Spherical Voronoi Diagram
jasondavies.com
jasondavies.com
SciPy can calculate spherical Voronoi diagrams, and MatPlotLib can display them with map projections. I haven't tried to display them as a rotatable globe, but years ago I did it in 2D for volcanos: https://news.ycombinator.com/item?id=21301942, https://imgur.com/closest-volcano-lsxjRXP (argh, Imgur has gotten really aggressive with autoplaying unrelated videos - at least they're silent).
It would be fun to do Turtle graphics with geodesic motions on the sphere. If one adds Loxodromic motions, even better.
The geodesic turtle on the globe would be a good way to play with other platonic solids.
I used for intersection crash analysis to make sure each crash was assigned to at most intersection. I combined this with a radius around each intersection so crashes too far away were also not attributed to an intersection.
More here: https://mark.stosberg.com/intersection-crash-analysis-with-q...
It is still not known for sure how he came up with mathematical details of the stretching and spacing of the latitudes.
The actual closed form was discovered much later and that too by sheer accident - by looking at log trig tables and noticing they match the Mercator scaling numerically, upto four places of decimals, not by working out the integral from first principles. Once the connection has been made it was formally derived of course.
100 years to solve an integral The history of the Mercator map and the integral of the secant https://liorsinai.github.io/mathematics/2020/08/27/secant-me...
https://news.ycombinator.com/item?id=24304311
https://news.ycombinator.com/item?id=43741273
Also, many find it very unintuitive that even in the absence of waves, winds, currents or such disturbances, you will have to constantly steer to follow constant bearing paths (North-South and equatorial East-West excluded). You have to steer even if you want to follow a latitude East-West, barring the Equator.
Its not that you can set your heading South by Southwest and be done with it, even if there are no disturbances.
Is this why airline routes look like they are "hopping" on a map?
When I was a kid, I thought this was meant to represent the Z axis (so until today)
Airline routes between touchpoints usually follow great circles, not constant bearing paths. Great circles are shortest paths and also they do not need any steering (modulo crosswinds etc).
The bearing keeps changing along the great circle path though *, this why sailors of earlier vintage preferred constant bearing paths -- easy to be sure that you are on track, just keep the ship pointed at the required angle with respect to North South direction.
In other words, the kid was damn right.
* Except for North South paths and along Equator paths.
That it was pushed into other usages was a function of cold war politics (makes Russia seem larger/more intimidating) and needs to be considered in that context.
Arguably, every classroom (and home with children) should have a globe.
Or even better is to build one. It is a lot of fun.
It is very instructive to understand why you need to shape the gores that you cut out of flat paper to stick them to the sphere. The boundaries of the gores need to curve so that there are no creases or no bits and pieces sticking out. Even then it is not going to be an exact fit on the globe, unless the flat material has some give.
One needs some interrupted equi-areal projection.
Interrupted sinusoidal is the one commonly used
https://www.jasondavies.com/maps/interrupted-sinusoidal/ (from the same site). Imagine running zipper fastners along the tears/boundaries. When one zips up one almost forms an exact sphere.
Except that Google Maps (yes maps, not Google Earth) has shown a globe for a decade now.
And, of course, pupils do have access to a globe. But somehow, people always frame it as an unfulfilled necessity that perpetuates "global power imbalances" or stuff like that.
Is this actually true, or was it just done "on autopilot" because before universal public education most people using maps of the entire world were navigators for whom Mercator made the most sense?
I don't know, I just hear a lot of conspiracy theories about the dominance of Mercator. If it's not Cold War politics then it's "white supremacists trying to make North America/Europe larger and Africa smaller", and I think laziness and just going with what worked in the past is a more likely explanation.
On a flat map I always have to ask myself how much of what I’m seeing is geometry and how much is projection distortion
My intent was to simplify the shapes of state borders as much as possible while retaining the topological (?) relationship between states. But there is no fancy math behind my map, it's just hand-drawn mess.
(I guess saying "The Incredibles" would have been shorter, but it feels weird to credit them with this style)
The idea that springs to my mind is to do Delaunay and Voronoi using spherical geometry. I think the article uses flat Euclidean geometry but if we tweak the fifth axiom we could do spherical or hyperbolic?
edit: Found the code. Looks like I instead ended up simply building a convex hull (which is the Delaunay triangulation) and deriving the Voronoi diagram from it.
Compute the Voronoi tessellation as usual and then project the linear separators back to the sphere (by connecting points on it to centre and extending the ray to the spherical surface).
I think it was this one: https://www.math.kent.edu/~zheng/papers/voronoi_paper_update... or this one http://nautilus.fc.ul.pt/jd/jd10sphere.pdf
From the looks of it the separators seem to be segments of great circles. That is what you would get as loci of angular bisectors.
Angular bisectors is what you would get when you use spherical geometry / arc length metric / Haversine metric.
You could still be right though. Euclidean would get you straight line bisectors, but when you project them back to the surface of the sphere, you get great circles again.
This connects with an important point (no pun intended):
People often use Haversine distance to find nearest neighbours on the globe. As long as you want to compute them modulo a hemisphere, this is unnecessary. Haversine distance and Euclidean restricted to the sphere are monotonic transforms of each other. Therefore the nearest point will be the same regardless of which one you used.
This makes me think he uses the Euclidean geometry of the surrounding three-dimensional space. But of course spherical geometry is induced by the surrounding Euclidean geometry, so the results are the same as using spherical geometry directly.