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The graphics samples below were obtained by programs written in Mathematica (including version 12.0) by Dr. Erik Perkerson and myself. These programs helped me understand the behavior of piecewise linear level curves of harmonic functions which come up in my research revolving around discrete uniformization theorems for m - connected planar Jordan domains.  

The new algorithms implemented in the above mentioned programs are based on the mathematical analysis in paper #25 on my Papers page (see also #23 with H.R. Arabnia and T.R. Taha as well as #26 with E. Perkerson). Note that the Mathematica programs are making an essential usage of the C program aCute (please write me if you wish to see more details) by creating acute triangulations of planar polygonal domains.

The first group depict two triangulated annuli and their conformal equivalent (approximation of) Eucildean annuli. The second group presents higher genus  examples and their decomposition along singular level curves of the real part of the uniformization map.

 

 



 

ConfImage1.jpg
Triang1.jpg
Triang2.jpg
Confimage2.jpg
Genus21Trian.jpg
genus2Trian.jpg
Genus21LevelCurves.jpg
genus2LevelCurves.jpg
genus4LevelCurv.jpg
genus4Trian.jpg
Genus3Sing.jpeg
Genus3SingLevel.jpeg
genus3LebelCurv.jpg
genus3Trian.jpg
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