How does one take a spherical object, like the earth, the night sky, or even the visual field of a fruit fly and display it on a 2D planar plot?
These transformations, generally developed to produce accurate maps of the earth, are called map projections. Because it is not possible to just cut the sphere and make it into a plane, which is topologically not possible,1 all the map projection transformations (and there are lot of them!) make some tradeoff or the other.23
I looked into this to figure how to represent receptive fields of the neurons in the visual space, which can be though of a sphere around the animal. Because insects (and fruit flies) only a small blind spot, their visual field occupies the whole space.
The two main constraints I had during my search was 1) simple to implement & 2) should be equal area (as it will be able to compare receptive field sizes visually). I narrowed it down to the following, that could be useful:
- Equirectangular projection: This is the simplest cylindrical projection - plotting the azimuth and elevation (spherical coordinate systems) as x and y. It works, but it distorts the area at extreme elevations.
- Hobo-Dyer projection: Might be a better cylindrical projection to try - it retains the simplicity of the equirectangular projection while having the added bonus of equal area.
- Strebe’s projection: Is an equal area map that looks cool! But it was specifically made for earth (distortions in areas without continent), plus a particular parameter needs to be iteratively solved.
- Winkel triple projection: Quite popular, seems to have straightforward equations to implement. A possible candidate to try out.
- Tobler hyperelliptical projection: A family of equal-area psuedocylindrical projections. A lot of the common projections (like Mollweide projection) falls under this family. A possible candidate to try out.
(in the process of trying out the above projections. will update this soon :) )
Footnotes
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This video nicely illustrates the point and provides a good intro to map projections. ↩
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The famous Mercator projection is made for navigation (angles and shapes are accurate), but it distorts areas. For example, the size of Greenland in the Mercator projection (Google Maps uses the Mercator projection) looks the same as Africa, but in reality, Africa is 14x the area of Greenland. ↩
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Looks like the Equal Earth projection is the one I need to use if I plan to generate my own maps, as it keeps the area equal with distortions to the shapes of continent to the minimum. ↩