Fractals/Conformal map

Conformal Maps

Here are examples of conformal maps^[1] applied to 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.^[2]

Conformal maps

The picture of a clock after a similarity transformation.

A conformal map is a transformation of the plane preserving angles. The plane can be parametrized by Cartesian coordinates where a point is denoted as $(x,y)\in \mathbb {R} ^{2}$ , but for conformal maps, it is better to understand it as the complex plane where points are denoted $x+i\,y\in \mathbb {C}$ .

In complex coordinates, the multiplication by a real number r corresponds to a homothety, by a unitary number $e^{i\theta }$ to a rotation of angle θ and by a generic complex number $z=r\,e^{i\theta }$ to a similarity mapping.

A holomorphic function is a conformal map because it is locally a similarity $f(z)=a\,(z-z_{0})+b+o(z-z_{0})$ with $a=f'(z_{0})$ the derivative and $b=f(z_{0})$ the value of f at z₀. The derivative is the local zoom factor of the transformation.

After similarities, which have constant derivative function, polynomials and in particular monomials $z\mapsto z^{k}$ are the simplest holomorphic functions. Its derivative is $z\mapsto k\,z^{k-1}$ , it is null at the origin; hence the associated map is conformal only away from the origin.

A problem we face to represent them is that a holomorphic function is in general not injective: Consider the monomial $z^{k}$ for example, k different points are mapped to the same value.

Considering the plane tiled by the picture of the clock, it becomes, when squared the following blurry picture:

A clock to the square. Notice the double cover of each point.

We 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).

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.

The negative real half plane to the square.

The positive real half plane to the square.

Seen from further away, we get the same big picture for the whole tiling.

The negative real half plane to the square.
Clock to the square: conformal but not injective.
The positive real half plane to the square.

Pull-back

To get a nice conformal picture, it is easier and more natural to consider not the direct image but the preimage.

This is exactly how a lot of information of earth is pictured. For example, a map setting to each point a temperature value is pictured by plotting every point of the globe with a color specifying the value of the temperature function at that point. The target space, 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.

The picture tiles no longer the domain of the application but its target space. The point z colored according to the pixel f(z).

Notice the duplication: points z and −z are colored similarly because they are both mapped to the same image z².

A polynomial of degree 4. Notice the zeros of its derivative. Simple zeros don't look anything special.

Likewise, the monomial of order k maps k different points to the same image.

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. We 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 we restrict ourselves to the real axis, we can figure out a sketch of the graph of a real function. We notice as well the inflexion points as a minimum or maximum of the zoom factor.

Inversion, poles

After holomorphic functions, locally conformal maps comprise as well meromorphic functions, and the position and order of their poles can be read-off.

The inversion $z\mapsto 1/z$ has a simple pole at zero. It is a Möbius transformation $z\mapsto {\frac {az+b}{cz+d}}$ with a, b, c and d four complexes such that $ad-bc\neq 0$ , 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.

Like zeros, poles can be of higher order than simple. Circles are only infinitesimally preserved in general. We can picture higher order poles as several simple poles coming together.

Logarithm and exponential

The exponential map $z\mapsto e^{z}$ . Notice the homothety and rotation invariances, corresponding to horizontal (real) and vertical (pure imaginary) translations invariance in the target lattice.

The logarithm map $z\mapsto \log(z)$ . Notice the unit disk being unwrapped vertically.

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 reciprocal of each other ( $\log(\exp(z))=z$ ). Indeed,

$\log(r\,e^{i\theta })=\log(r)+i\theta$ maps (r, θ) to (x = log(r), y = θ) and $\exp(x+i\,y)=\exp(x)\,e^{i\,y}$ maps (x, y) to (r = exp(x), θ = y).

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.

Changing the basis of the lattice, one can obtain spiral variations. As poles and zeros, logarithmic singularities can be added.

Essential singularity

Analytic functions suffer from another type of singularities, for example the essential singularity $z\mapsto e^{1/z}$ is zero for $z=0^{-}$ , with an accumulation of zeros, and an accumulation of poles for $z=0^{+}$ .

Convergence radius

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:

The tangent function has a simple pole every kπ/2. It has an essential singularity at infinity.
Its Taylor series at 0 of order 7 is a good approximation in its disk of convergence.