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| Here are examples of [[conformal map]]s understood as [[deforming]]{{disambiguation needed|date=October 2013}} pictures. This technique is a generalization of [[domain coloring]] where the domain space is not colored by a fixed infinite [[color wheel]] but by a finite picture tiling the plane. A pedagogical interest is to have a flow of pictures coming from a webcam to allow more interactivity and a richer feedback loop.<ref>[http://images.math.cnrs.fr/Applications-conformes.html Images des mathématiques] CNRS 2009</ref>
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| == Conformal maps ==
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| [[Image:ConformId.jpg|thumb|left|The picture of a clock.]]
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| [[Image:ConformSimilitude.jpg|thumb|The picture of a clock after a similarity transformation.]]
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| A [[conformal map]] is a transformation of the plane preserving angles. The plane can be parametrized by [[Cartesian coordinate plane|Cartesian coordinates]] where a point is denoted as <math>(x,y)\in\mathbb{R}^2</math>, but for conformal maps, it is better to understand it as the [[complex plane]] where points are denoted <math>x+i\,y\in\mathbb{C}</math>.
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| In complex coordinates, the multiplication by a real number ''r'' corresponds to a [[homothety]], by a unitary number <math>e^{i\theta}</math> to a [[rotation]] of angle ''θ'' and by a generic complex number <math>z=r\,e^{i\theta}</math> to a [[similarity (geometry)|similarity mapping]].
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| A [[holomorphic function]] is a conformal map because it is locally a similarity <math>f(z)=a\,(z-z_0)+b+o(z-z_0)</math> with <math>a=f'(z_0)</math> the derivative and <math>b=f(z_0)</math> the value of ''f'' at ''z''<sub>0</sub>. The derivative is the local zoom factor of the transformation.
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| After similarities, which have constant derivative function, [[polynomial]]s and in particular [[monomial]]s <math>z\mapsto z^k</math> are the simplest holomorphic functions. Its derivative is <math>z\mapsto k\,z^{k-1}</math>, it is null at the origin; hence the associated map is conformal only away from the origin.
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| A problem we face to represent them is that a holomorphic function is in general not injective: Consider the monomial <math>z^k</math> for example, ''k'' different points are mapped to the same value.
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| Considering the plane tiled by the picture of the clock, it becomes, when squared the following blurry picture:
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| [[Image:ConformRoot.jpg|center|A clock to the square. Notice the double cover of each point.]]
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| {{Clr}}
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| You see that the central disk is globally preserved, mapped to itself, but each point (except zero) is covered twice, rendering the picture blurry. For example +1 (at 3 oclock) and −1 (9:00) are both sent to +1 (to the middle right), ''+i'' (noon) and ''i'' (6:00) are sent to −1 (to the middle left).
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| In order to get an injective application, we can either restrict ourselves, for example, to the positive real half plane, or to the negative real half plane.
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| [[Image:ConformRoot2.jpg|thumb|left|The negative real half plane to the square.]]
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| [[Image:ConformRoot1.jpg|thumb|right|The positive real half plane to the square.]]
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| {{Clr}}
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| Seen from further away, you get the same big picture for the whole tiling.
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| <gallery> | |
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| Image:ConformRoot2bis.jpg|The negative real half plane to the square.
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| Image:ConformRootbis.jpg|Clock to the square: conformal but not injective.
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| Image:ConformRoot1bis.jpg|The positive real half plane to the square.
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| </gallery> | |
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| ==Pull-back==
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| [[Image:WOA09_sea-surf_SAL_AYool.png|thumb|right|Salinity pictured as a preimage]]To get a nice conformal picture, it is easier and more natural to consider not the ''direct image'' but the ''[[preimage]]''.
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| This is exactly how a lot of information is pictured on earth. For example a map setting to each point a temperature value, is pictured by plotting every point of the globe with a certain color specifying the value of the temperature function at that point. The target space, that is the temperature space, is painted from blue for small values, to red for large values. The same graphing technique is used here but the target space is not 1-real dimensional, it is not a line but the whole plane.
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| The picture tiles no longer the [[domain (mathematics)|domain]] of the application but its [[range (mathematics)|target]] space. The point ''z'' colored according to the pixel ''f''(''z'').
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| {{Clr}}
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| [[Image:Conformz2.jpg|thumb|center|The pull-back by <math>z\mapsto z^2</math>.]]
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| Notice the duplication: points ''z'' and −''z'' are colored similarly because they are both mapped to the same image ''z''<sup>2</sup>.
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| {{Clr}}
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| [[Image:ConformPoly.jpg|thumb|right|A polynomial of degree 4. Notice the zeros of its derivative. Simple zeros don't look anything special.]]
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| [[Image:Conformz3.jpg|thumb|left|The pull-back by <math>z\mapsto z^3</math>.]]
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| Likewise, the monomial of order ''k'' maps ''k'' different points to the same image.
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| A lot of useful information can be understood concerning the conformal map by picturing its pull-back. Since the forward zoom factor is the derivative, the pull-back zoom factor is the 'inverse' of the derivative. Therefore something very special takes place at the zeros of the derivative of the function, the zoom factor becomes infinite, and it shows. Moreover the degree of the zero can be read by the number of time a feature repeats itself around the singularity. You notice as well when the derivative is real and positive, the picture is "standing up", and when it is real and negative, the picture is "upside-down". When you restrict yourself to the real axis, you can figure out a sketch of the [[Graph of a function|graph]] of a real function. You notice as well the inflexion points as a minimum or maximum of the zoom factor.
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| {{clr}}
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| ==Inversion, poles==
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| After holomorphic functions, locally conformal maps comprise as well [[meromorphic]] functions, and the position and order of their poles can be read-off.
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| [[Image:ConformInversion.jpg|thumb|left|The inversion <math>z\mapsto 1/z</math>.]]
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| [[Image:ConformInversionZoom.jpg|thumb|The inversion (detailed view).]]
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| The inversion <math>z\mapsto 1/z</math> has a simple pole at zero. It is a [[Möbius transformation]] <math>z \mapsto \frac{az+b}{cz+d}</math> with ''a'', ''b'', ''c'' and ''d'' four complexes such that <math>ad - bc \neq 0</math>, therefore it maps circles and lines to circles and lines. In particular horizontals and verticals become circles through zero. Inversion exchanges interior and exterior of the unit disk. | |
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| {{clr}}
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| [[Image:ConformalTwoPoles.jpg|thumb|right|Two poles at ±1/2.]]
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| [[Image:ConformPole2.jpg|thumb|left|A double pole <math>z\mapsto 1/z^2</math>.]]
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| Like zeros, poles can be of higher order than simple. Circles are only infinitesimally preserved in general. You can picture higher order poles as several simple poles coming together.
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| {{clr}}
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| ==Logarithm and exponential==
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| [[Image:ConformLog.jpg|thumb|left|The exponential map <math>z\mapsto e^z</math>. Notice the homothety and rotation invariances, corresponding to horizontal (real) and vertical (pure imaginary) translations invariance in the target lattice.]]
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| [[Image:ConformExp.jpg|thumb|right|The logarithm map <math>z\mapsto \log(z)</math>. Notice the unit disk being unwrapped vertically.]]
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| A very important map in [[complex analysis]] and cartography is the transformation from [[cartesian coordinates]] ''(x,y)'' to [[polar coordinates]] ''(r,θ)''. This transformation is realized by the couple of functions [[logarithm]]/[[Exponential function|exponential]] reciprocal of each other (<math>\log(\exp(z))=z</math>). Indeed,
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| <math>\log(r\,e^{i\theta})=\log(r)+i\theta</math> maps (''r'', ''θ'') to (''x'' = log(''r''), ''y'' = ''θ'') and <math>\exp(x+i\,y)=\exp(x)\,e^{i\, y}</math> maps (''x'', ''y'') to (''r'' = exp(''x''), ''θ'' = y). | |
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| In picture, the logarithm unwraps circles centered at the origin into vertical lines, and maps rays to horizontal lines. The exponential on the contrary wraps vertical lines into concentric circles, and maps horizontal lines to rays through the origin. Notice that the logarithm goes to infinity at zero, but at a much slower pace than the inversion.
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| {{clr}}
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| Changing the basis of the lattice, one can obtain spiral variations. As poles and zeros, logarithmic singularities can be added.
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| <gallery>
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| Image:ConformSpirale.jpg
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| Image:ConformSpirale2.jpg
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| Image:ConformSpirale3.jpg
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| Image:ConformSpirales.jpg
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| </gallery>
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| ==Essential singularity==
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| [[Image:ConformSingulariteEssentielle.jpg|thumb|The essential singularity <math>z\mapsto e^{1/z}</math>.]]
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| [[Analytic function]]s suffer from another type of singularities, for example the [[essential singularity]]<math>z\mapsto e^{1/z}</math> is zero for <math>z=0^-</math>, with an accumulation of zeros, and an accumulation of poles for <math>z=0^+</math>.
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| ==Convergence radius==
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| Analytic functions are summable in [[power series]]. At a given point, its [[Taylor series]] admits a [[convergence radius]]. Comparison of pull-backs of the function and its truncated Taylor series allows us to illustrate this notion:
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| <gallery>
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| Image:ConformTan.jpg|The tangent function has a simple pole every ''kπ''/2. It has an essential singularity at infinity.
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| Image:ConformTanTaylor7.jpg|Its Taylor series at 0 of order 7 is a good approximation in its disk of convergence.
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| </gallery> | |
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| == See also ==
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| * [[Domain coloring]]
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| ==External links==
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| <references /> | |
| * The pictures were deformed using this [http://math.univ-lyon1.fr/~mercat/downloads/ComplexMapStill.jar Java applet]; Another [http://math.univ-lyon1.fr/~mercat/downloads/ComplexMap.jar version] is available for [[Mac OS X]] that deforms the video flux from the webcam. <code>java -d32 -jar ComplexMap.jar</code>. That's great fun. Using [http://jtem.de/ Java Tools for Experimental Mathematics] library.
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| * [http://math.fullerton.edu/mathews/c2003/ConformalMappingMod.html Conformal Mapping Module by John H. Mathews]
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| * [http://virtualmathmuseum.org/galleryCM.html interactive visualizations of many conformal maps]
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| * [http://www.3d-xplormath.org/j/applets/en/index.html Java applets for visualization of conformal maps]
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| * [http://demonstrations.wolfram.com/ConformalMaps/ Conformal Maps] by Michael Trott, [[Wolfram Demonstrations Project]].
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| * [http://www-m10.ma.tum.de/twiki/bin/view/Lehre/Bsp10_6 Java applet] by Jürgen Richter-Gebert using [[Cinderella (software)|Cinderella]].
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| {{DEFAULTSORT:Conformal Pictures}}
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| [[Category:Conformal mapping|*]]
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| [[Category:Cartographic projections]]
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| [[Category:Angle]]
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| [[Category:Complex analysis]]
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| [[Category:Complex numbers]]
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| [[fr:Application conforme]]
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