Finding Surface for Complex ParametricPlot3D

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SUMMARY

The discussion focuses on visualizing complex parametric plots in 3D using Mathematica's ParametricPlot3D. The user seeks to find a surface that, when projected onto the xy-plane, reveals iso displacement lines of the complex plot. The conversation highlights the challenge of representing complex data in a 4D context and explores the potential connection to Riemann surfaces. The user also notes that higher values of k may lead to self-intersecting surfaces, complicating the visualization process.

PREREQUISITES
  • Understanding of complex analysis and Riemann surfaces
  • Familiarity with Mathematica's ParametricPlot3D functionality
  • Knowledge of 3D plotting techniques and visualization
  • Basic concepts of iso displacement lines in complex functions
NEXT STEPS
  • Research advanced techniques in Mathematica for 3D complex visualizations
  • Explore the mathematical foundations of Riemann surfaces
  • Learn about contour and density plots in 3D contexts
  • Investigate methods for handling self-intersecting surfaces in parametric plots
USEFUL FOR

Mathematicians, graduate students in complex analysis, and researchers interested in advanced 3D visualizations of complex functions.

zxh
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Normally, complex plot visualization is a 4D problem. Coloring re or I am separately isn't very intuitive.

Otoh, using Mathematica's ParametricPlot3d gives all the information and leaves us with the z-axis unused. Example for z→z+k/z:

Yudkowski.jpg


I'm trying to find a surface which when projected onto the xy-plane gives the iso displacement lines of the complex parametric plot. (Mathematica uses polygons for computation here).

An alternative would be to compute the grid line density for the z-value.
I looked at a density/contour plot, but it appears quite different:

image.jpg


How would you go about this? Is this connected to riemann surfaces? It seems for higher k's the surface would intersect/overlap itself/ project non-injective.
 
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nobody? is this graduate stuff?
 

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