Hilbert and Peano Space filling Curves and Beyond:
From Squares and Cubes to Surfaces (bidimensional Manifolds) and tridimensional Manifolds
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
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Contents:
1-The bidimensional Peano Surjection:
Giuseppe 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... ∈ [0,1] with Ai ∈ {0,1,2}
Let's X(T) and Y(T) being two real functions of T defined as
:
X(T) = 0.B1B2B3... ∈ [0,1] with Bi ∈ {0,1,2}
Y(T) = 0.C1C2C3... ∈ [0,1] with Ci ∈ {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 two 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" -as little spheres- is the trajectory of P(T) when T varies from 0 (lower left corner) to 1-epsilon (upper right corner).
Here are the four first bidimensional Peano curves with an increasing number of digits
{2,4,6,8}:
[See the used color set to display the parameter T]
2-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... ∈ [0,1] with Ai ∈ {0,1,2}
Let's X(T), Y(T) and Z(T) being three real functions of T defined as:
X(T) = 0.B1B2B3... ∈ [0,1] with Bi ∈ {0,1,2}
Y(T) = 0.C1C2C3... ∈ [0,1] with Ci ∈ {0,1,2}
Y(T) = 0.D1D2D3... ∈ [0,1] with Di ∈ {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 three 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).
Here are the three first tridimensional Peano curves with an increasing number of digits
{3,6,9}:
[See the used color set to display the parameter T]
3-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.
See a special C1(T) curve 
in order to understand the geometrical meaning of the 4 transformations 
and of their order 
.
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
:
4-The tridimensional Hilbert Curves:
Let's C1(T) being
a parametric curve
defined by means of 3 real functions of T
(T ∈ [0,1])
X1(T) ∈ [0,1], Y1(T) ∈ [0,1] and Z1(T) ∈ [0,1]
such as
:
X1(T=0)=0 Y1(T=0)=0 Z1(T=0)=0 (lower left foreground corner)
X1(T=1)=1 Y1(T=1)=0 Z1(T=1)=0 (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)} ∈ [0,1]x[0,1]x[0,1] --> Ci+1(T) = {Xi+1(T),Yi+1(T),Zi+1(T)} ∈ [0,1]x[0,1]x[0,1]
if T ∈ [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 ∈ [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 ∈ [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 ∈ [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 ∈ [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 ∈ [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 ∈ [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 ∈ [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 
.
Here are the four first tridimensional 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 tridimensional curves using different generating curves and in
particular "open" knots
:
[More information about Peano Curves and Infinite Knots -in english/en anglais-]
[Plus d'informations à propos des Courbes de Peano et des Nœuds Infinis -en français/in french-]
5-Tridimensional Surfaces (Bidimensional Manifolds):
Many surfaces -bidimensional manifolds- in a tridimensional space
can be defined using a set of three equations:
X = Fx(u,v)
Y = Fy(u,v)
Z = Fz(u,v)
with:
u ∈ [Umin,Umax]
v ∈ [Vmin,Vmax]
For example:
Fx(u,v) = R.sin(u).cos(v)
Fy(u,v) = R.sin(u).sin(v)
Fz(u,v) = R.cos(u)
with:
u ∈ [0,pi]
v ∈ [0,2.pi]
defines a sphere with R as the radius and the origin of the coordinates as the center.
[Umin,Umax]*[Vmin,Vmax] then defined a bidimensional rectangular domain D.
v ^
|
V |...... ---------------------------
max | |+++++++++++++++++++++++++++|
| |+++++++++++++++++++++++++++|
| |+++++++++++++++++++++++++++|
| |+++++++++++++++++++++++++++|
| |+++++++++++++++++++++++++++|
| |+++++++++++++++++++++++++++|
| |+++++++++++++++++++++++++++|
| |+++++++++++++++++++++++++++|
| |+++++++++++++++++++++++++++|
V |...... ---------------------------
min | : :
| : :
O------------------------------------------------->
U U u
min max
Let's define a curve that fills the [0,1]x[0,1] square
:
^
|
1 |---------------
| +++++ +++++ |
| + + + + |
| + +++++ + |
| + + |
| +++++ +++++ |
| + + |
| +++++ +++++ |
0 O------------------------>
0 1
Obviously one can define a mapping between the [0,1]x[0,1] square and the [Umin,Umax]*[Vmin,Vmax] domain
:
v ^
|
V |...... ---------------------------
max | | ++++++++ ++++++++ |
| | + + + + |
| | + + + + |
| | + +++++++++++ + |
| | + C + |
| | ++++++++ ++++++++ |
| | + + |
| | + + |
| | ++++++++ ++++++++ |
V |...... ---------------------------
min | : :
| : :
O------------------------------------------------->
U U u
min max
Then it suffices to display only the points {Fx(u,v),Fy(u,v),Fz(u,v)}
with {u,v} on the preceding curve C to fill the surface with C....
Here are some examples of this process
:
| Surface | ==> | C Curve | ==> | Surface filling Curve |
 | ==> |  | ==> |  |
| | ==> | | ==> |  |
| | ==> |  | ==> |  |
| | ==> |  | ==> |  |
| | ==> |  | ==> |  |
| | ==> |  | ==> |  |
| | ==> | | ==> |  |
| | ==> | | ==> |  |
6-Tridimensional Manifolds:
Many tridimensional manifolds in a tridimensional space
can be defined using a set of three equations:
X = Fx(u,v,w)
Y = Fy(u,v,w)
Z = Fz(u,v,w)
with:
u ∈ [Umin,Umax]
v ∈ [Vmin,Vmax]
w ∈ [Wmin,Wmax]
[Umin,Umax]*[Vmin,Vmax]*[Wmin,Wmax] then defined a tridimensional rectangular domain D.
It is obvious to generalize the preceding bidimensional process in the tridimensional space...
Here are some examples of this process
:
| Tridimensional Manifold | ==> | C Curve | ==> | Tridimensional Manifold filling Curve |
7-Beyond:
Instead of using space-filling curves in order to fill the {u,v} and {u,v,w} bi- and tridimensional domains,
obviously one can use any means available and, for example, the bi-
and the tridimensional brownian motions respectively....
Here are some examples of these extended processes
:
Copyright © Jean-François COLONNA, 2023-2024.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2023-2024.
