
Let's T being a real number defined using the base 3:
T = 0.A1A2A3... E [0,1] with Ai E {0,1,2}
Let's X(T), Y(T) and Z(T) being 3 real functions of T defined as:
X(T) = 0.B1B2B3... E [0,1] with Bi E {0,1,2}
Y(T) = 0.C1C2C3... E [0,1] with Ci E {0,1,2}
Y(T) = 0.D1D2D3... E [0,1] with Di E {0,1,2}
with:
Bn = A3n-2 if A3+A6+...+A3n-0 is even
Bn = 2-A3n-2 otherwise
Cn = A3n-1 if A2+A5+...+A3n-1 is even
Cn = 2-A3n-1 otherwise
Dn = A3n if A1+A4+...+A3n-2 is even
Dn = 2-A3n otherwise
These 3 functions X(T), Y(T) and Z(T) are the coordinates of a point P(T) inside the [0,1]x[0,1]x[0,1] cube.
The displayed "curve" is the trajectory of P(T) -displayed as little spheres- when T varies from 0 (lower left corner) to 1-epsilon (upper right corner).
See some related pictures (possibly including this one):
See the used color set to display the parameter T.
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.
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