Log polar mapping

As this is logarithmic infinitely small scale is infinitely far down the z-direction. The inner yellow cylinders inner volume is mapped from the entire positive half-space. The outer blue cylinder rings volume is mapped from the entire negative half-space. As the latter is heavily distorted it's better to stick just to the former for visualizations. The positive z-axis remains on the z axis. That's perhaps the easiest to comprehend part. The xy-plane maps to the yellow cylinders wall. The negative z axis maps to the outermost blue cylinder wall. Outside that there is forbidden space not being mapped to from any real valued xyz-position. This forbidden space is safely ignorable.

This page is about a generalization of log-polar mapping to 3D space.
Specifically usable as one of the visualization methods for gemstone metamaterial factories and
as one of the distorted visualization methods for convergent assembly.

Displaying many scales and their relation simultaneously

This can be done by generalizing log polar mapping to 3D like so:

• x'(x,y,z) = pi/2 - atan2( z, sqrt(pow(x,2) + pow(y,2)) ) * cos(atan2(y,x)))
• y'(x,y,z) = pi/2 - atan2( z, sqrt(pow(x,2) + pow(y,2)) ) * sin(atan2(y,x)))
• z'(x,y,z) = log( sqrt(pow(x,2) + pow(y,2) + pow(z,2)) ) / log(base)

Related

• Distorted visualization methods for convergent assembly
• Visualization methods for gemstone metamaterial factories
• Challenges in the visualization of gem-gum factories

External links

• https://en.wikipedia.org/wiki/Log-polar_coordinates
• 1996 – An Introduction to the Log-Polar Mapping – Helder J. AraujoJorge Miranda DiasJorge Miranda Dias – ResearchGate

Not log polar mapping but mercator projection to the extreme.
This should be locally similar to log polar mapping in that math in the limit becomes identical.
https://mrgris.com/projects/merc-extreme/