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The article mentions a lot of interesting properties of the complex exponential map. I find it especially neat that horizontal and vertical translation map to u
by pierrec 7y ago
The article mentions a lot of interesting properties of the complex exponential map. I find it especially neat that horizontal and vertical translation map to uniform scaling and rotation. A mapping that shuffles transformations around like that can be used in creative ways.
I'd like to add another interesting property, that naturally stems from the above: the mapping transforms regularly repeating shapes into self-similar shapes. Any geometry that repeats along the x axis (with a given period P) will be mapped to a geometry that is self-similar under scaling by the same value P. You can create mappings in more dimensions that have the same properties, for example in 3 dimensions: https://www.osar.fr/notes/logspherical/ https://www.osar.fr/notes/logspherical/
- roywiggins 7y agoAha! I knew such a thing (a spherical log transform) had to be possible, but my SDF skills were not up to snuff. Very happy to see it done!
- ttoinou 7y agoVery good blog article thanks for sharing. It's not really the 3D equivalent, though? Only adapted from 2D
- pierrec 7y agoThe section titled "Log-polar Mapping in 3D" is, as you say, simply adapting the same map to 3D. The section "Log-spherical Mapping" is about the proper 3D equivalent. However, while maps in 2D space can be neatly represented with complex numbers, there is no equivalent for 3D space, so things can't be as neat. But the geometric properties are there: using the log-spherical map, translation along one axis maps to uniform scaling (in all 3 axes).