Lab - Domain Coloring of Complex Functions

Lab - Domain Coloring of Complex Functions

Konstantin Poelke Freie Universit?t Berlin ABV Visualization 2014

Outline

Recap

Implementing Your Color Schemes Python Hacking Implementing Domain Coloring

Quizzes Quiz 1 Quiz 2 Quiz 3 Quiz 4 Quiz 5

FU Berlin, Lab - Domain Coloring, ABV Vis 2014

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Recap: Domain Coloring

1. Discretize domain U C into pixels 2. Define color scheme function col : C {} RGB 3. For each pixel (i, j) compute domain point zij U 4. Compute color cij := col f (zij) 5. Assign color to pixel: img[i][j] = cij

f

col

U

V

FU Berlin, Lab - Domain Coloring, ABV Vis 2014

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Recap: Notions from Complex Analysis

Recall or look up the following definitions and notions from complex analysis.

Holomorphic functions

Zeros of a holomorphic function and their multiplicities

Singularities:

Poles:

1 (z-p)k

Essential

singularities:

exp(

1 z

),...

Conformality: preservation of angles and orientation

Complex Logarithm, principal branch, complex powers and roots

Branching: log z, z, ...

Riemann Surface

FU Berlin, Lab - Domain Coloring, ABV Vis 2014

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Outline

Recap

Implementing Your Color Schemes Python Hacking Implementing Domain Coloring

Quizzes Quiz 1 Quiz 2 Quiz 3 Quiz 4 Quiz 5

FU Berlin, Lab - Domain Coloring, ABV Vis 2014

, 5

IPython

We will use the IPython environment Python programming language in Mathematica-like notebooks Easy programming in browser Heavily used in scientific community Many additional python libraries available, e.g. for: visualization, numerical computations, CAS, ODE/PDE solving, integration, symbolic differentiation,...

FU Berlin, Lab - Domain Coloring, ABV Vis 2014

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Installing IPython

On Linux-Systems IPython usually comes with your distribution:

> sudo apt-get install ipython3

Cross-Platform: There are (I)Python bundles for easy installation of all dependencies, e.g. Anaconda for Windows, Mac or Linux:



Whatever you choose, try to get one with Python 3.x support!

FU Berlin, Lab - Domain Coloring, ABV Vis 2014

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Starting IPython

IPython is pre-installed in PC pool Open terminal and type

user@pc> ipython3 notebook -pylab

This starts the IPython notebook viewer for Python 3 The -pylab option loads the matplotlib stack for visualization Open a browser (e.g. Iceweasel) and go to



FU Berlin, Lab - Domain Coloring, ABV Vis 2014

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