May 21, 2024 6:28 am

Domain Coloring

Domain Coloring

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Domain Coloring

Developer’s Description

By VoiceSync
With this application you can select a predefined formula or create your own one, you can also modify at your convenience a predefined one, a multiple 3d object animation or fixed image panel is displayed with generated color domain. Usage is simple and intuitive: * Select a predefined formula from the list or * Modify any formula or build a new one, use ‘z’ as te complex variable, press to evaluate * Predefined functions are: sin, cos, tan , exp, log, acos, asin, atan * Operators: () * / – + ^ * examples: z 1/z * sin(1/z) log(z)/z * click on the animation panel to switch between fixed image/animation * images can be saved in PNG loos less format at any resolution (render time may be long)
Representation of complex functions with the method of domain coloring. The online version of these applets is slow. I recommend to download all the files for better performance.

Domain Coloring Editor’s review

FromVoiceSync:Domain coloring is a technique for visualizing functions of a complex variable. You can select one of the preset functions or create/modify your own, rendering is performed combining multi threading so all available cores of CPU are used and a fast zCompiler to achieve near real time display. Results can be exported to a .png or .jpg file in selectable resolution up to 32k width

Code generates HSV color representation of complex function.
domainColoring.m : this MATLAB function returns different representations of complex function.
veryComplex.m : contains collection various complex functions and their visualization using domainColoring.m.
testDomainColoring.m : simple script that demonstrates the use of domainColoring.m

Reference:

Domain coloring (cf. section 5.2) is a technique that makes use of a color scheme to visualize complex valued functions. The left image shows a reference color scheme that represents the complex plane. The right image shows a typical color plot of a meromorphic function. The black spots denote zeros, here with multiplicity 2 (the bigger one on the left) and 1 (the one on the right). The white spot in the upper third is a double pole whereas the punctual spot at the bottom is a simple pole. This can also be seen from the multiplicity and order of the colors around these points – the double zeros and poles are surrounded by the complete color circle twice in contrast to the simple zeros and poles.

Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in \({\mathbb R}^3\) from explicitly given branch points with corresponding branch…

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