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Spherical Voronoi Diagram

jasondavies.com

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Re: Spherical Voronoi Diagram

#11

Post by red_trumpet »

srean wrote: post #5
wood_spirit wrote: post #2

Beautiful :) 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?

Curious about why you think he is using Euclidean. 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 surfac…

> This implementation uses a randomised incremental algorithm to compute the 3D convex hull of the spherical points.

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.

Re: Spherical Voronoi Diagram

#13

Post by mkl »

For me in Firefox Android, rotating the globe scrolls the page at the same time, so it's pretty hard to use.

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).

Re: Spherical Voronoi Diagram

#16

Post by zokier »

Reminds me of this little map I created some time ago: https://ibb.co/TPVMCR3

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.

Re: Spherical Voronoi Diagram

#17

Post by WillAdams »

kayo_20211030 wrote: post #12

Thanks and very fun. The graphic "The United States of Voronoi" is another reminder of why the mercator projection is so counter-intuitive at times.

The Mercator projection is perfectly suited to its intended use --- maintaining angles for navigation.

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.

Re: Spherical Voronoi Diagram

#18

Post by srean »

kayo_20211030 wrote: post #12

Thanks and very fun. The graphic "The United States of Voronoi" is another reminder of why the mercator projection is so counter-intuitive at times.

The problem that Mercator was trying to solve and the solution he came up with is super interesting to me. Very neat.

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.

Re: Spherical Voronoi Diagram

#19

Post by kayo_20211030 »

zokier wrote: post #14
kayo_20211030 wrote: post #12

Thanks and very fun. The graphic "The United States of Voronoi" is another reminder of why the mercator projection is so counter-intuitive at times.

I'm not really seeing the connection to projections here?

I reckon that OP did the work and that the map is correct. The CA/NV border seems weird. Under Mercator, you're insisting that the angles are correct in the projection. It just looks a little strange - with all those straight lines that you'd expect to see on a plane, but not on a sphere. The conservation of angles threw me off.

Re: Spherical Voronoi Diagram

#20

Post by srean »

WillAdams wrote: post #17
kayo_20211030 wrote: post #12

Thanks and very fun. The graphic "The United States of Voronoi" is another reminder of why the mercator projection is so counter-intuitive at times.

The Mercator projection is perfectly suited to its intended use --- maintaining angles for navigation. 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.

> 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.

https://en.wikipedia.org/wiki/Sinusoidal_projection

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