## 0977: "Map Projections"

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eran_rathan
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### Re: 0977: "Map Projections"

Pfhorrest wrote:
eran_rathan wrote:The earth isn't a sphere, its more properly modeled with an ellipsoid (if you want a regular geometric shape to make your math easier - otherwise, you need to be using the geoid - current one is Geoid-09, based on the WGS84(2004 revision), until they come up with the new one that was started last year.)

You know, a map that would be really interesting would be some kind of rectangular projection of the geoid, with the latitude and longitude lines drawn on it. Since lat and long are lines on a sphere approximating the geoid, if we project the actual geoid onto a rectangle rather than that sphere, the lat and long lines should appear distorted in proportion to how the geoid varies from that sphere, giving us a visualization of exactly how nonsphereical the geoid is in what geographic areas. I suppose it would end up like a sort of topographic map in the end.

Not exactly what you were thinking, but there are a few decent images of the geoid kicking around, such as this, and for excellent visualization, there is this, with massive exaggeration, natch:

(from The University of Stuttgart, Geodesy Dept.)
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Capt. Jon
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### Re: 0977: "Map Projections"

Now people have explored projections i reckon it is time to post some of Layla Curtis' work:

chrth
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### Re: 0977: "Map Projections"

Actually, I have one last projection question.

The Mercator projection uses the equator as the, um, focus? -- not really sure what the cartographic term is. Couldn't you use the Prime Meridian (or any other longitude) and do a similar projection?

eran_rathan
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### Re: 0977: "Map Projections"

chrth wrote:Actually, I have one last projection question.

The Mercator projection uses the equator as the, um, focus? -- not really sure what the cartographic term is. Couldn't you use the Prime Meridian (or any other longitude) and do a similar projection?

Yes, but the distortion is wicked.
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chrth
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### Re: 0977: "Map Projections"

eran_rathan wrote:Yes, but the distortion is wicked.

I'm not sure what the problem is!

I'm trying to think what longitude would have both the least distortion and the greatest. Poles at -30 long and +150 long would probably have the least distortion since they're set in oceans, so that would put your "equator" at -120 long/+60 long.

Unfortunately, due to how "top heavy" the Northern Hemisphere is, there doesn't appear to be a good set of "skew line" (i.e., the "equator" line is not longitudinal or the equator based on current convention) pole co-ordinates that would have least distortion.

PS: My head hurts now because I was staring at a mercator projection that I was mentally de-projecting into a globe and then re-projecting it

CatCube
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### Re: 0977: "Map Projections"

chrth wrote:Actually, I have one last projection question.

The Mercator projection uses the equator as the, um, focus? -- not really sure what the cartographic term is. Couldn't you use the Prime Meridian (or any other longitude) and do a similar projection?

Maybe I'm wrong, but are you thinking of the Transverse Mercator projection? http://en.wikipedia.org/wiki/Transverse_Mercator_projection

Post edited: I quoted the wrong post by chrth

marioveen
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### Re: 0977: "Map Projections"

This is my favorite map, just invented by a Dutch guy. It is a map of the Netherlands based on TIME rather than distance and depending on the time it takes to get somewhere from your reference point the size and shape of the map changes. E.g. at night some places become very distant because they are not reachable by night trains: http://www.timemaps.nl/

eran_rathan
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### Re: 0977: "Map Projections"

CatCube wrote:
chrth wrote:Actually, I have one last projection question.

The Mercator projection uses the equator as the, um, focus? -- not really sure what the cartographic term is. Couldn't you use the Prime Meridian (or any other longitude) and do a similar projection?

Maybe I'm wrong, but are you thinking of the Transverse Mercator projection? http://en.wikipedia.org/wiki/Transverse_Mercator_projection

Post edited: I quoted the wrong post by chrth

Yep. The issue is that if you get outside of the fairly narrow bands that the TM projection uses, you start to get some really funky stuff going on - hence why there are so many UTM zones - each zone is conformal, but the distortion increases as you head farther and farther from the central meridian of the zone. There is a decent discussion of it at wikipedia.
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chrth
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### Re: 0977: "Map Projections"

CatCube wrote:
chrth wrote:Actually, I have one last projection question.

The Mercator projection uses the equator as the, um, focus? -- not really sure what the cartographic term is. Couldn't you use the Prime Meridian (or any other longitude) and do a similar projection?

Maybe I'm wrong, but are you thinking of the Transverse Mercator projection? http://en.wikipedia.org/wiki/Transverse_Mercator_projection

Post edited: I quoted the wrong post by chrth

I don't think so. I would expect the projection I'm thinking of would have the same characteristics of the Mercator, namely the distance between each degree growing greater as you head towards the "poles". It would just be the longitudinal degree distance that is growing greater, not the latitudinal one.

The transverse *seems* to be doing something else in addition (and the description of how the two are generated bears this out). What I'm looking for should use the exact same method as the Mercator projection to project, just using a different starting point.

AvatarIII
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### Re: 0977: "Map Projections"

how useful would a map projection that is simply the spherical earth imagined as a cube and then unfolded, so you'd have 4 squares in a row, centred on the equator, and then right-angled isosceles triangle above and below each square pointing towards the poles

obviously you'd get distortion close to the edges of each square side, but that could be accounted for by the edges mostly being centred to oceans, having a quick lok you could easily have the 4 sides be africa/europe, asia/australasia, the pacific, the americas
Last edited by AvatarIII on Fri Nov 18, 2011 2:54 pm UTC, edited 2 times in total.

steve waterman
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### Re: 0977: "Map Projections"

Just found this...was looking for another like it...but this is much better and more types are included...

Here is a fabulous applet for virtually all well-known projections

just scroll to page bottom...and well, see how different projections compare...

as you may know...a globe has circles as its Tissot ellipses
the closer to having equally sized ( area ) ellipses ..than the less area distortion
Tissot's reveals some angular info as well

you can scroll left and right and the map "rotates" in 2d

http://www.jhlabs.com/java/maps/proj/

there are almost 100 projections available

One day, perhaps they will acknowledge that a waterman projection exists.

I shall write them and see...
Last edited by steve waterman on Mon Nov 21, 2011 9:58 pm UTC, edited 1 time in total.
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PM 2Ring
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### Re: 0977: "Map Projections"

AvatarIII wrote:how useful would a map projection that is simply the spherical earth imagined as a cube and then unfolded, so you'd have 4 squares in a row, centred on the equator, and then right-angled isosceles triangle above and below each square pointing towards the poles

obviously you'd get distortion close to the edges of each square side, but that could be accounted for by the edges mostly being centred to oceans, having a quick lok you could easily have the 4 sides be africa/europe, asia/australasia, the pacific, the americas

Why a cube? (I think such things have already been discussed in this thread). Why not a dodecahedron, which is the most spherical of the Platonic solids?

http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/Foldout/Dodecahedron/dodecahedron.html

And some cubes:
http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/projPoly2.html

CatCube
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### Re: 0977: "Map Projections"

chrth wrote:I don't think so. I would expect the projection I'm thinking of would have the same characteristics of the Mercator, namely the distance between each degree growing greater as you head towards the "poles". It would just be the longitudinal degree distance that is growing greater, not the latitudinal one.

The transverse *seems* to be doing something else in addition (and the description of how the two are generated bears this out). What I'm looking for should use the exact same method as the Mercator projection to project, just using a different starting point.

Well, I'm not sure what you're trying to describe, then. The only difference between the Mercator and the Transverse Mercator is that the Mercator is a vertical cylinder centered on a particular latitude (conventionally the equator), while the Transverse Mercator is a horizontal cylinder centered on a line of longitude.

truenorth
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### Re: 0977: "Map Projections"

I just noticed that North America looks a bit like a Raptor. Don't say you haven't been warned.

(side note: when Google-imaging "raptor", you get way more images of ATVs than of the fighter jet, or even the dinosaur)

eran_rathan
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### Re: 0977: "Map Projections"

chrth wrote:
CatCube wrote:
chrth wrote:Actually, I have one last projection question.

The Mercator projection uses the equator as the, um, focus? -- not really sure what the cartographic term is. Couldn't you use the Prime Meridian (or any other longitude) and do a similar projection?

Maybe I'm wrong, but are you thinking of the Transverse Mercator projection? http://en.wikipedia.org/wiki/Transverse_Mercator_projection

Post edited: I quoted the wrong post by chrth

I don't think so. I would expect the projection I'm thinking of would have the same characteristics of the Mercator, namely the distance between each degree growing greater as you head towards the "poles". It would just be the longitudinal degree distance that is growing greater, not the latitudinal one.

The transverse *seems* to be doing something else in addition (and the description of how the two are generated bears this out). What I'm looking for should use the exact same method as the Mercator projection to project, just using a different starting point.

Maybe a Foucault Sinusoidal?
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dbosst
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### Re: 0977: "Map Projections"

Never cared about map projections until I saw the quincuncial projection ...
Now its my desktop background ...
WTF XKCD?

chrth
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### Re: 0977: "Map Projections"

Ok, in looking at the Transverse some more ... and taking the picture and rotating/re-orienting it to my expectations ... I think that is what I'm talking about. It's just kind of hard to process because of the whole South America thing - I wouldn't expect that what is on a north-south line now would end up at a right angle - but I looked at a real globe and yes, I can see that happening.

It's weird that the projection done longitudinally "appears" to distort more, but if I had to guess, that's due to exposure to the regular mercator projection my entire life.

steve waterman
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### Re: 0977: "Map Projections"

I had completely forgotten to replace any graphic of the projection - at actual size and resolution.

This is a portion of Africa at actual size and 100 percent resolution on a 36 inch waterman map.

It seems the low resolution, that I have shown, is so unclear to even detect that country names and border lines are included.

I had been concerned too, about putting up too good a resolution on the net, and did not notice that my "at actaul size and res" had got deleted somehow.

Too be clear...these are the same map...one is at 100 percent and the other "fuzzy" percent, does not matter.

Thanks Alex B, for the heads-up on this.

steve
Last edited by steve waterman on Mon Nov 21, 2011 9:59 pm UTC, edited 1 time in total.
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### Re: 0977: "Map Projections"

Anyone else having a Comic Sans moment with this Peters thing? It's so weird to discover a whole subculture that hates something you always thought was totally innocuous.

AndyS wrote:This one *really* says something about the person :

http://www.nytimes.com/slideshow/2011/1 ... os-17.html

Boy, she's gonna be sorry when Australia gets nuked like in The Martian Chronicles.
FiveMinute.net: Because stuff is long and life is short

steve waterman
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### Re: 0977: "Map Projections"

CZeke wrote:Anyone else having a Comic Sans moment with this Peters thing? It's so weird to discover a whole subculture that hates something you always thought was totally innocuous.

AndyS wrote:This one *really* says something about the person :

http://www.nytimes.com/slideshow/2011/1 ... os-17.html

Boy, she's gonna be sorry when Australia gets nuked like in The Martian Chronicles.

Coincidentally, Daan Strebe, its author, is the person that objected at wiki, and single-handed, got the waterman map removed from its pages.
Done, even before I was notified, actually wiki still has not notified me ...just poof...not "citeable" I think...one can read the "discussion at wiki", I cannot bear to, again.

This is however, my very first time seeing his "skinny" production/projection. It does, happily and haphazardly in fact, violate two of MY deal breakers.
Antarctica does not appear like that on the globe.
Parts of the globe are missing in the projection.
I wonder what a 1000 mile edged square around the south pole plots to ???...it would be cosmic relief, for me personally.
"While statistics and measurements can be misleading, mathematics itself, is not subjective."
"Be careful of what you believe, you are likely to make it the truth."
steve

CatCube
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### Re: 0977: "Map Projections"

chrth wrote:Ok, in looking at the Transverse some more ... and taking the picture and rotating/re-orienting it to my expectations ... I think that is what I'm talking about. It's just kind of hard to process because of the whole South America thing - I wouldn't expect that what is on a north-south line now would end up at a right angle - but I looked at a real globe and yes, I can see that happening.

It's weird that the projection done longitudinally "appears" to distort more, but if I had to guess, that's due to exposure to the regular mercator projection my entire life.

Yeah, it's weird to look at because at first you intuitively expect the longitude to behave the way latitude does in the Mercator. However, since lines of longitude aren't parallel, their projections come together at the poles, where the latitude lines on the Mercator never come together (because latitude lines never meet). So you have the longitude lines far from the central meridian at the equator really far out, and that same longitude still has to be really close to the middle at the upper latitudes. This type of map is really good for showing a thin strip (6° of longitude at a time for the UTM grid, for example), but it's a bad projection to use for much further out.

What a world map in Transverse Mercator is really useful for is comparing to the Mercator, to show you how map projections can confound your expectations, and it provides an immediately-graspable view of extreme map distortions.

Strebe
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### Re: 0977: "Map Projections"

steve waterman wrote:Coincidentally, Daan Strebe, its author, is the person that objected at wiki, and single-handed, got the waterman map removed from its pages.
Done, even before I was notified, actually wiki still has not notified me ...just poof...not "citeable" I think...one can read the "discussion at wiki", I cannot bear to, again.

This is however, my very first time seeing his "skinny" production/projection. It does, happily and haphazardly in fact, violate two of MY deal breakers.
Antarctica does not appear like that on the globe.
Parts of the globe are missing in the projection.
I wonder what a 1000 mile edged square around the south pole plots to ???...it would be cosmic relief, for me personally.

I see my name has been used in vain.

And now you are wrong yet again; no parts of the globe are missing on that particular projection of mine.

I like your map, Mr. Waterman. That doesn’t mean I should have overlooked its lack of qualification for inclusion in Wikipedia. That doesn’t mean I ought to ignore the kinks in the graticule. That doesn’t mean I should pretend interruptions come without cost. That doesn’t mean I should prefer your map and obsessions over all else. Can you not yearn for a more vibrant world outside the confines of your own head?

Best regards, and with apologies to Mr. Munroe for this intrusion,
— daan Strebe

Harry Voyager
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### Re: 0977: "Map Projections"

keithl wrote:So, what projection do they use for world maps at Admundsen-Scott south pole station? With Mercator, the bathroom down the hall is about 100 kilometers away ...

Since it is run by the US, and primarily served by the US armed forces, that I'm pretty sure they use the UTM, which has convenient polar cutouts mapped using the Universal Polar Stereographic coordinate system.

This sort of thing is why the US Army Corp of Engineers went to all the trouble of developing UTM in the first place.

"We need to deliver 15 tonns of anti-bacterial soap where???"

steve waterman
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### Re: 0977: "Map Projections"

Strebe wrote:
steve waterman wrote:Coincidentally, Daan Strebe, its author, is the person that objected at wiki, and single-handed, got the waterman map removed from its pages.
Done, even before I was notified, actually wiki still has not notified me ...just poof...not "citeable" I think...one can read the "discussion at wiki", I cannot bear to, again.

This is however, my very first time seeing his "skinny" production/projection. It does, happily and haphazardly in fact, violate two of MY deal breakers.
Antarctica does not appear like that on the globe.
Parts of the globe are missing in the projection.
I wonder what a 1000 mile edged square around the south pole plots to ???...it would be cosmic relief, for me personally.

I see my name has been used in vain.

Mr. Waterman, we’re not rivals. It’s sad you don’t understand that. It’s sad you couldn’t sit back and bask in the moment of fame Mr. Munroe so generously afforded, but instead chose to co-opt his forum for self-promotion.
http://xkcd.com/977/

The contents of your many postings are sad, being riddled with errors, paranoia, and aggrandizement of your map/projection/thing. It’s sad you chose to try to bias what few readers you might have left against me (what do I have to do with any of this?),
http://xkcd.com/978/

despite the many, many hours I have spent engaging you in good faith. I have encouraged you to publish your work in a peer-reviewed journal so that people have something reliable to refer to, offered to help however necessary, and tendered constructive comments on your attempts to do so. I have spent excruciating hours engaging your correspondence on kinematics.

I tried. I submitted work I believe twice, to one of those, that you had provided in a list. Then eventually said that they would need to give to an expert as there was no one there qualified, to judge. The only gave me one reference name...that was you, Daan Strebe. This was circular now, and to keep my focus, I gave up and moved on. It seems that someone posted to wiki that I cannot use xkcd either as reference...nor the work at Carlos Furuti, Izidor Hafner, and numerous other collaborative works...seen via my site. No waterman polyhedra, nor collaborators either for sphere packing...which indeed derives the polyhedron required for the this projection.

nor on-line waterman polyhedral generator http://paulbourke.net/geometry/waterman/gen/index.html
nor http://www.antiprism.com/programs/waterman.htmlAntiprism[/url]
Also, there are specific routines in Mupad and Mathematica and a few documented at MathWorld.
nor http://www.ac-noumea.nc/maths/polyhedr/Waterman_.htmMaurice Starck
nor http://paulbourke.net/geometry/waterman/index.htmlPaul Bourke...
including a aqrt 10,000,00 done several years ago with a supercomputer, and the image can be downloaded.
nor http://www.watermanpolyhedron.com/2011IH2.htmlwaterman gored globes
nor http://www.software3d.com/UsersModels.php#watermanGreat Stella..has virtually millions of watermans with their nets drawn by the software automatically... that, with no inventory, like Mark Newbold's applet, the next below, are quite amazing pieces of software....the applet is wik-hed fast and almost instantaneously generates the new poly...from just a single value...indeed, not one object in any inventory.

...none of thes eare valid for the wiki and pass the wiki test/filter/judgment/acceptance...as i understand their rules for "citable".
nor http://dogfeathers.com/java/ccppoly.html
nor http://www.saintjohnsabbey.org/wenninger/setten/index.html
at his own polyhedral site http://www.saintjohnsabbey.org/wenninger/index.html
he is a monk...and one rightfully assumes a man of trust.
http://en.wikipedia.org/wiki/Magnus_Wenninger
nor [url][his most generous endorsement in the polyhedral arena.[/color]
nor http://www.watermanpolyhedron.com/gsstels.html

Publish or perish, seems to be the wiki manta. This likely would do squat either as verification...as "citable"...

Yet all you can muster in your moment of fame is spite and a grab at free advertising. It’s all just so… sad.

You see this all the above as self-promotion, whereas to me, this is my body of work..and I also have some Physics stuff too...

And now you are wrong yet again; no parts of the globe are missing on that particular projection of mine.

Then why is now the first time I hear from you that you have a projection, or any through this thread that i first see it. let me guess, it IS on wiki and nicely secured with citation. I would have looked if I had EVER heard on such a thing.

So, due to the huge distortion to Antarctica's real shape in the Strebe map, and with no actual numbers, and no documentation ever seen, I could not tell exactly how poor the polar region was. No matter, i still would like to see your map with a square of 100 miles edge around the south pole on the globe plotted on new projection. I suspect you will fail to post such a request, and merely avoid it, by switching focus as diversion and never will actually respond to it as a posted graphic.

I like your map, Mr. Waterman.

Thank you Daan.

That doesn’t mean I should have overlooked its lack of qualification for inclusion in Wikipedia.

You are correct...it is not "citable" under the wiki rules.

That doesn’t mean I ought to ignore the kinks in the graticule. That doesn’t mean I should pretend interruptions come without cost. That doesn’t mean I should prefer your map and obsessions over all else. Can you not yearn for a more vibrant world outside the confines of your own head?

Best regards, and with apologies to Mr. Munroe for this intrusion
— daan Strebe

upcoming. posting to #0977 forum.....[ not -self promoting...just some more work and those results ]

on Monday...I will post 4 brand new projections, never seen before, on a cuboctahedron - akin to Fuller's..though my poles will always go through a face center, as is true for those below too. This first set will have its pieces symmetrically placed...say, like a totem pole. These will have numerous land breaks. I will also post a set made from each, that rearranges those pieces to minimize the land breaks ( is the waterman 1...from sphere packing )

on Wednesday... same for the Cahill octahedron..none as per his butterfly...and again two sets of 4 each...symmetrical and rearranged
( is the waterman 2...from sphere packing )

on Friday...8 more from the waterman 3

on Monday...8 more waterman 5...akin to my...but all different

( waterman 1 and 4 generate the same polyhedron...with edges of 1 and respectively. )

on Wednesday...8 more for the waterman 6...just for comparison sale.

So, a total of 40 new projections, with 8 Fuller cuboctahedron projections and 8 Cahill octahedron projections.

Will you post your map here or the links to it please Daan ? I would like to see some math...and the polar square thingie eh ?
Last edited by steve waterman on Mon Nov 21, 2011 9:23 pm UTC, edited 1 time in total.
"While statistics and measurements can be misleading, mathematics itself, is not subjective."
"Be careful of what you believe, you are likely to make it the truth."
steve

esperanto41
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### Re: 0977: "Map Projections"

I have posted on my website a detailed and profusely illustrated critique of Buckminster Fuller's 1943 and 1954 Dymaxion maps, as compared to B.J.S. Cahill's 1909 Butterfly World Map, which he continued to develop for the next three decades. As stated therein, "I love Bucky, but Cahill's map is a lot better. Here's how."

Strebe
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### Re: Strebe projection

steve waterman wrote:Will you post your map here or the links to it please Daan ? I would like to see some math...and the polar square thingie eh ?

An image more conventional than a tattoo can be seen here:

Code: Select all

http://www.mapthematics.com/ProjectionsList.php?Projection=180#Strebe%201995

steve waterman wrote:Then why is now the first time I hear from you that you have a projection, or any through this thread that i first see it. let me guess, it IS on wiki and nicely secured with citation. I would have looked if I had EVER heard on such a thing.

I am not responsible for your uninformed proclamations about my projection. I don’t promote my projection. It’s not in Wikipedia because I haven’t published it. It’s only one of many projections I have devised; so far I have chosen to publish only a couple. For that matter, even the ones I have published aren’t in Wikipedia.

Generating formulæ:
[imath]\begin{align*}
x &= \frac {2D}{k_g} \cos \varphi_p \sin \lambda_p \\
y &= k_g D \sin \phi_p \\
k_g &= 1.35 \\
D &= \sqrt {\frac {2}{1 + \cos \varphi_p \cos \lambda_p}} \\
\sin \varphi_p &= \frac {2 \sin^{-1}\frac{\sqrt{2} y_e}{2} + r y_e}{\pi} \\
\lambda_p &= \frac {\pi x_e}{4r} \\
r &= \sqrt {2 - y_e^2} \\
x_e &= 2 k_g \frac {\lambda (1+\cos \theta)}{\sqrt {4\pi + \pi^2}} \\
y_e &= 2 \frac {\sqrt \pi \sin \theta}{k_g \sqrt {4+\pi}} \\
\text{where } \theta \text{ is solved iteratively:} \\
\theta + \sin \theta \cos \theta + 2 \sin \theta &= \frac{1}{2} \left(4+\pi\right) \sin \varphi \\
\end{align*}[/imath]

If you cannot see the TeΧ formulæ properly:
Spoiler:
Strebe 1995 equal-area projection

The map is precisely equal-area.

Regards,
— daan Strebe

scarletmanuka
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Location: Perth, Western Australia

### Re: 0977: "Map Projections"

Capt. Jon wrote:Now people have explored projections i reckon it is time to post some of Layla Curtis' work:

Awesome. I imagine using a map like that as reference material would make history very confusing.

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Joined: Fri Jul 15, 2011 5:44 am UTC

### Re: 0977: "Map Projections"

My holy damn humping a dead rabbit, Randall you sir are the worlds finest troll. After many years of readership, Thank you.

AvatarIII
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### Re: 0977: "Map Projections"

PM 2Ring wrote:
AvatarIII wrote:how useful would a map projection that is simply the spherical earth imagined as a cube and then unfolded, so you'd have 4 squares in a row, centred on the equator, and then right-angled isosceles triangle above and below each square pointing towards the poles

obviously you'd get distortion close to the edges of each square side, but that could be accounted for by the edges mostly being centred to oceans, having a quick lok you could easily have the 4 sides be africa/europe, asia/australasia, the pacific, the americas

Why a cube? (I think such things have already been discussed in this thread). Why not a dodecahedron, which is the most spherical of the Platonic solids?

http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/Foldout/Dodecahedron/dodecahedron.html

And some cubes:
http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/projPoly2.html

why a cube? just for simplicity I guess, I was trying to think of the best way to project a globe with a little distortion and as few separate sides as possible. the waterman has 9 sides, I thought perhaps it might be simpler with just 6. thanks for the links though
this is as close to what I was imagining as possible. although i imagined Africa and s.America more centralised on square sides. (shifted 2 "squares" to the right perhaps)

Copper Bezel
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### Re: 0977: "Map Projections"

Are these actually real people? Is that the assumption we're working on? I mean, I really assumed that Waterman was a (non-Waterman) troll until the bit from Strebe, who seems likely to be himself, whoever that is, and now we have Gene Keyes, too (esperanto41)?

It's a bit funny; I'm not one of the people who know things about maps, but I ran into the Cahill-Keyes projection while reading the thread, possibly from the one time that Fake Steve mentioned it himself without realizing that's where I'd seen it, and intended to mention it to troll Fake Steve until seeing Keyes' own post. It really does do all the things Fake Steve wants, but slightly better. It does not, of course, come with a promotional poem like the Waterman, but I think that's largely preferable in this sort of situation. = ) If the real criterion here is what makes the most accurate-but-attractive wall art, this one really has to win out for the sane semi-consistency of north-south and east-west alone.

Pfhorrest wrote:You know, a map that would be really interesting would be some kind of rectangular projection of the geoid, with the latitude and longitude lines drawn on it. Since lat and long are lines on a sphere approximating the geoid, if we project the actual geoid onto a rectangle rather than that sphere, the lat and long lines should appear distorted in proportion to how the geoid varies from that sphere, giving us a visualization of exactly how nonsphereical the geoid is in what geographic areas.

It's not far enough from a sphere to really make any difference at an ordinary map size, though.

SpringLoaded12 wrote:You're like a modern-day Holden Caulfield, except that no one would read a book about you.

I. Love. This.
So much depends upon a red wheel barrow (>= XXII) but it is not going to be installed.

she / her / her

steve waterman
Posts: 1610
Joined: Mon Nov 14, 2011 4:39 pm UTC

### Re: Strebe projection

Strebe wrote:
steve waterman wrote:Will you post your map here or the links to it please Daan ? I would like to see some math...and the polar square thingie eh ?

An image more conventional than a tattoo can be seen here:

Code: Select all

http://www.mapthematics.com/ProjectionsList.php?Projection=180#Strebe%201995

steve waterman wrote:Then why is now the first time I hear from you that you have a projection, or any through this thread that i first see it. let me guess, it IS on wiki and nicely secured with citation. I would have looked if I had EVER heard on such a thing.

Strebe wrote:I am not responsible for your uninformed proclamations about my projection. I don’t promote my projection. It’s not in Wikipedia because I haven’t published it. It’s only one of many projections I have devised; so far I have chosen to publish only a couple. For that matter, even the ones I have published
aren’t in Wikipedia.

Thanks so much for posting both the math and your projection. This gives me a chance to give critique regarding MY personal criteria

On the good side
1 equal area...thus you get double points for this, this is excellent
2 land sinuses....good in general
3 symmetry...very good
4 interruptions...very good
5 obvious equator and prime meridian
6 pole is a single point...very good

1 the shrinking in between 30 and 45 degrees
2 the expanding in between 0 and 15, and between 75 and 90
4 the shape of Antarctica...very bad

For me, this is a very good compromise projection as it gets bonus points for being equal area.
it does very well it most arenas of my concern. However, it is the poor of Antarctica, but far more importantly is my concern below...

Greenland's northern land ends at about 83 degrees...leaving 7 degrees of water to the actual pole....which, visually, seems to be non-existent above Greenland in your projection. Is there a larger version some place where a more detailed look where your northern 75 to 90 section can be seen a little close-up ?

Strebe wrote:Generating formulæ:
[imath]\begin{align*}
x &= \frac {2D}{k_g} \cos \varphi_p \sin \lambda_p \\
y &= k_g D \sin \phi_p \\
k_g &= 1.35 \\
D &= \sqrt {\frac {2}{1 + \cos \varphi_p \cos \lambda_p}} \\
\sin \varphi_p &= \frac {2 \sin^{-1}\frac{\sqrt{2} y_e}{2} + r y_e}{\pi} \\
\lambda_p &= \frac {\pi x_e}{4r} \\
r &= \sqrt {2 - y_e^2} \\
x_e &= 2 k_g \frac {\lambda (1+\cos \theta)}{\sqrt {4\pi + \pi^2}} \\
y_e &= 2 \frac {\sqrt \pi \sin \theta}{k_g \sqrt {4+\pi}} \\
\text{where } \theta \text{ is solved iteratively:} \\
\theta + \sin \theta \cos \theta + 2 \sin \theta &= \frac{1}{2} \left(4+\pi\right) \sin \varphi \\
\end{align*}[/imath]

If you cannot see the TeΧ formulæ properly:
Spoiler:
Strebe_1995.png

The map is precisely equal-area.

Regards,
— daan Strebe
"While statistics and measurements can be misleading, mathematics itself, is not subjective."
"Be careful of what you believe, you are likely to make it the truth."
steve

PM 2Ring
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Location: Sydney, Australia

### Re: Strebe projection

Strebe wrote:An image more conventional than a tattoo can be seen here:
http://www.mapthematics.com/ProjectionsList.php?Projection=180#Strebe%201995

Generating formulæ:
Spoiler:
[imath]\begin{align*}
x &= \frac {2D}{k_g} \cos \varphi_p \sin \lambda_p \\
y &= k_g D \sin \phi_p \\
k_g &= 1.35 \\
D &= \sqrt {\frac {2}{1 + \cos \varphi_p \cos \lambda_p}} \\
\sin \varphi_p &= \frac {2 \sin^{-1}\frac{\sqrt{2} y_e}{2} + r y_e}{\pi} \\
\lambda_p &= \frac {\pi x_e}{4r} \\
r &= \sqrt {2 - y_e^2} \\
x_e &= 2 k_g \frac {\lambda (1+\cos \theta)}{\sqrt {4\pi + \pi^2}} \\
y_e &= 2 \frac {\sqrt \pi \sin \theta}{k_g \sqrt {4+\pi}} \\
\text{where } \theta \text{ is solved iteratively:} \\
\theta + \sin \theta \cos \theta + 2 \sin \theta &= \frac{1}{2} \left(4+\pi\right) \sin \varphi \\
\end{align*}[/imath]

Thanks, daan. That's a rather nice looking projection.

Would you be able to post some info or a link explaining the meaning of the variables in your formulæ for those of us who are cartography newbies? Obviously lambda & phi are longitude & latitude, respectively, and x & y refer to coördinates on the projection, but what do the various subscripts mean?

steve waterman,
Would you please repair the quotes in your last couple of posts? In their current form, it's very hard to distinguish between your material and the material you are responding to.

Thanks.

PS. Using red in your posts makes it look like a moderator has modified your text, since that colour is reserved for their use on these forums.

steve waterman
Posts: 1610
Joined: Mon Nov 14, 2011 4:39 pm UTC

### Re: Strebe projection

PM 2Ring wrote:
Strebe wrote:An image more conventional than a tattoo can be seen here:
http://www.mapthematics.com/ProjectionsList.php?Projection=180#Strebe%201995

Generating formulæ:
Spoiler:
[imath]\begin{align*}
x &= \frac {2D}{k_g} \cos \varphi_p \sin \lambda_p \\
y &= k_g D \sin \phi_p \\
k_g &= 1.35 \\
D &= \sqrt {\frac {2}{1 + \cos \varphi_p \cos \lambda_p}} \\
\sin \varphi_p &= \frac {2 \sin^{-1}\frac{\sqrt{2} y_e}{2} + r y_e}{\pi} \\
\lambda_p &= \frac {\pi x_e}{4r} \\
r &= \sqrt {2 - y_e^2} \\
x_e &= 2 k_g \frac {\lambda (1+\cos \theta)}{\sqrt {4\pi + \pi^2}} \\
y_e &= 2 \frac {\sqrt \pi \sin \theta}{k_g \sqrt {4+\pi}} \\
\text{where } \theta \text{ is solved iteratively:} \\
\theta + \sin \theta \cos \theta + 2 \sin \theta &= \frac{1}{2} \left(4+\pi\right) \sin \varphi \\
\end{align*}[/imath]

Thanks, daan. That's a rather nice looking projection.

Would you be able to post some info or a link explaining the meaning of the variables in your formulæ for those of us who are cartography newbies? Obviously lambda & phi are longitude & latitude, respectively, and x & y refer to coördinates on the projection, but what do the various subscripts mean?

steve waterman,
Would you please repair the quotes in your last couple of posts? In their current form, it's very hard to distinguish between your material and the material you are responding to.

PS. Using red in your posts makes it look like a moderator has modified your text, since that colour is reserved for their use on these forums.

Thanks...got that fixed now...and went back and changed that i could.
Last edited by steve waterman on Mon Nov 21, 2011 10:04 pm UTC, edited 1 time in total.
"While statistics and measurements can be misleading, mathematics itself, is not subjective."
"Be careful of what you believe, you are likely to make it the truth."
steve

Strebe
Posts: 6
Joined: Sat Nov 19, 2011 9:48 pm UTC

### Re: Strebe projection

PM 2Ring wrote:Thanks, daan. That's a rather nice looking projection.

Would you be able to post some info or a link explaining the meaning of the variables in your formulæ for those of us who are cartography newbies? Obviously lambda & phi are longitude & latitude, respectively, and x & y refer to coördinates on the projection, but what do the various subscripts mean?

Thanks, PM 2Ring. I have just posted an explanation here:

Code: Select all

http://mapthematics.com/forums/viewtopic.php?f=8&t=223

Regards,
— daan Strebe

Manabu
Posts: 30
Joined: Tue Nov 30, 2010 1:57 am UTC

### Re: 0977: "Map Projections"

I know it was already said, but it should be said again: this comic is missing the best projection of all!

The Mollweide Projection!!

Gall-Peters gives equal-area projections a bad name... but, on the other hand, at least pointed the flaws of Mercator projection to the large public. I liked the following resolution, calling for a ban on all rectangular coordinate maps for general use:
In 1989, seven North American professional geographic organizations (including the American Cartographic Association, National Council for Geographic Education, Association of American Geographers, and the National Geographic Society) adopted a resolution that called for a ban on all rectangular coordinate maps.

WHEREAS, the earth is round with a coordinate system composed entirely of circles, and

WHEREAS, flat world maps are more useful than globe maps, but flattening the globe surface necessarily greatly changes the appearance of Earth's features and coordinate systems, and

WHEREAS, world maps have a powerful and lasting effect on peoples' impressions of the shapes and sizes of lands and seas, their arrangement, and the nature of the coordinate system, and

WHEREAS, frequently seeing a greatly distorted map tends to make it "look right,"

THEREFORE, we strongly urge book and map publishers, the media and government agencies to cease using rectangular world maps for general purposes or artistic displays. Such maps promote serious, erroneous conceptions by severely distorting large sections of the world, by showing the round Earth as having straight edges and sharp corners, by representing most distances and direct routes incorrectly, and by portraying the circular coordinate system as a squared grid. The most widely displayed rectangular world map is the Mercator (in fact a navigational diagram devised for nautical charts), but other rectangular world maps proposed as replacements for the Mercator also display a greatly distorted image of the spherical Earth.

Djehutynakht wrote:I managed to get into a debate about the Galls-Peters and the Mercator today. I feel accomplished.

I like this subject.

I hope you mentioned Mollweide Projection. You did it, right?

steve waterman
Posts: 1610
Joined: Mon Nov 14, 2011 4:39 pm UTC

### Re: 0977: "Map Projections"

Here are my four new Waterman cuboctahedron projections...

Since there are so few pieces, then, at this stage , the symmetry of the layout will seem random.

These 4 cuboctahedron based projections are directly related to my root 1
http://paulbourke.net/geometry/waterman/index.html

so root 1 re Fuller - however, unlike Fuller , all poles go through a face center.
Fuller also does not have his pole at a vertex, as does Cahill.

I have yet to be able to take pictures of these 4 projections above, with a minimal land break arrangement... to make some jpgs,
and will have to post those tomorrow.

On Wednesday, i will post my root 2 set ( see same page as above )
...the octahedron...as was Cahill's. None of these new 8 projections will be his butterfly pattern.
Thsse 4 cuboctahedron based projections are directly related to my root 2
"While statistics and measurements can be misleading, mathematics itself, is not subjective."
"Be careful of what you believe, you are likely to make it the truth."
steve

steve waterman
Posts: 1610
Joined: Mon Nov 14, 2011 4:39 pm UTC

### Re: 0977: "Map Projections"

Now the cuboctahedron re-arranged....from my previous cuboctahedron post...

The equator is in red...the prime meridian/date-line is in blue

[img]http://www.watermanpolyhedron.com/images/watermanprojection1f.jpg[img]
"While statistics and measurements can be misleading, mathematics itself, is not subjective."
"Be careful of what you believe, you are likely to make it the truth."
steve

steve waterman
Posts: 1610
Joined: Mon Nov 14, 2011 4:39 pm UTC

### Re: 0977: "Map Projections"

I have decided to place these new projections onto my site instead of here in this forum...
After failing to have the first graphic above, not show up in the previous post...I believe, for sure, that file is sitting there yet just is being ignored.

These will take a couple of days to complete and I will post that address here at this forum where all new projections
can be better organized and formatted than here.
"While statistics and measurements can be misleading, mathematics itself, is not subjective."
"Be careful of what you believe, you are likely to make it the truth."
steve

esperanto41
Posts: 2
Joined: Mon Nov 21, 2011 4:47 am UTC

### Re: 0977: "Map Projections"

I have just posted a 40 MB pdf version of the Cahill-Keyes Megamap, 1/1,000,000 and 132' x 65' (40 m x 20 m), which can be easily opened and scrolled if you use the Chrome browser. It can be zoomed in pre-set increments, from 9% to 100% and beyond. (When first opening the giant pdf, you must scroll it down and to the right.) Available for free from this page on my website:
http://www.genekeyes.com/MEGAMAP-BETA-1/Megamap-Beta-1.html
PS: esperanto41 is Gene Keyes, and vice versa.

PM 2Ring
Posts: 3715
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Location: Sydney, Australia

### Re: Strebe projection

Strebe wrote:Thanks, PM 2Ring. I have just posted an explanation here:
http://mapthematics.com/forums/viewtopic.php?f=8&t=223

Thanks again. That "projection of a projection" technique is a neat idea; I guess it could be used to make vast numbers of variant projections. And you could do all sorts of weird & wonderful things if you use a different surface for the intermediate stage, rather than using a sphere. Reading your explanation in the above link reminded me of a projection technique that I read about a few years ago, which involved a couple of stereographic projections, but using a pseudosphere as the intermediate surface. But I can't for the life of me remember what the application was.

BTW, there seems to be a minor typo in your LaTeX. On line 2, you've used \phi_p instead of \varphi_p.