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In [[Mathematics|mathematics]], a '''de Rham curve''' is a certain type of [[Fractal|fractal]] [[Curve|curve]] named in honor of [[Biography:Georges de Rham|Georges de Rham]].
The [[Cantor function]], Cesàro curve, Minkowski's question mark function, the [[Lévy C curve]], the [[Blancmange curve|blancmange curve]] and the [[Koch snowflake|Koch curve]] are all special cases of the general de Rham curve.

==Construction==
Consider some [[Complete metric space|complete metric space]] <math>(M,d)</math> (generally <math>\mathbb{R}</math><sup>2</sup> with the usual euclidean distance), and a pair of [[Contraction mapping|contracting map]]s on M:

:<math>d_0:\ M \to M</math> 
:<math>d_1:\ M \to M.</math> 

By the Banach fixed point theorem, these have fixed points <math>p_0</math> and <math>p_1</math> respectively.  Let ''x'' be a [[Real number|real number]] in the interval <math>[0,1]</math>, having binary expansion

:<math>x = \sum_{k=1}^\infty \frac{b_k}{2^k},</math>

where each <math>b_k</math> is 0 or 1. Consider the map

:<math>c_x:\ M \to M</math>

defined by

:<math>c_x = d_{b_1} \circ d_{b_2} \circ \cdots \circ d_{b_k} \circ \cdots,</math>

where <math>\circ</math> denotes [[Function composition|function composition]]. It can be shown that each <math>c_x</math> will map the common basin of attraction of <math>d_0</math> and <math>d_1</math> to a single point <math>p_x</math> in <math>M</math>. The collection of points <math>p_x</math>, parameterized by a single real parameter ''x'', is known as the de Rham curve.

==Continuity condition==
When the fixed points are paired such that

:<math>d_0(p_1) = d_1(p_0)</math>

then it may be shown that the resulting curve <math>p_x</math> is a continuous function of ''x''. When the curve is continuous, it is not in general differentiable.

In the remaining of this page, we will assume the curves are continuous.

==Properties==
De Rham curves are by construction self-similar, since 
:<math>p(x)=d_0(p(2x))</math> for <math>x \in [0, 1/2]</math> and 
:<math>p(x)=d_1(p(2x-1))</math> for <math>x \in [1/2, 1].</math>

The self-symmetries of all of the de Rham curves are given by the [[Monoid|monoid]] that describes the symmetries of the infinite binary tree or [[Cantor set]].  This so-called period-doubling monoid is a subset of the [[Modular group|modular group]].

The [[Image (mathematics)|image]] of the curve, i.e. the set of points <math>\{p(x), x \in [0,1]\}</math>, can be obtained by an [[Iterated function system]] using the set of contraction mappings <math>\{d_0,\ d_1\}</math>. But the result of an iterated function system with two contraction mappings is a de Rham curve if and only if the contraction mappings satisfy the continuity condition.

==Classification and examples==

===Cesàro curves===
[[Image:Cesaro-0.3.png|thumb|right|Cesàro curve for ''a''&nbsp;=&nbsp;0.3&nbsp;+&nbsp;''i''&nbsp;0.3]]
[[Image:Cesaro-0.5.png|thumb|right|Cesàro curve for ''a''&nbsp;=&nbsp;0.5&nbsp;+&nbsp;''i''&nbsp;0.5]]

'''Cesàro curves''' (or '''Cesàro-Faber curves''') are De Rham curves generated by [[Affine transformation|affine transformation]]s conserving orientation, with fixed points <math>p_0=0</math> and <math>p_1=1</math>.

Because of these constraints, Cesàro curves are uniquely determined by a [[Complex number|complex number]] <math>a</math> such that <math>|a|<1</math> and <math>|1-a|<1</math>.

The contraction mappings <math>d_0</math> and <math>d_1</math> are then defined as complex functions in the [[Complex plane|complex plane]] by:

:<math>d_0(z) = az</math>
:<math>d_1(z) = a + (1-a)z.</math>

For the value of <math>a=(1+i)/2</math>, the resulting curve is the [[Lévy C curve]].

===Koch&ndash;Peano curves===
[[Image:Koch-Peano-0.37.png|thumb|right|Koch&ndash;Peano curve for ''a''&nbsp;=&nbsp;0.6&nbsp;+&nbsp;''i''&nbsp;0.37]]
[[Image:Koch-Peano-0.45.png|thumb|right|Koch&ndash;Peano curve for ''a''&nbsp;=&nbsp;0.6&nbsp;+&nbsp;''i''&nbsp;0.45]]

In a similar way, we can define the Koch&ndash;Peano family of curves as the set of De Rham curves generated by affine transformations reversing orientation, with fixed points <math>p_0=0</math> and <math>p_1=1</math>.

These mappings are expressed in the complex plane as a function of <math>\overline{z}</math>, the [[Complex conjugate|complex conjugate]] of <math>z</math>:

:<math>d_0(z) = a\overline{z}</math>
:<math>d_1(z) = a + (1-a)\overline{z}.</math>

The name of the family comes from its two most famous members. The [[Koch snowflake|Koch curve]] is obtained by setting:

:<math>a_\text{Koch}=\frac{1}{2} + i\frac{\sqrt{3}}{6},</math>

while the [[Peano curve]] corresponds to:

:<math>a_\text{Peano}=\frac{(1+i)}{2}.</math>

===General affine maps===
[[Image:Curve_0.33_-0.38_-0.18_-0.42.png|thumb|right|Generic affine de Rham curve]]
[[Image:Curve_-0.10_-0.80_-0.30_-0.60.png|thumb|right|Generic affine de Rham curve]]
[[Image:Curve_0.00_0.60_0.18_0.60.png|thumb|right|Generic affine de Rham curve]]
[[Image:Curve_-0.35_0.00_-0.35_0.00.png|thumb|right|Generic affine de Rham curve]]

The Cesàro-Faber and Peano-Koch curves are both special cases of the general case of a pair of affine linear transformations on the complex plane. By fixing one endpoint of the curve at 0 and the other at one, the general case is obtained by iterating on the two transforms

:<math>d_0=\begin{pmatrix}
1 & 0 & 0 \\ 
0 & \alpha &\delta \\ 
0 & \beta & \epsilon
\end{pmatrix}</math>

and
:<math>d_1=\begin{pmatrix}
1&0&0 \\ 
\alpha & 1-\alpha&\zeta \\ 
\beta&-\beta&\eta
\end{pmatrix}.</math>

Being affine transforms, these transforms act on a point <math>(u,v)</math> of the 2-D plane by acting on the vector

:<math>\begin{pmatrix}
1 \\
u \\
 v \end{pmatrix}.</math>

The midpoint of the curve can be seen to be located at <math>(u,v)=(\alpha,\beta)</math>; the other four parameters may be varied to create a large variety of curves.

The [[Blancmange curve|blancmange curve]] of parameter <math>w</math> can be obtained by setting <math>\alpha=\beta=1/2</math>, <math>\delta=\zeta=0</math> and <math>\epsilon=\eta=w</math>. That is:

:<math>d_0=\begin{pmatrix}
1&0&0 \\ 
0 & 1/2&0 \\ 
0&1/2&w
\end{pmatrix}</math>

and
:<math>d_1=\begin{pmatrix}
 1&0&0 \\ 
1/2 & 1/2&0 \\ 
1/2&-1/2&w
\end{pmatrix}.</math>

Since the blancmange curve of parameter <math>w=1/4</math> is the parabola of equation <math>f(x)=4x(1-x)</math>, this illustrate the fact that in some occasion, de Rham curves can be smooth.

===Minkowski's question mark function===
Minkowski's question mark function is generated by the pair of maps

:<math>d_0(z) = \frac{z}{z+1}</math>

and 

:<math>d_1(z)= \frac{1}{z+1}.</math>

== Generalizations ==
It is easy to generalize the definition by using more than two contraction mappings. If one uses ''n'' mappings, then the ''n''-ary decomposition of ''x'' has to be used instead of the binary expansion of real numbers. The continuity condition has to be generalized in:
:<math>d_i(p_{(n-1)})=d_{(i+1)}(p_0)</math>, for <math>i=0 \ldots n-2.</math>

Such a generalization allows, for example, to produce the [[Sierpiński arrowhead curve]] (whose image is the [[Sierpiński triangle]]), by using the contraction mappings of an iterated function system that produces the Sierpiński triangle.

== See also ==
* [[Iterated function system]]
* [[Refinable function]]
* [[Modular group]]
* [[Fuchsian group]]

== References ==
* Georges de Rham, ''On Some Curves Defined by Functional Equations'' (1957), reprinted in ''Classics on Fractals'', ed. Gerald A. Edgar (Addison-Wesley, 1993), pp. 285&ndash;298.
* Georges de Rham,  ''Sur quelques courbes definies par des equations fonctionnelles''. Univ. e Politec. Torino. Rend. Sem. Mat., 1957, 16, 101 &ndash;113
* Linas Vepstas, ''[http://linas.org/math/de_Rham.pdf A Gallery of de Rham curves]'', (2006).
* Linas Vepstas, ''[http://linas.org/math/chap-takagi.pdf Symmetries of Period-Doubling Maps]'', (2006). ''(A general exploration of the modular group symmetry in fractal curves.)''

[[Category:Fractal curves]]

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