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| In [[mathematics]], a '''de Rham curve''' is a certain type of [[fractal]] [[curve]] named in honor of [[Georges de Rham]].
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| The [[Cantor function]], Césaro curve, [[Minkowski's question mark function]], the [[Lévy C curve]], the [[blancmange curve]] and the [[Koch snowflake|Koch curve]] are all special cases of the general de Rham curve.
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| ==Construction==
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| Consider some [[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:
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| :<math>d_0:\ M \to M</math>
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| :<math>d_1:\ M \to M.</math>
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| 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]] in the interval <math>[0,1]</math>, having binary expansion
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| :<math>x = \sum_{k=1}^\infty \frac{b_k}{2^k},</math>
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| where each <math>b_k</math> is 0 or 1. Consider the map
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| :<math>c_x:\ M \to M</math>
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| defined by
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| :<math>c_x = d_{b_1} \circ d_{b_2} \circ \cdots \circ d_{b_k} \circ \cdots,</math>
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| where <math>\circ</math> denotes [[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.
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| ==Continuity Condition==
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| When the fixed points are paired such that
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| :<math>d_0(p_1) = d_1(p_0)</math>
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| 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.
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| In the remaining of this page, we will assume the curves are continuous.
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| ==Properties==
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| De Rham curves are by construction self-similar, since
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| :<math>p(x)=d_0(p(2x))</math> for <math>x \in [0, 0.5]</math> and
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| :<math>p(x)=d_1(p(2x-1))</math> for <math>x \in [0.5, 1].</math>
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| The self-symmetries of all of the de Rham curves are given by the [[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]].
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| 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.
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| ==Classification and examples==
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| ===Césaro curves===
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| [[Image:Cesaro-0.3.png|thumb|right|Césaro curve for ''a'' = 0.3 + ''i'' 0.3]]
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| [[Image:Cesaro-0.5.png|thumb|right|Césaro curve for ''a'' = 0.5 + ''i'' 0.5]]
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| '''Césaro curves''' (or '''Césaro-Faber curves''') are De Rham curves generated by [[affine transformation]]s conserving [[orientation (mathematics)|orientation]], with fixed points <math>p_0=0</math> and <math>p_1=1</math>.
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| Because of these constraints, Césaro curves are uniquely determined by a [[complex number]] <math>a</math> such that <math>|a|<1</math> and <math>|1-a|<1</math>.
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| The contraction mappings <math>d_0</math> and <math>d_1</math> are then defined as complex functions in the [[complex plane]] by:
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| :<math>d_0(z) = az</math>
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| :<math>d_1(z) = a + (1-a)z.</math>
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| For the value of <math>a=(1+i)/2</math>, the resulting curve is the [[Lévy C curve]].
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| ===Koch–Peano curves===
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| [[Image:Koch-Peano-0.37.png|thumb|right|Koch–Peano curve for ''a'' = 0.6 + ''i'' 0.37]]
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| [[Image:Koch-Peano-0.45.png|thumb|right|Koch–Peano curve for ''a'' = 0.6 + ''i'' 0.45]]
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| In a similar way, we can define the Koch–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>.
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| These mappings are expressed in the complex plane as a function of <math>\overline{z}</math>, the [[complex conjugate]] of <math>z</math>:
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| :<math>d_0(z) = a\overline{z}</math>
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| :<math>d_1(z) = a + (1-a)\overline{z}.</math>
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| The name of the family comes from its two most famous members. The [[Koch snowflake|Koch curve]] is obtained by setting:
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| :<math>a_\text{Koch}=\frac{1}{2} + i\frac{\sqrt{3}}{6},</math>
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| while the [[Peano curve]] corresponds to:
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| :<math>a_\text{Peano}=\frac{(1+i)}{2}.</math>
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| ===General affine maps===
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| [[Image:Curve_0.33_-0.38_-0.18_-0.42.png|thumb|right|Generic affine de Rham curve]]
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| [[Image:Curve_-0.10_-0.80_-0.30_-0.60.png|thumb|right|Generic affine de Rham curve]]
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| [[Image:Curve_0.00_0.60_0.18_0.60.png|thumb|right|Generic affine de Rham curve]]
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| [[Image:Curve_-0.35_0.00_-0.35_0.00.png|thumb|right|Generic affine de Rham curve]]
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| The Césaro-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
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| :<math>d_0=\begin{pmatrix}
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| 1 & 0 & 0 \\
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| 0 & \alpha &\delta \\
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| 0 & \beta & \epsilon
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| \end{pmatrix}</math>
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| and
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| :<math>d_1=\begin{pmatrix} | |
| 1&0&0 \\
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| \alpha & 1-\alpha&\zeta \\
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| \beta&-\beta&\eta
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| \end{pmatrix}.</math>
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| Being [[affine transform]]s, these transforms act on a point <math>(u,v)</math> of the 2-D plane by acting on the vector
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| :<math>\begin{pmatrix} | |
| 1 \\
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| u \\
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| v \end{pmatrix}.</math>
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| 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.
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| The [[blancmange curve]] of parameter <math>w</math> can be obtained by setting <math>\alpha=\beta=\epsilon=1/2</math>, <math>\delta=\zeta=0</math> and <math>\eta=w</math>. That is:
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| :<math>d_0=\begin{pmatrix}
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| 1&0&0 \\
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| 0 & 1/2&0 \\
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| 0&1/2&w
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| \end{pmatrix}</math>
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| and
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| :<math>d_1=\begin{pmatrix}
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| 1&0&0 \\
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| 1/2 & 1/2&0 \\
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| 1/2&-1/2&w
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| \end{pmatrix}.</math>
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| 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.
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| ===Minkowski's question mark function===
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| [[Minkowski's question mark function]] is generated by the pair of maps
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| :<math>d_0(z) = \frac{z}{z+1}</math>
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| and
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| :<math>d_1(z)= \frac{1}{z+1}.</math>
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| == Generalizations ==
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| 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#Representing_real_numbers|binary expansion of real numbers]]. The continuity condition has to be generalized in:
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| :<math>d_i(p_{(n-1)})=d_{(i+1)}(p_0)</math>, for <math>i=0 \ldots n-2.</math>
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| 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.
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| == See also ==
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| * [[Iterated function system]]
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| * [[Refinable function]]
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| * [[Modular group]]
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| * [[Fuchsian group]]
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| == References ==
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| * Georges de Rham, ''On Some Curves Defined by Functional Equations'' (1957), reprinted in ''Classics on Fractals'', ed. Gerald A. Edgar (Addison-Wesley, 1993), pp. 285–298.
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| * Linas Vepstas, ''[http://linas.org/math/de_Rham.pdf A Gallery of de Rham curves]'', (2006).
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| * 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.)''
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| [[Category:Fractal curves]]
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Wilber Berryhill is the title his mothers and fathers gave him and he totally digs that title. I am an invoicing officer and I'll be promoted soon. To perform lacross is the thing I love most of all. Mississippi is the only place I've been residing in but I will have to move in a yr or two.
Have a look at my blog post :: are psychics real; kjhkkb.net,