[HN Gopher] A flat map with the least error possible: The Gott-G...
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A flat map with the least error possible: The Gott-Goldberg-
Vanderbei projection
Author : westcort
Score : 164 points
Date : 2024-04-22 03:00 UTC (20 hours ago)
(HTM) web link (vanderbei.princeton.edu)
(TXT) w3m dump (vanderbei.princeton.edu)
| kristopolous wrote:
| Is there any reason why the rotation animation doesn't just use
| CSS rotation? The code looks rather complicated and this old
| laptop seems to really be unhappy to do what appears to just be
| two images doing a standard rotation.
|
| Is it not that?
| axblount wrote:
| It's performing the projection in real time. It has to because
| you can change the perspective of the projection by clicking on
| the map.
| kristopolous wrote:
| Yeah sure, that feature makes sense. But when you start
| rotating it by pressing the spin/pause button in the
| interface, don't things become much simpler?
|
| Maybe the answer is "no" but I really can't understand why.
| kzrdude wrote:
| I think you're right.. it looks that way visually. You'd
| want to render at increased resolution to improve quality
| and then do a simple rotate?
| DougBTX wrote:
| Probably just because it is harder to implement, the CSS
| would be an optimisation of a specific case on top of the
| general code. Classic cost (in time and overall complexity)
| vs performance tradeoff.
| kristopolous wrote:
| How would projection code get you rotational animation
| for free?
| DougBTX wrote:
| The projection logic has an orientation parameter (which
| you'd probably always want, so that you can pick initial
| rotation), so the rotation animation itself is just
| updating the param on a timer.
|
| Basically line 182 plus the calls to redraw:
| gamma -= 0.003;
|
| To do it with CSS, at a minimum it would require adding
| support for rendering to two different canvas elements.
| Someone wrote:
| > To do it with CSS, at a minimum it would require adding
| support for rendering to two different canvas elements.
|
| It might into trouble because rotating a rectangular
| <img> area varies it in height and width [1], but I would
| investigate animating a 'rotate' transformation.
| https://developer.mozilla.org/en-
| US/docs/Web/CSS/transform-f....
|
| [1] seems to have a demo without that problem, so it
| seems this can be made to work
| https://imagekit.io/blog/image-rotation-html-and-css-
| imageki...
| brianshaler wrote:
| Here's an interactive WebGL implementation:
| https://brian.sh/around/index.html
|
| It doesn't have a satellite image option, but that would
| arguably be simpler than drawing continents like it does
| now. It uses a seldom-redrawn base canvas (menu -> debug
| > show 2d texture) then the discs are drawn with a simple
| shader.
| alanbernstein wrote:
| Before I pushed the rotate button, I expected the rotation to
| be along a different axis, not the one aligned with the
| projection. i didn't look at the code, does it support that?
| kbrosnan wrote:
| The paper is from 2012 if the visualization is from that era JS
| makes sense.
| zilti wrote:
| At that point, just render a globe.
| 4gotunameagain wrote:
| Just because it is two discs it does not mean that it could be
| replaced with a pinpoint camera render.
|
| Hemispherical projections have different properties.
|
| https://map-projections.net/compare.php?p1=azimuthal-equidis...
| GuB-42 wrote:
| That would be just another projection. Most likely a
| perspective or orthographic projection.
| 317070 wrote:
| But, for what definition of "error"?
| lancebeet wrote:
| This article explains how they measure the error.
|
| https://www.scientificamerican.com/article/the-most-accurate...
|
| >Previously, Goldberg and I identified six critical error types
| a flat map can have: local shapes, areas, distances, flexion
| (bending), skewness (lopsidedness) and boundary cuts. >The
| Goldberg-Gott error score (sum of squares of the six normalized
| individual error terms)[...]
| dmurray wrote:
| This is a nice generalised metric, and it's mentioned a few
| times that the authors' projection is good for a set of maps
| of the solar system.
|
| But a generalised metric misses some points: practical maps
| of the world emphasise continuity at the points the mapmaker
| subjectively considers important. The standard Mercator
| projection has the London meridian at the centre, not purely
| because Europe was considered important but because the
| antipodal meridian through the Pacific, not passing through
| any population centres, is considered unimportant. Other
| projections like Goode-homolosine [0] are even more
| opinionated.
|
| This map emphasises the polar areas, which are front and
| centre, and introduces a boundary cut along the equator,
| cutting populous countries like Brazil, Kenya and Indonesia
| in two. (It's ridiculous to say there's no boundary cut
| because you can turn the map over - in the same way you can
| fold a Mercator map or roll it into a cylinder, though
| admittedly other projections like Winkel-tripel don't have
| this property).
|
| [0]
| https://en.m.wikipedia.org/wiki/Goode_homolosine_projection
| jonathanlydall wrote:
| I didn't realize until another comment mentioned it, but
| before you start spinning, you can click a part of the
| world to center the map on it.
| dmurray wrote:
| Wherever you centre it, though, there's a boundary cut
| which is an entire great circle (compared to Mercator
| which has a boundary cut half as long).
| KineticLensman wrote:
| > But a generalised metric misses some points: practical
| maps of the world emphasise continuity at the points the
| mapmaker subjectively considers important.
|
| There is an old military complaint that battles always seem
| to take place at the very corner of the map. I guess that
| modern C2 systems have eliminated this problem.
| aidenn0 wrote:
| Choosing the poles to be in the Pacific and Indian Oceans
| at the equator nearly completely solves this problem; the
| Russian Far East is cut, but no other populated landmasses
| are.
| hoseja wrote:
| For the OP projection, the best centre seems to be around the
| Isle of Man. Edit: Just north of Cornwall gets most of the land
| except SEA, southern tip of South America and
| Australia+Antarctica.
|
| I just like properly split sinusoidal map the most though [0]
| Sinusoidal map is where you start at the pole and unwrap the
| circles of latitude.
|
| [0]
| https://en.wikipedia.org/wiki/File:Usgs_map_sinousidal_equal...
| jsjohnst wrote:
| Respectfully, disagree completely. Like somebody else stated,
| clicking in the Arabian Sea just to the west of India neatly
| splits the two circles without cutting continents. One circle
| becomes Africa / Europe / Asia / Australia, the other becomes
| North & South America.
|
| Example: https://imgur.com/rgeg1Lc
| hoseja wrote:
| Oh yea, that's a good one. Too bad you have to reset after
| each try, makes it hard to experiment.
| mci wrote:
| If your metric is the length of the land that the map boundary
| cuts, then the best centre lies somewhere in the Indian Ocean.
| Proven here: [1]
|
| [1] https://marcinciura.wordpress.com/2015/11/17/slicing-
| earth-c...
| hoseja wrote:
| I seem to have found this one, basically:
| https://marcinciura.wordpress.com/min-land-4/ But my metric
| wasn't as much intersection as maximum land area in one half.
|
| edit: I didn't even know about this:
| https://en.wikipedia.org/wiki/Land_and_water_hemispheres
| kuschku wrote:
| Gott-Goldberg-Vanderbei may have a lower error, but its
| usefulness is also significantly reduced.
|
| My favourite for world maps is still Winkel Tripel
| (https://en.wikipedia.org/wiki/Winkel_tripel_projection). Winkel-
| Tripel was given one of the best ranks by Gott and Goldberg,
| before they developed the projection in the OP.
|
| Winkel Tripel used to be the standard until Google Maps came
| along and pushed everyone back to using Mercator for data
| visualization and political maps.
| defrost wrote:
| The triple was|is a damn fine world projection.
|
| My favourite for "local area" usage was the old New Zealand Map
| Grid .. _not_ a polyconic projection, rather a custom complex
| polynomial optimised to reduce grid error in toto (by multiple
| metrics) for the North and South Islands of New Zealand.
|
| As a topographic grid projection it was aligned with the "spine
| of best fit" of the two islands, rather than stright up
| North|South aligned, and weighted to minimise the N|S and E|W
| distortion within the land region of interest as distance from
| the centre zone increased.
|
| https://www.linz.govt.nz/guidance/geodetic-system/coordinate...
|
| There were very few (three ?) in use about the world pre WGS84
| .. and like many things went the way of the Dodo, the Krasovsky
| 1940 ellipsoid, the Bessel 1841, and all those tens and tens of
| other ellipsoids, datums, and projections of days yore.
| bradrn wrote:
| > My favourite for "local area" usage was the old New Zealand
| Map Grid .. not a polyconic projection, rather a custom
| complex polynomial optimised to reduce grid error in toto (by
| multiple metrics) for the North and South Islands of New
| Zealand.
|
| Paper link:
| https://www.linz.govt.nz/resources/research/conformal-
| mappin...
| defrost wrote:
| That's the paper alright, I've got a physical copy in a
| filing cabinet somewhere :-)
|
| IIRC, for anyone else looking for a curio, there was one
| state|county|region in the USofA that used a custom
| conformal projection for a local grid system pre 1980s ..
| somewhere outside the regular PLA mapping grids.
| saalweachter wrote:
| I don't think this is the one you're thinking of, but in
| NYC the subway maps (which are not particularly
| geographically accurate) are drawn with Manhattan
| vertical, rather than on its 20-something degree tilt.
| defrost wrote:
| > subway maps (which are not particularly geographically
| accurate)
|
| Subway maps and bus maps in cities are very typically not
| geographically accurate map _projections_ .. the focus
| tends to be on a compact stylisation that represents
| _connections_ for travel plans.
|
| Historically Roman Road maps were on a scroll and very
| linear, today the UK underground tube maps are displayed
| in a manner that aids commuters in making connections.
|
| I'm not all that familiar with NYC subway maps (only
| there for a short time in the 80s .. and who can recall
| their youth?) but I remember them being more "graphic"
| rather than "geographic".
|
| I'm thinking more of a local land projection map that
| would have been used for local surveying in the days
| prior to global GPS .. but thanks for the thought all the
| same!
| theluketaylor wrote:
| The NYC MTA map is pretty geographic by metro map
| standards. There are some obvious distortions, like
| Manhattan being wider than it actually is and it's not
| actually vertical as noted before. It also includes a lot
| of surface street and neighbourhood references. Most
| metro maps just focus on station names.
|
| Compared with something like the London Tube map or Paris
| Metro map it's extremely geographic.
|
| https://new.mta.info/map/5256
| lupire wrote:
| Tilt relative to what? Rotating a map doesn't make it
| inaccurate.
| playworker wrote:
| Oblig: https://xkcd.com/977/
| nox101 wrote:
| And West Wing: https://www.youtube.com/watch?v=vVX-PrBRtTY
| jschrf wrote:
| IIRC the use of Web Mercator is primarily for ease of
| generating 256x256 tiles.
|
| The ideal projection is simply 3D, as it accounts for all
| scales, and the geoid if so inclined.
| zokier wrote:
| > The ideal projection is simply 3D, as it accounts for all
| scales
|
| Unless you have 3D display that is not really true, it is
| still projected to 2D; perspective projection is still
| projection and it is not obvious that it's in any way "ideal"
| for maps
| TheBicPen wrote:
| Sure, but it can be projected in a way that minimizes
| distortion of the current region you're looking at. 3D
| effectively allows you to choose the best projection from
| an infinite family of projections, while 2D restricts you
| to a single projection regardless of your location of
| interest.
| orangeboats wrote:
| My personal favorite map projection is the Equal Earth
| projection
| (https://en.wikipedia.org/wiki/Equal_Earth_projection) but it
| seems like it is relatively unknown. Though in general I have a
| soft spot for all equal-area projects, except the abomination
| Gall-Peters.
| taeric wrote:
| I'm assuming someone has made a graph of error versus utility
| in map projections? Not being able to draw straight lines is a
| fairly useful thing to do.
|
| If you want accurate, it is also silly to insist on it being a
| static 2d projection? Having a globe is not exactly difficult.
| lalaithion wrote:
| A globe is just using physics to create a perspective
| projection on your retina.
|
| https://en.wikipedia.org/wiki/General_Perspective_projection
| taeric wrote:
| Right? And that isn't hard and can get you accurate images.
|
| I get that this was a lot harder in the past. But today,
| this isn't difficult at all.
| Retric wrote:
| Not quite, you have two retina and local light sources. So
| you get more information from a globe than any 2d
| projection.
|
| However by rotating a perspective projection your brain can
| better recreate the underlying 3D nature of earth.
| liotier wrote:
| To me, Kavrayskiy VII still feels like the most balanced
| compromise among general-purpose pseudocylindrical projection -
| more so than Winkel Tripel, and of course miles ahead of
| Mercator but that isn't even a contest.
| trylfthsk wrote:
| I'm partial to the Pierce Quincuncial [0] projection myself.
| Actually, any conformal projection that tiles really.
|
| [0]https://en.wikipedia.org/wiki/Peirce_quincuncial_projection
| contravariant wrote:
| Technically Mercator tiles.
| akdor1154 wrote:
| Their justification for no boundary cut error is kinda dodgy..
| they say they have none because this projection is really two
| discs back to back, 'you can just stretch a string over the edge
| of the disk'.
|
| That's cool but by that argument can't i just fold a Mercator map
| in half and also have no boundary cut?
| dmurray wrote:
| I would say worse than "kinda dodgy" - it's completely
| intellectually dishonest, and their paper should not have got
| past peer review if it claims this has no boundary cut but the
| Mercator projection has a big one.
| DiogenesKynikos wrote:
| Equally, one could just glue together the edges of a
| butterfly map, eliminating the boundary penalty. I think this
| is cheating.
|
| The problem statement is: find a mapping from the surface of
| a sphere to R2 that minimizes a particular penalty function.
| This paper maps each hemisphere to R2, and then argues that
| the normal boundary penalty term can be ignored.
|
| However, if you just look at what the map does to South
| America and Africa, where there's a massive discontinuity at
| the equator, it's absurd to argue that the boundary penalty
| should be ignored. This map is useless for equatorial
| regions, and the penalty function should reflect that.
| a1o wrote:
| In the online version you can click and it moves the map so
| it doesn't have to cut things anywhere specifically - but
| it will always cut somewhere.
|
| https://imgur.com/rgeg1Lc
| DiogenesKynikos wrote:
| You can always make good local maps of a sphere. I think
| that's just a basic property of Riemannian manifolds:
| they locally look like flat space.
|
| The problem that this paper is trying to solve, however,
| is the creation of a minimally distorted (according to
| some penalty function) global map.
| zokier wrote:
| > their paper should not have got past peer review
|
| As far as I can tell its not published anywhere nor received
| any peer review.
|
| https://xkcd.com/2304/
| antiquark wrote:
| Yes, dodgy. Same argument could be made for the dymaxion map,
| which can be folded into an icosahedron, then you can easily
| stretch the string over the polyhedron.
|
| https://en.wikipedia.org/wiki/Dymaxion_map
| bunabhucan wrote:
| Google maps just repeats if you dont set the limits:
|
| https://i.stack.imgur.com/UhosY.jpg
| zokier wrote:
| > That's cool but by that argument can't i just fold a Mercator
| map in half and also have no boundary cut?
|
| You need to both fold it in half _and_ glue the ends together,
| basically creating a torus (or two-sided cylinder) shape
| dotancohen wrote:
| I've seen maps of the heavens using this projection, with the
| added stipulation of the celestial equator being on a separate
| bar. So the two circles would represent e.g. +45 degrees north
| and south of the celestial equator, and the bars would represent
| from 45 north to 45 south (or thereabout, I'm not sure about the
| actual degrees).
|
| Here's one that I just found online:
|
| https://fineartamerica.com/featured/vintage-stars-map-celest...
| fish44 wrote:
| if you click the map - just West of the tip of India- it creates
| a much better division - without slicing populated continents in
| half
| MichaelZuo wrote:
| Huh, it looks so much better that it seems strange its not the
| default.
| Turing_Machine wrote:
| That's pretty good. Nice find.
|
| I agree with the other commenter that this would be a good
| default.
| curtisf wrote:
| Am I understanding correctly that this is just two 'azimuthal
| equidistant projections' center on antipodal points, side by
| side?
|
| (but envisioned as being glued to opposite sides of a single
| disk)
| mxfh wrote:
| Yes they are.
|
| It's probably increased accessibility of applied map projection
| plotting libraries vs. the knowledge of theory and history as
| formal requirement for making up stuff like this. See also
| Gall-Peters. Formalizing and marketing Map Projectsions are two
| separate skill sets.
|
| https://twitter.com/mxfh/status/1363807641932337153
|
| _Physplaining_ [2] describes this quite well, if there is an
| established body of resarch and astrophysic specialist
| "rediscover" a specialist area that got reduced exposure with
| in the era of digital print and publishing.
|
| [1]
| https://www.mappingasprocess.net/blog/2021/2/17/a-radically-...
|
| [2]
| https://www.mappingasprocess.[net/blog/2021/2/21/perfecting-...
| kingkawn wrote:
| Hilarious to cut Africa in half and present it as errorless
| lupire wrote:
| Why does the HN title have a false editorialized claim that is
| not in the linked page?
| crazydoggers wrote:
| The linked paper makes the claim "We believe it is the most
| accurate flat map of Earth yet."
| btbuildem wrote:
| If we're doing strangely discontinuous maps, I'd like to submit
| Fuller's Dymaxion Map [1] -- at least that one keeps the
| continents contiguous, while truly minimizing deformations.
|
| 1: https://en.wikipedia.org/wiki/Dymaxion_map
| WorldMaker wrote:
| The Dymaxion projection came up in my own recent reading
| because one was sent in the "Cosmic Call":
| https://en.wikipedia.org/wiki/Cosmic_Call
|
| The blog series I was reading:
| https://blog.plover.com/aliens/dd/intro.html
|
| The Cosmic Call map was specifically pages 19-20:
| https://blog.plover.com/aliens/dd/p19.html
| contravariant wrote:
| If strange discontinuities are not a problem, what about the
| Euler spiral?
|
| Okay in the limit it has no area, but if you see it as a
| limiting process of arbitrarily thin strips then the distortion
| goes to 0 as the width decreases.
| brianshaler wrote:
| I made an interactive WebGL implementation of this without
| realizing it was a thing: https://brian.sh/around/index.html
|
| I don't really agree with the claims in the articles linked in
| OP, and don't find it to be a generally useful projection, even
| for the tool I made using it. It was novel as a representation
| that included daylight context (instead of just "what time is it
| there?" it helped express "is it getting dark there?") that
| preserved area better than a globe and was more intuitive than a
| day/night waveform on a rectilinear projection. But ultimately,
| if you're showing anything that has to do with populations
| (cities, people) pretty much any projection will waste large
| amounts of space on oceans and unpopulated land regions. That is
| to say, before choosing a favorite map projection, I think it's
| probably better to not to use a map projection at all unless
| you're going for a hike or setting sail.
|
| Somebody beat me to the obligatory xkcd, but this West Wing bit
| is my go-to for map projection discussions: https://youtu.be/vVX-
| PrBRtTY
|
| I like framing map projections by what they prioritize or
| sacrifice--fidelity in axis, position, size--and what projection
| is "best" depends entirely on which characteristics are more
| important. I disagree with OP's claim about this projection being
| "the most accurate flat map of the Earth yet" though haven't put
| a ton of thought into the physical, back-to-back definition of
| "flat" vs on-screen.
| karmakaze wrote:
| 'of land'. For less-accurate needs I prefer the ones that still
| show _(roughly North-top, South-bottom and)_ lat /long lines and
| non-blue space where water is separated by the mapping.
|
| Maybe this mapping is most useful for accurately tracking global
| warming effects at the poles.
| mourner wrote:
| The problem with map projections in the digital age is what works
| well on world scale doesn't on street scale and vice versa. As
| explained in detail in this post:
| https://www.mapbox.com/blog/adaptive-projections
| jccalhoun wrote:
| This map we made does the best in this metric we invented!
| mbostock wrote:
| Here's the D3 implementation (which is just an interrupted
| azimuthal equidistant projection):
|
| https://observablehq.com/@d3/azimuthal-equidistant-hemispher...
| modeless wrote:
| Map projections are fun but have limited relevance in the age of
| computers. Almost all on-screen interactive maps should use
| perspective projection, the only intuitive projection. (I feel
| like people dismiss perspective projection as not a "real" map
| projection, but it most definitely is.) If you need to judge
| relative sizes or distances or draw straight lines or whatever
| other things you might want from a map projection, software tools
| can help you do it more easily and more accurately.
| contravariant wrote:
| Which perspective projection do you mean? Stereographic, gnomic
| or just a 3D rendering of a globe?
| modeless wrote:
| Those are special cases of perspective projection, but they
| are non-physical because they ignore occlusion. I prefer
| perspective projections that correspond at least roughly to
| what a physical camera could see, or, more importantly, an
| eye. That's what I mean by "intuitive".
| cozzyd wrote:
| Orthographic?
| aidenn0 wrote:
| Having the boundary at the equator is a strange choice. If you
| instead have the boundary go through the poles and the cook
| strait (e.g. click roughly at the edge of the northern hemisphere
| ~45 degrees down and left on the interactive map), then you end
| up only cutting off the Russian Far East and Antarctica, all
| other major landmasses are preserved.
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(page generated 2024-04-22 23:01 UTC)