Thursday, August 11, 2011

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



2 comments:

  1. Hi, this 'Azimuthal Equidistant Projection Centered Near Khartoum. Map, 1942.' is the neatest map I've located on the web showing the entire 'pangaean' world island with as little distortions as possible - what depends on choice of center. Only it's very irritating that it seems to have been copied only in part, leaving off parts of the South American pacific coast. Is there a copy of the whole thing, or could someone upload that somewhere?

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  2. Thanks for the great post on your blog, it really gives me an insight on this topic.

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