3 ms·

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.

by kayo_20211030 4mo ago

zokier 4mo ago
I'm not really seeing the connection to projections here?
- kayo_20211030 4mo ago
  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.
WillAdams 4mo ago
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.
- srean 4mo ago
  > 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/ 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 https://en.wikipedia.org/wiki/Sinusoidal_projection
- Analemma_ 4mo ago
  > 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. 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.
  - deleted 4mo ago
    [deleted]
- stephbook 4mo ago
  The Mercator hate is a remnant of the turn of the decade. The unprivileged "global south" is smaller than Greenland!! 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.
- UltraSane 4mo ago
  Google Maps is the modern globe.
srean 4mo ago
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-mercator.html https://liorsinai.github.io/mathematics/2020/08/27/secant-me... https://news.ycombinator.com/item?id=24304311 https://news.ycombinator.com/item?id=24304311 https://news.ycombinator.com/item?id=43741273 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.
- ghurtado 4mo ago
  > you will have to constantly steer to follow constant bearing paths 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)
  - srean 4mo ago
    Very likely no. 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.

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.

zokier 4mo ago

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

kayo_20211030 4mo ago
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.

WillAdams 4mo ago

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.

srean 4mo ago
> 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/ 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 https://en.wikipedia.org/wiki/Sinusoidal_projection
Analemma_ 4mo ago
> 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. 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.
- deleted 4mo ago
  [deleted]
stephbook 4mo ago
The Mercator hate is a remnant of the turn of the decade. The unprivileged "global south" is smaller than Greenland!! 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.
UltraSane 4mo ago
Google Maps is the modern globe.

srean 4mo ago

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-mercator.html https://liorsinai.github.io/mathematics/2020/08/27/secant-me... https://news.ycombinator.com/item?id=24304311 https://news.ycombinator.com/item?id=24304311 https://news.ycombinator.com/item?id=43741273 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.

ghurtado 4mo ago
> you will have to constantly steer to follow constant bearing paths 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)
- srean 4mo ago
  Very likely no. 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.