What’s your favorite map projection?

What’s your favorite projection? And what does that say about you? Cartoon by Randall Munroe, from: http://xkcd.com/977/

OK, a number of you guys sent me this link to a pretty humorous cartoon which lampoons the various map projections, so I figured I may as well make an official blog posting about it!  According to the premise of the cartoon, the map projection that you prefer supposedly indicates what sort of person you are, or at least gives certain clues about your personality.  I like that better than astrological signs.  And it just might be more accurate.  Uh oh!  I’m in trouble.  I am attracted to the Waterman butterfly projection! 

If you mouse over the cartoon on its original website, the bubble shows up with the text “What’s that?  You think I don’t like the Peters map because I don’t like having my cultural assumptions challenged?  Are you sure you’re not:: ….puts on sunglasses:: ….projecting?”

Thanks, Jonathan Halabi, Diana Morgan, Keith Miyake, and kuschk (The Basement Geographer) for sending the link. 

Here’s another something about the Waterman butterfly projection, which is based on the 1909 Cahill projection, shown above. “From the cover of 1919 pamphlet by Cahill, ‘The Butterfly Map.’  His Butterfly World Map, like Buckminster Fuller's later Dymaxion Map of 1943 and 1954, enabled all continents to be uninterrupted, and with reasonable fidelity to a globe.  Cahill demonstrated this principle by also inventing a rubber-ball globe which could be flattened under a pane of glass in the ‘Butterfly’ form, then return to its ball shape.”

From Wikipedia at http://en.wikipedia.org/wiki/Waterman_butterfly_projection:

“The Waterman ‘Butterfly’ World Map Projection was created by Steve Waterman and published in 1996.  It is an octahedral transformation of a globe, reviving the butterfly map principle first developed by Bernard J.S. Cahill (1866–1944) in 1909.  Cahill and Waterman maps can each be shown in various profiles, typically linked at the north Pacific or north Atlantic oceans.

Whereas Cahill's approach was that of an architect, Waterman derived his design from his work on close-packing of spheres.  This involved the interpretation of a spherical extraction from cubic closest packed spheres, into a corresponding convex hull. Then for its projection; straight lines were used to define each 5 × 5 section onto this convex hull.

Projection employed an equal length delineation approach for the equator.  Latitudes were drawn in three straight line sections (in each octant): from pole to fold-line, fold-line to largest line parallel to equator, and then from there to the equator.  The largest line parallel to the equator also has equal length delineations.  One particular Waterman polyhedron best served to minimize land sinuses (breaking up of land masses) and was therefore chosen.

Like Buckminster Fuller's 1943 Dymaxion Projection, an octahedral butterfly map can show all the continents uninterrupted if its octants are divided at the proper meridian, that is, 20° W, and joined, for example, at the North Atlantic, as in the 1996 version.”

There have been a number of criticisms leveled at the Waterman projection, though.  “I didn't appreciate the free-floating Antarctica in the Waterman butterfly projection, so I made a slight alteration.” From the Map Porn blog (great blog name!) at http://www.reddit.com/r/MapPorn/comments/mcd7p/i_didnt_appreciate_the_freefloating_antarctica_in/

            Here is a really nice critique of the Waterman projection, with some suggestions for improvements: http://www.genekeyes.com/WATERMAN-REVIEW/Waterman-review.html

Gene Keyes hand-drawn version of the Keyes-Cahill map.  From: http://www.genekeyes.com/Cahill-Keyes-2.html

Waterman’s own site, at http://www.watermanpolyhedron.com/, has results on his 20+ years of research on the “close-packing of spheres,” PLUS original theories in theoretical physics! 

We Have a Winner!

Map Projection Identification Contest Prize – The book “Mapping Curiosities,” by the Royal Geographical Society, 2009, Pomegranate Europe, Warwick, UK.  The book’s cover shows “Map of the World showing the extent of the British Empire in 1886.”  Map by Sir John Charles Ready Colomb (English, 1838-1890) from “Imperial Federation: Naval and Military” (London, 1886).

As most of you know, we ran a Map Projection Identification Contest last week - see  http://geographer-at-large.blogspot.com/2011/09/name-that-map-projection.html.  Despite several joke-y contest entries (for instance – “It’s the John Boehner hairpiece projection!”), we have a winner.  This was also despite the little snafu with the map projection image itself - the genius blogger that I am posted the image and left the name of the projection as the name of the image, for all the world to see if they only right-clicked on the image.  I was afraid I would be deluged with bogus correct answers, based on the tell-tale and rather ridiculously large clue I left on the contest page itself.  This was mercifully brought to my attention within a few hours of posting by a very honest geography doctoral student, who sent me an e-mail saying “psst…you should quickly change the name of the image because clicking on it gives it away!!!!” and of course made me feel like a foolish idiot in the process – which I’m sure was not his intention, but was very effective at doing so nonetheless!  And also mercifully, apparently no one looked very closely at the post until after I had changed the name of the image.

In any event - and why should this post be any different than my usual overly-wordy texts? – we have a winner! 

Here’s the FIRST, FASTEST, CORRECT response to the Map Projection Contest question:

From Jenn Brisbane, GISc analyst at the NYC DoITT (Department of Information Technology and Telecommunications):

“It's the Craig Retroazimuthal map projection - also known as the Mecca projection - because it was created to help Muslims figure out which direction to pray in. This map projection preserves directionality! (the following is from Wikipedia):

‘The Craig retroazimuthal map projection was created by James Ireland Craig in 1909.  It is a cylindrical projection preserving the direction from any place to another, predetermined place while avoiding some of the bizarre distortion of the Hammer retroazimuthal projection.  It is sometimes known as the Mecca projection because Craig, who'd worked in Egypt as a cartographer, created it to help Muslims find their qibla.’”

I then followed up with Jenn, told her she had won the prize, and asked how she managed to figure it out.  Here’s what she said:

“Well, after google image searching ‘map projections’ till my eyes glazed over, I googled ‘weird map projections’ and ‘unusual map projections’ and came across a powerpoint presentation for some class.  One of the slides showed the Hinks' Retroazimuthal Projection which looked somewhat similar to the image on your blog.  I googled more about that projection and found a paper by Walter Tobler about retro-azimuthal maps where he mentions the Craig projection (which I google imaged searched and found it matched your image) and the Hammer projection - an equally silly-looking projection!  The Tobler paper is titled ‘Qibla, and Related, Map Projections.’

‘The Qibla problem - determination of the direction to Mecca - has given rise to retro-azimuthal map projections, an interesting, albeit unusual and little known, class of map projections. Principal contributors to this subject were Craig and Hammer, both writing in 1910.’

Is there really a prize?? Exciting! 

Alright now, who answered ‘The John Boehner's hairpiece projection’?  Was that Andrew??  I think that answer deserves a prize as well!!”

jenn

I received a couple of other correct responses, but too late! The early bird, etc., etc. 

Honorary Mentions: 

“I believe it to be a Craig retroazimuthal projection.  The analytic function is similar to the inflow equation for a helicopter rotor. My near-east friends reminded me that it can be centered (as the equations are for the center of a helicopter hub) at any important place, such as Mecca.” from U-know-who (this is from my brother Wayne, the ace helicopter designer for NASA, also the originator of the John Boehner’s hairpiece projection entry. Also, I think he is counting on none of us knowing doodley squat about helicopter rotor design inflow equations, and therefore feels free to make any possibly spurious analogies he wants to!  Not that he would ever do that, of course!)

And from Andrew (“Salty Language”) Maroko, Geography/GISc Professor extraordinaire:

“Son of a bitch!!!! Is this it?  http://en.wikipedia.org/wiki/Craig_retroazimuthal_projection”

A few other worthies made good guesses, some close, but no cigar.  And of course, a big shout out to Keith Miyake, who pointed out my drastic oversight in having the mystery map projection image actually entitled with the correct response.  (although apparently most people didn’t notice it, or if they did, were too polite or blasé to tell me I made a mistake). 

The contest prize was the book “Mapping Curiosities,” by the Royal Geographical Society in the UK, and the book contains reproductions of many well-known cartographic examples from the RGS collections (such as Mercator’s 16^th century world map, and Claudius Ptolemy’s 2^nd century world map) as well as a number of more obscure maps (such as the route map of the 1953 expedition from Kathmandu to Mount Everest, and an aviation map of France for World War I night flying)..See http://geographer-at-large.blogspot.com/2011/07/my-visit-to-royal-geographical-society.html for narrative on my recent visit to the Royal Geographical Society headquarters in London. 

The book’s Introduction states “When the Royal Geographical Society was founded in 1830, one of its aims was expressly stated to be the formation of a ‘complete collection of maps and charts from the earliest periods of rude geographical delineations to the most improved of the present time.’….. In the 1840’s, through its association with the era of African discovery, events in Africa began to stimulate further public interest in mapping, and it is recorded that the Map Room was ‘daily visited by intelligent strangers as well as by members generally.’  It was at this time that manuscript maps by African explorers like Livingstone also began to come to the Map Room.  During the remainder of the nineteenth century, extensive efforts were made to complete the existing holdings of maps and to extend the scope of the collection with commitments made by the India Office, Army Map Service, US Army, and other institutions to provide copies of all their mapping output.…By early to mid twentieth century, annual receipts numbers some 9,000-10,000 sheets of maps and approximately 30 atlases every year, supplied by a wide range of international governmental departments, academics, and individuals with a passion for maps and mapping.  Today, the Society holds the world’s largest private map collection, including over 1 million sheets of maps and charts, 3,000 atlases, 40 globes (as gores or mounted on stands) and 1,000 gazetteers.  Rare antiquarian maps, atlases, and gazetteers continue to be presented to the Society….with the earliest printed items dating back to the fifteenth century.  In addition, the collection includes manuscript materials from the mid-sixteenth century onwards, aerial photography, and contemporary satellite images, making this one of the world’s most comprehensive resources relating to maps and mapping for a wide variety of users.”   See www.rgs.org/collections for more maps and info.  

The Craig Retro-Azimuthal Projection, from the original post about the contest.  http://geographer-at-large.blogspot.com/2011/09/name-that-map-projection.html

Also see the Map of the Week 9-19-2011 post for further details about Mecca-centric map projections.  http://geographer-at-large.blogspot.com/2011/09/map-of-week-9-19-2011-mecca-at-center.html

Name That Map Projection!

Map Projection Identification Contest! 

OK, I am trying something new and exciting!  A contest!  With a nice prize, even!  The way to win the contest and the prize is to correctly identify the map projection used in the map shown above.  You have to tell me what it is, details about it, and what possible purpose it is particularly good for.  Whoever sends me the most correct, complete response the fastest will win the prize, and the winner and correct answer will be announced in my September 19^th blog posting on Map of the Week! 

I was inspired to do this because (1.) I love quizzes and contests!  And (2.) my posting on the oblique case map projections, for some (very!) odd reason, has become one of the most oft-viewed of my blog posts.  http://geographer-at-large.blogspot.com/2011/08/fun-with-map-projections-oblique-case.html

This does not necessarily mean that the oblique case map projection post is very beloved and popular with readers, of course.  Or even that it is very excellent.  Or that there is inordinate interest, all of a sudden, in the obscure topic of oblique case projections.  It could just mean that some crazy (err, I mean uber-discerning) professor or two has assigned the post as reading for a class on mapping science, cartography, or GISc.  That’s the only reasonable explanation I can come up with, anyway, to explain the mad number of views!  Totally unexpected!  So I figured if there is so much interest in strange map projections, perhaps I better have a contest on map projection identification. Fun, right? 

Please send your responses to me via e-mail by no later than 6:00 PM EDT on September 16^th (GMT -5) at YahMonForReal@aol.com. Subject line: Map projection contest.  Please!  No copying from Wikipedia (or anywhere else, for that matter!)  I will know!  Instant disqualification! 

UPDATE: See the posts Map of the Week 9-19-2011: Mecca at the Center of the World
for info on these Mecca-Centric projections, at http://geographer-at-large.blogspot.com/2011/09/map-of-week-9-19-2011-mecca-at-center.html  and 
We Have a Winner at
http://geographer-at-large.blogspot.com/2011/09/we-have-winner.html for the correct answer to the contest, details about the projection, and how the mystery map projection was solved.

Fun with Map Projections, Oblique Case

Lambert cylindrical equal-area projection with oblique orientation from: https://www.cia.gov/library/publications/the-world-factbook/geos/xx.html 

Isn’t this a fantastic map projection?  It’s called a “Cylindrical Equal-Area Projection, Oblique Case.”  Kristen Grady sent it to me, (it's apparently what the CIA uses to map global data) and I figured it was a good opportunity to think a little bit about map projections, and of course, share my thoughts with you, dear readers. 

First of all, a simple definition of a map projection: “A cartographical map projection is a formal process which converts (mathematically speaking, maps) features between a spherical or ellipsoidal surface and a projection surface, often flat.” From: http://www.progonos.com/furuti/MapProj/Dither/CartDef/MapDef/mapDef.html#Oblique

One of the reasons that I believe the topic of map projections is so confusing to people (both those trying to teach it and those trying to learn it) is that there are so many different ways to classify the types of projections, and by necessity, every projection falls into several different categories simultaneously. 

Projections are classified by which property of reality they preserve the best.  All projections distort the globe, but some are better at depicting “true” area (equal area or equivalent projections); distance (equidistant); direction; and shape (conformal); etc.  There are also so-called “compromise” map projections, which don’t preserve any property entirely, but minimize distortion in all. 

Projections are also categorized by which “projection surface,” or “developable surface” is used to create the flat map from the spherical globe.  For instance, azimuthal (a flat plane touching the globe with one point of tangency); cylindrical (wrapped around the globe in a cylindrical shape, with one line of tangency); conic (yes, you guess it, a cone sitting on the globe like a conical birthday hat, with two lines of tangency). 

 These are the three most common, but there are also sinusoidal, pseudo-cylindrical, such as the now-familiar, but still strange-seeming Goode’s interrupted homolosine (shown at left), and “others,” such as Buckminster Fuller’s Dymaxion, which projects onto the surface of a polyhedron.  

Buckminster Fuller's projection on a polyhedron. Projections are also classified as to how the developable surface is oriented on the globe.  There are normal (aligning with the earth’s axis), transverse (at right angles to the earth’s axis), and oblique (aligning with a Great Circle, but not the Equator or a meridian), and each different orientation has different lines of latitude or longitude where scale is “true.”

 

Above, left: cylindrical "normal": Above right: cylindrical transverse; lower left: cylindrical oblique.
Then, to complicate matters even further, the "developable surface" can be "tangent," considered to have points or lines of tangency where it is touching the globe, or it can be "secant," which means the developable surface seems to be slicing through the globe. Secant projects have the advantage of additional points of tangency, and therefore more areas on the map that are "true."
And, then, we have projection classification by method of light source: from where, in our imaginary transfer of data from the face of the globe to the projection surface, is the light coming?  There’s Gnomonic (light projected from the center of the globe to projection surface); Stereographic (light projected from the antipode of point of tangency); or Orthographic (light projected from infinity).

            The above is a vast simplification of projection categories, and omits a great deal of detail and nuance.  But overall, it hits the highlights.  Some pretty good projection typologies are given at http://www.progonos.com/furuti/MapProj/Dither/ProjTbl/projTbl.html and at http://egsc.usgs.gov/isb/pubs/MapProjections/projections.html and at http://www.kartografie.nl/geometrics/map%20projections/body.htm

So, to return to our interesting “cylindrical equal-area projection, oblique case,” what this means when we break it down is that the developable surface is a cylinder (like many more common projections, including the Mercator, the Lambert, and the Miller).  It s an area-preserving projection, like the Gall-Peters, so areas are shown “true” to size (with many caveats), but “true” shape and distance are sacrificed.  And the developable surface is oriented in an oblique fashion, resulting in those nice sinuous lines of lat-long that are so unusual to see on a world map.  A projection using the Oblique Case usually implies that the cylinder, for instance, is oriented so that the equator is not the line of tangency, as in the “normal” projection, nor is it transverse, where a line of longitude would be the line of tangency.  In an oblique case, the line of tangency is one of the Great Circles.  An oblique map has neither the polar axis nor the equatorial plane aligned with the projection system.

Other Cool Oblique Case Projections:

Mollweide Projection, Oblique Case

The Atlantis Map (Bartholomew, 1948), an Oblique Mollweide projection, features the Atlantic Ocean as the major focus.

“If neither Equator nor the central meridian are aligned with and centered on the map axes, the result is commonly called an oblique projection (or, more properly, an oblique map).  Although general properties of the original projection (like area and shape equivalence) still hold, those depending on the graticule orientation are generally not preserved.  A common reason for tilting a projection is moving a large, important area to the places of lesser distortion. The Atlantis map (Bartholomew, 1948) presents the Atlantic Ocean in a long, continuous strip aligned with the map's major dimension. Also clearly showing the Arctic ‘ocean’ as a rather small extension of the larger Atlantic, it is an oblique Mollweide projection centered at 30°W, 45°N.” From: http://www.progonos.com/furuti/MapProj/Dither/ProjObl/projObl.html#ObliqueRectangular

The Hammer-Aitoff Projection, Oblique Case

“The Hammer-Aitoff map projection is an equal-area map projection which displays the world on an ellipse.  This map was created from a hemisphere of the equatorial case of Lambert's Azimuthal Equal-Area projection using a trick similar to the method by which the Miller Cylindrical projection was made from the Mercator projection.  This projection is also popular in the oblique case.  The basic idea behind the Hammer-Aitoff projection is simple enough: first, place the whole world in one hemisphere by the simple expedient of dividing all longitudes by two, then to compensate for that, stretch the map out twice as wide.”  From: http://www.quadibloc.com/maps/meq0801.htm  

Oblique Mercator

“This is used to show regions along a Great Circle other than the Equator or a meridian, that is having their general extent oblique to the Equator.  This kind of map can be made to show as a straight line the shortest distance between any two preselected points along the selected Great Circle.  Distances are true only along the Great Circle (the line of tangency for this projection) or along two lines parallel to it.  The map is conformal, and rhumb lines are curved.  Developed by Rosenmund, Laborde, Hotine et al.”

Space Oblique Mercator

“This new space-age conformal projection was developed by the USGS for use in Landsat images because there is no distortion along the curved groundtrack under the satellite.  Such a projection is needed for the continuous mapping of satellite images, but it is useful only for a relatively narrow band along the groundtrack.  Space Oblique Mercator maps show a satellite’s groundtrack as a curved line that is continuously true to scale as orbiting continues.  Developed in 1973-79 by A. P. Colvocoresses, J. P. Snyder, and J. L. Junkins.”  From: http://www.geomart.com/articles/mapprojections.htm

This Space Oblique projection is also used in the Cassini Spacecraft’s radar altimetry and synthetic aperture radar (SAR) imaging, which recently mapped Saturn’s moon Titan, for instance.

Mosaic of Cassini RADAR image coverage of Titan.  The Cassini image strips are made available in map projected form and use the Oblique Cylindrical projection oriented along its individual flyby ground track.  Mosaics of multiple image strips  transformed to a common global map projection are being made. From: http://www.isprs.org/proceedings/XXXVII/congress/4_pdf/173.pdf  “RADARGRAMMETRY ON THREE PLANETS,” by R.L. Kirk and E. Howington-Kraus, Astrogeology Program, U.S. Geological Survey, The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008.

The Math Involved:

            Now, I know there are math geeks out there (several of whom were in my recent classes, and always clamoring for “more math”!) so this is for you, guys:

An oblique form of the cylindrical equal-area projection is given by the equations:

         

And, here are some tres cool maps that use unusual projections which I found when poking around on the Internet:

Azimuthal Equidistant Projection Centered Near Khartoum. Map, 1942.

  This is a map of Human Migration, using the Dymaxion Projection.  The choice of this particular projection highlights the interconnectedness of the land masses of the world, and makes it easy to see the flow of humankind over the centuries.  

Another one based on the Dymaxion Projection! and I suspect for the same reason as the Human Migration map. This one portrays a prediction of the energy grid of the future, with various types of energy sources named.

A map of Faunal provinces based on Maurer’s star projection presents all lands minus Antarctica.  Since climate is a great determinant of faunistic features, most regional borders follow latitudes and are, therefore, roughly concentric in polar-aspect maps. 

Cassini's projection is a transverse aspect of the plate carrée, here with central meridians 70°E and 110°W.

And, just for kicks, here is the Cassini Projection, developed by César-François Cassini de Thury in 1745.  This is the son of the guy for whom the Cassini spacecraft was named (Giovani-Domenico Cassini), even though NASA is not using his projection to map the data, but rather an oblique conformal projection.  But Cassini (the elder) was the first to observe and document Saturn’s moons, and that’s probably why they named the spacecraft after him.  He also began the work to create a comprehensive topographic map of France, work that was completed by his son and grandson over a century later.  He used the then-new method of triangulation and his own method of determining longitude, which meant that this was the first truly accurate map of France.  Unfortunately, this accuracy resulted in a much smaller France than the Sun King Louis XIV (who commissioned the survey) was expecting.  Apparently the King joked that Cassini had taken more of France from him than he had won in all his wars. It was the first topo map of an entire country.

           Cassini’s projection is the transverse aspect of the equi-rectangular projection, in that the globe is first rotated so the central meridian becomes the “equator,” and then the normal equi-rectangular projection is applied.  It maintains scale along the central meridian and all lines parallel to it, and is neither equal-area nor conformal.  It is best suited for mapping areas predominantly north-south in extent near the central meridian.   Formerly used by the Ordnance Survey in Great Britain, Cassini is still used in Cyprus, Czechoslovakia, Denmark, Germany, and Malaysia.  

An application of the Cassini Projection - Present and fossil teeth suggest several migration waves in the past, when reduced sea levels created bridges between now isolated Japanese and Aleutian islands. From: 

Christy G. Turner II, Teeth and Prehistory in Asia, Scientific American 260 (2), February 1989. 

And to end up on a humorous note, here is a part of a poem about, of all things, maps! and their use (or not!) in navigation

From Lewis Carroll's 1876 The Hunting of the Snark - an Agony in Eight Fits:

Fit the First - The Landing

...Navigation was always a difficult art,

Though with only one ship and one bell:

And he feared he must really decline, for his part,
Undertaking another as well

Fit the Second - The Bellman's Speech

...He had bought a large map representing the sea,

Without the least vestige of land:

And the crew were much pleased when they found it to be

A map they could all understand.

"What's the good of Mercator's North Poles and Equators,

Tropics, Zones, and Meridian Lines?

So the Bellman would cry: and the crew would reply

"They are merely conventional signs!

"Other maps are such shapes, with their islands and capes!

But we've got our brave Captain to thank"

(So the crew would protest) "that he's bought us the best –

A perfect and absolute blank!"...

UPDATE - September 10, 2011: Map Monkey has instituted a map projection identification contest! For details, see http://geographer-at-large.blogspot.com/2011/09/name-that-map-projection.html