A Modified Azimuthal Equidistant/Stereographic Projection

[quadibloc]

I'm feeling a little lazy, and so it may take me some time to implement a new map projection of which I have thought.

So I am hastening now to post a description of it, so that the world will not have to wait until I have implemented it to make use of it.
Essentially, this projection is intended to be used in the polar case, and it belongs to the azimuthal class of projections.

A parallel is chosen. Between the North Pole and that parallel, the map in this projection is drawn according to the Stereographic projection. South of that parallel, a modified Azimuthal Equidistant projection is used. The scale along the meridians of the modified Azimuthal Equidistant matches that of the Stereographic at the parallel where the two projections are joined. So the projection is conformal until the chosen parallel, and then it is gradually stretched in the east-west direction south of it.

What is the intended application of this projection?

Of late, people claiming that the Earth is flat have chosen to answer objections coming out of travel across the Pacific and the existence of time zones and the International Date Line and so on by using the polar case of the Azimuthal Equidistant projection as a map of the world. But this clearly can't be what the world really looks like. Maps of the world based on the globe are accurate locally everywhere, and so the world can't be stretched out in an east-west direction everywhere but the North Pole.

This new projection I have devised is an attempt to show what would have to be done to make a flat Earth slightly less implausible.

For the Western Hemisphere, use a map with a standard parallel of 50 degrees North latitude. Because people can freely travel in Canada and the United States, the errors from the globe-based maps must be kept to a tiny minimum in this region. In the high Arctic, well, it's very hard to travel there, so the Globe Earth Conspiracy could be taking advantage of the fact that distances travelled in subzero cold seem longer. In South America... well, perhaps the Andes mountains are really as wide as the Himalayas, but our globe maps conceal their full extent, which would allow Chile and Argentina to be in their proper time zones. Also, the Amazon tropical rainforest is not an easy place for people to travel.

For the Eastern hemisphere... from 10 degrees West longitude to 30 degrees East longitude, one could use a map with a transitional parallel at 55 degrees North latitude. Possibly 50 degrees North would do as well.

From 100 degrees East longitude to 160 degrees East longitude, a transitional parallel at 40 degrees South latitude could be used. In this way, a journey by road from Perth to Sydney would involve the expected distance, and yet cross the correct number of time zones. Japan, Taiwan, the Phillipines and Korea would need to be pasted in from a map on a larger scale, of course, to keep errors within those countries small.

Between 30 degrees East and 100 degrees East, well, travel is difficult and restricted, so these areas need not in reality bear much resemblance to what we see in atlases based on the Globe Earth "hoax"!

There might be other details, such as recognizing that as South Africa was once a Western nation with convenient travel, it would have to be dealth with on a projection like the one for Australia, and so most of Africa north of it would have to be squashed vertically.

I think that a best-effort map of what a flat Earth would have to look like would only serve to show how ridiculous the notion really is. However, just in case agents of the Globe Earth Conspiracy come to get me, I thought it would be better to have the description in the public eye before I finished my program for it.

[Milo]

What do you think of just centering an azimuthal projection somewhere in Africa?

Equal-area:

azea+25+20.png (393.39 KiB) Viewed 1586 times

Equidistant:

azed+25+20.png (252.37 KiB) Viewed 1586 times

With country borders:

azed+25+20_borders.png (280.82 KiB) Viewed 1586 times

Here the center is 25 E, 20 N. I originally had it on the equator, but shifted it north a little because you mentioned North America as a priority. I'm just eyeballing it, anyway, the point isn't based on some deep reasoning. (It'd be cool if I could play around with the dataset/algorithm used to compute Point Nemo, but it seems the entire library has been discontinued, and may never have been publicly available.)

North America still doesn't come out great, with Florida and Alaska in particular having visible torsion relative to the central states, but frankly it's probably close enough to look plausible to anyone who seriously thinks that a flat earth is a reasonable idea. New Zealand also gets distorted, which is a problem - it's small but it's still a first-world country, so people are going to notice if something's wrong.

Really though, how do flat-earthers explain compass directions? What is the compass pointing at? If you're willing to acknowledge one of humanity's oldest and most widely-used navigation technologies as not being a hoax, then a proper flattening would need to project meridians (or "magnetic meridians", going from the magnetic south pole to the magnetic north pole along a path which follows magnetic north at every point) as straight lines, and also be conformal (so that compass bearings other than 0°/180° are correct), which probably means a Mercator projection, or Lambert conic so the north pole can be an actual point (but that trashes Antarctica even more, although it would be easy to write off Antarctica as the secret homebase of the evil conspiracy and draw it however you feel like). At least, it would need to satisfy these conditions anywhere people regularly travel, but I think it would be hard to do that everywhere you need to without just extending it to most of the world.

For the Western Hemisphere, use a map with a standard parallel of 50 degrees North latitude.

For the Eastern hemisphere... from 10 degrees West longitude to 30 degrees East longitude, one could use a map with a transitional parallel at 55 degrees North latitude. Possibly 50 degrees North would do as well.

From 100 degrees East longitude to 160 degrees East longitude, a transitional parallel at 40 degrees South latitude could be used.

How would you use different transitional parallels in different places? If you try to fit that into a single world map, you'll have blatant discontinuities. If you're willing to split the world into multiple regional maps without worrying about how those maps fit together, you can probably do better by abandoning the commitment to polar aspect.

You could probably make something pretty decent by starting with something like the projections I have above and widening the South Atlantic so the American torsion isn't so bad. Tweak Australia and New Zealand too a little, nobody's going to notice some extra water. Besides, you need somewhere to put the sunken city of Atlantis.

[Milo]

And so I'm not just derailing the thread, I tried my hand at making the projection you described, to the best of my understanding. Transition parallel is 50 N. Stereographic north of there, equidistant south of there. Is this what you had in mind?

quadibloc-flat.png (380.89 KiB) Viewed 1581 times

Frankly, any difference from the plain equidistant projection is extremely minor. Am I missing something?

Here's the code I used (with a Linux binary, but you should be able to easily compile it on any computer that has a C compiler and libpng):

quadibloc-flat.zip

(8.01 KiB) Downloaded 209 times

[quadibloc]

Frankly, any difference from the plain equidistant projection is extremely minor. Am I missing something?

Possibly. The idea is to strech the Azimuthall Equidistant radially, so that it's conformal at the transitional parallel instead of at the central pole. (Stretch away, of course, from the transitional parallel - stretching from the pole only changes the scale, not the projection.)
Presumably, flat-Earthers think the compass is pointing to the North Magnetic Pole in the center of the map, and not anything in the southern outer edge.
It's because they've switched to a projection with the pole in the center that they can have the Sun going around in circles above the Earth instead of going below the Earth when it sets.
Basically, I guess that flat-Earthers don't think too much about any part of the Earth except whatever city they happen to live in. If you never go more than a few miles from home, then there's no reason for the Earth not to be flat. It's only when you have to acknowledge the existence of all those people who live in the rest of the world that you run into problems.
However, I have realized now that if I were to move the transitional parallel down to 40 degrees South Latitude, then the portion of the projection including Australia, in order to be at the correct scale, would have to be so small as to put Australia in the Northern Hemisphere, and then the Sun would be visible from the wrong direction.
I think I have finally come up with a workable solution for Australia.
It just requires that everything in Australia is parity-reversed so that it could be flipped upside down, and yet the Sun would still rise in the East and set in the West. So there would be a barrier surrounding Australia such that people entering the area, whether by sea or by air, would be transformed on their journey without noticing it.
Perhaps they would also be converted to antimatter, and we could call this barrier the "Negative Zone". Of course, this was secretly hinted at in the pages of one popular comic book...
Of course, though, that would mean that the scientific experiments which showed the breakdown of CP-symmetry, leaving only CPT-symmetry as preserving the laws of physics, would have to also be a hoax by the Globe Earth Conspiracy.

[Milo]

Possibly. The idea is to strech the Azimuthal Equidistant radially, so that it's conformal at the transitional parallel instead of at the central pole. (Stretch away, of course, from the transitional parallel - stretching from the pole only changes the scale, not the projection.)

Yes, I did that. (Then almost thought I forgot, double-checked, and found I did it right after all. Maybe I got turned around and didn't explain it quite right in the post, but the actual code is correct.)

The formulas I'm using are:
radius_plane/2 = tan(radius_sphere/2) [if radius_sphere <= boundary]
radius_plane/2 = tan(boundary/2) + (radius_sphere - boundary) / (cos(boundary) + 1) [if radius_sphere >= boundary]
The derivatives match up at radius_sphere = boundary.

It's because they've switched to a projection with the pole in the center that they can have the Sun going around in circles above the Earth instead of going below the Earth when it sets.

How does that work? If the sun goes around in a circle above the world, then it shouldn't ever set, just become a little bit dimmer when it's on the far side from you. And at the center of the circle, illumination would remain practically constant and you wouldn't have a day/night cycle at all.

Ancient cultures that believed the world to be flat described the sun as rising above the ground, travelling west, and then setting again (presumably either beyond the edge of the world or into some sort of cavern dug for that purpose by the gods), which makes sense locally, but doesn't account for time zones. Claiming the sun is always aboveground doesn't make sense even locally, unless you're using some weird non-Euclidean geometry to divert the solar rays, which defeats the purpose of having a flat world.

I considered putting a cosmic lampshade on the sun, so it doesn't shine evenly in all directions, meaning it can be night even if the sun is overhead, but that still doesn't actually solve the problem that we can obviously see the sun rising and setting at some point. Also, then the sun should have phases like the moon.

Actually, I think I came up with a better argument to use in the future if I ever run into any flat-earthers. If you acknowledge (A) that the sun rises and sets across the horizon, and (B) it does so at different times in different places, then the earth pretty much has to be round. Maybe not spherical (you could argue it's a cylinder, which is geometrically Euclidean just like the plane), but definitely not a flat plane.

Even prehistorical cavemen were aware of (A), so the only reason ancient cultures believed the world was flat was because they hasn't explored widely enough to notice (B) (and even if your empire spans a large east-west extent, without realtime communication the difference is a lot less noticeable, as a few hours of time zone shift is lost next to the months of travel time it took you to get there).

However, I have realized now that if I were to move the transitional parallel down to 40 degrees South Latitude,

That means you're running the stereographic projection out to 140 degrees from the center. Even leaving aside what you're doing beyond that, the stereographic projection's distortion becomes pretty blatant when you do that.

quadibloc-flat2.png (347 KiB) Viewed 1561 times

South America looks pretty terrible, while the oceans are ridiculously huge and North America is tiny.

Besides, 40 degrees south lies beyond almost everything except Tasmania, much of New Zealand, and the very tip of South America. Aside from managing to depict Antarctica in finite space (but still far too large), this has no meaningful advantage over the plain stereographic.

then the portion of the projection including Australia, in order to be at the correct scale, would have to be so small as to put Australia in the Northern Hemisphere,

There are no hemispheres. If the world isn't a sphere, then it also can't be divided into hemispheres.

(Hmm, does that mean that we should technically be talking about the Northern Hemiellipsoid?)

I think I have finally come up with a workable solution for Australia.
It just requires that everything in Australia is parity-reversed so that it could be flipped upside down, and yet the Sun would still rise in the East and set in the West. So there would be a barrier surrounding Australia such that people entering the area, whether by sea or by air, would be transformed on their journey without noticing it.

...What?

Seriously, I have no idea what you're talking about now.

If we're still going with the "sun travels in circles around the north pole" idea, then it doesn't matter how far any given landmass is from the north pole, so long as its north direction is oriented correctly, then its east and west directions are also oriented correctly.

If the sun is moving in some other trajectory, then a lot more than just Australia is going to end up wrong.

[quadibloc]

...What?

Seriously, I have no idea what you're talking about now.

If we're still going with the "sun travels in circles around the north pole" idea, then it doesn't matter how far any given landmass is from the north pole, so long as its north direction is oriented correctly, then its east and west directions are also oriented correctly.

If the sun is moving in some other trajectory, then a lot more than just Australia is going to end up wrong.

I am still going with the "Sun travels in circles arount the North Pole".
But if it is a circle, and not some strange oval, then if I have North America and Europe on a projection so that 50 degrees North latitude is correct in scale in both directions... and then put Australia on a projection to make 40 degrees South latitude correct in scale...
then I've made the portion of the Earth north of that so small that I've moved Australia into the Northern Hemisphere. So now the circle in which the Sun is moving is south of Australia instead of north of it.
However, after posting that which you quoted, I realized that it would not work.
Observations of sunspots from the mirror-reversed Australia would not be consistent with observations elsewhere.
Among other things. That was what first occurred to me because I was putting myself into the mind-set of a flat-earther.
A more obvious thing that would be wrong in such a scenario is that Australians would look south at night, and see a mirror-reversed Big Dipper instead of the Southern Cross. But from a flat-earth perspective, the constellations don't even bear thinking about.
After all, no matter where you are at night, you can look up and see almost half the sky.
And so the fact that our star maps are perfectly consistent and correct by putting the stars on the inside of a globe... is an inarguable fact.
And it doesn't matter how far south you go, the relationship of the night sky, and its nightly motion, to where you are is just that which is demanded by the Earth being a globe. The night sky proves that the Earth is a globe and can't possibly be flat.

[Milo]

I am still going with the "Sun travels in circles arount the North Pole".
But if it is a circle, and not some strange oval, then if I have North America and Europe on a projection so that 50 degrees North latitude is correct in scale in both directions... and then put Australia on a projection to make 40 degrees South latitude correct in scale...
then I've made the portion of the Earth north of that so small that I've moved Australia into the Northern Hemisphere. So now the circle in which the Sun is moving is south of Australia instead of north of it.

Ah, now I get it.

So why not just move Australia south? If you're arbitrarily projecting different continents with different projections to ensure they have roughly the correct size and shape, then that's a thing you can do.

Except that Australia can't be taken in isolation. There's a continuum from mainland Asia to Indonesia to New Guinea to Australia. Indonesia has the fourth-highest population of any nation in the world, so people are definitely going to notice if its neighbors aren't where they're supposed to be.

Observations of sunspots from the mirror-reversed Australia would not be consistent with observations elsewhere.

I seriously doubt that flat-earthers know enough about astronomy to even know what a sunspot is.

Just to begin with, observing sunspots requires you to trust equipment that shows you things you can't see with the naked eye, unless you want to go blind.

And it doesn't matter how far south you go, the relationship of the night sky, and its nightly motion, to where you are is just that which is demanded by the Earth being a globe. The night sky proves that the Earth is a globe and can't possibly be flat.

The stars could be located on a rotating celestial sphere in a geocentric model.

Again, what disproves this is that the night sky doesn't look the same from everywhere. From any single vantage point, the celestial sphere model appears to work. (Ancient Greek philosophers weren't dumb. Then again, Ancient Greek philosophers figured out the earth is round.)

Trickier is the planets (let's stick to the ones the ancients could see with the naked eye: Mercury, Venus, Mars, Jupiter, Saturn), which, even in a geocentric model, are still clearly orbiting the sun, not the earth. (Which is not necessarily absurd: after all, the moon orbits the earth in a heliocentric model. However, the moon's orbital period is lower than the earth's orbital period, whereas the planets' orbital periods are much higher than the sun's apparent "orbital period" in a geocentric model - actually the earth's rotational period. Also most planets are farther away from the sun than the earth is.) The Ancient Greek philosophers seemed to think the celestial sphere model still holds up, but I haven't checked their math.

Or the stars could be alien spacecraft flying a daily patrol. You know, why not.

[PeteD]

However, I have realized now that if I were to move the transitional parallel down to 40 degrees South Latitude, then the portion of the projection including Australia, in order to be at the correct scale, would have to be so small as to put Australia in the Northern Hemisphere, and then the Sun would be visible from the wrong direction.
I think I have finally come up with a workable solution for Australia.
It just requires that everything in Australia is parity-reversed so that it could be flipped upside down, and yet the Sun would still rise in the East and set in the West. So there would be a barrier surrounding Australia such that people entering the area, whether by sea or by air, would be transformed on their journey without noticing it.

Wouldn't compasses then point the wrong way in Australia?

This all reminds me of Terry Pratchett's Discworld novels, where the world is a flat disc (on the backs of four elephants, which in turn stand on the back of a turtle, but that's not important right now). The sun orbits the disc in a vertical plane such that it rises above the disc in the morning and sets below the disc in the evening, while the disc rotates about its "Hub", i.e. the equivalent of our North Pole. This results in several differences compared to our Roundworld that are mentioned in the novels:

1. At every point on the Discworld, there are two days per rotation on which the sun passes directly overhead. Both of these days are considered to be midsummer, and there are two midwinters in-between.

2. One of the two summers per rotation has hot mornings when the sun is close and cool evenings when it's far away; the other of the two summers has cool mornings and hot evenings.

3. In one of the two summers per rotation, the sun rises in the north and sets in the south; in the other of the two summers, it rises in the south and sets in the north. In one of the two winters per rotation, the sun rises in the east and sets in the west; in the other of the two winters, it rises in the west and sets in the east. (Except north, south, east and west are replaced by "Hubward", "Rimward", "turnwise" and "widdershins".)

4. There are no time zones because the sun rises above and sets below the disc at the same time everywhere.

5. Every day is an equinox.

6. Seasons vary with longitude/azimuth but not with latitude/radius.

That should be enough to convince any rational person that that's not the world that we live on.

It's because they've switched to a projection with the pole in the center that they can have the Sun going around in circles above the Earth instead of going below the Earth when it sets.

If instead the sun circled above the "equator", then as Milo says, it would get closer and further away (and therefore appear to get larger and smaller) but would never set and there would be no night.

[Milo]

1. At every point on the Discworld, there are two days per rotation on which the sun passes directly overhead. Both of these days are considered to be midsummer, and there are two midwinters in-between.

2. One of the two summers per rotation has hot mornings when the sun is close and cool evenings when it's far away; the other of the two summers has cool mornings and hot evenings.

I decided to check the math on this.

According to my calculations, the average daily insolation is:
(atan(cos(season)/(sun_distance/your_distance-sin(season)))-atan(cos(season)/(sun_distance/your_distance+sin(season))))/(your_distance^2*sin(2*season))

At the winters (season=0, season=pi), this simplifies to:
1/(sun_distance^2 + your_distance^2)

At the summers (season=pi/2, season=pi*3/2), this simplifies to:
1/(sun_distance^2 - your_distance^2)

Summer is hotter than winter. Checks out!

[Milo]

Here's another projection to consider.

This projection is azimuthal equidistant until the equator, and then pseudoazimuthal equidistant-equal-area (analogous to the sinusoidal projection, but in azimuthal rather than cylindrical form: parallels are at the same spacing as in an equidistant projection, but rescaled to make the projection equal-area) beyond the equator.

lobed-equidistant.png (221.75 KiB) Viewed 1532 times

There's a lot of opportunity to fiddle both with the choice of the projection (for example, using the azimuthal equal-area projection for the northern hemisphere, so the projection is equal-area everywhere) and with the locations of the lobes, but I hope you get the idea.

Here's a crude attempt to make the same projection look like it covers a full disc:

lobed-equidistant-shh.png (295.44 KiB) Viewed 1532 times