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I have been using ArcGIS and QGIS regularly for about 2 years now, and just recently I was struck by this new thought. If a data set in say, in ArcMap, has been georeferrenced with a Geographic Coordinate System (which is the representation in the Earth´s surface), and is hence not projected, how can ArcMap display such data set on a plane (i.e. the screen)?

  • They're projected to X/Y on the earths surface - as a globe - They're just not projected to a 2D plane... the software knows how to plot LAT/LON onto a projected coordinate system... but if your question is how is THAT done... well... I'll defer to the programmers... – DPSSpatial May 6 '15 at 18:23
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    It's not a silly question, but it has somewhat of a silly answer: to draw something on a screen or 2D graphical output device, all you need are ordered pairs of numbers to pass to its driver. They will be converted to horizontal and vertical displacements on the output. Guess what? Geographic coordinates are ordered pairs of numbers. So if you totally ignore what they mean, and just send them along to the display software, you will get a picture. That's tantamount to projecting the data, so what you really want to know is what projection this process is equivalent to. – whuber May 6 '15 at 18:36
  • that's very instructive whuber♦. So what type projection is the process equivalent to? – Rodrigues May 6 '15 at 18:52
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    plate carree, but standard plate carre multiplies radian values with the sphere's radius or ellipsoid's semimajor axis. – mkennedy May 6 '15 at 19:05
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When geographic coordinates are plotted "without projection", they are really being projected via the Simple Cylindrical (aka, Equirectangular, or Plate Carrée) projection. (It goes by many different names.)

Geographic coordinates, as latitudes and longitudes, are said to be unprojected because they define positions on a (curved) sphere or ellipsoid – they have not yet been "thrown forth" (projected) onto a plane.

Map coordinates, usually as northings and eastings, are said to be projected because they define positions on a (flat) plane – they have been "thrown forth" (projected), from a sphere, somehow.

The problem of the somehow is the nature of the map projection. Some projections have direct analogies in terms of straight physical projecting rays from the curved surface to a "developable surface", i.e., one that either is or can be unwrapped into a flat map. Those are said to be azimuthal, conic or cylindrical projections.

Other projections, most really, have no direct physical "projecting ray" analogy and must be realized purely mathematically. (Of course, even the simple "projecting ray" methods have mathematical forms.) The mathematical form of any map projection can be generalized in this form:

(N, E) = ƒ(R, φ, λ)

meaning northings and eastings are some function of Earth's radius, latitude and longitude. (For an ellipsoid, there are two parameters defining its size & shape, but let's focus on a sphere.)

The simple cylindrical projection happens to be the most trivial mathematically:

N = k R φ

E = k R λ

where k is some convenient constant that helps convert degrees into pixels or millimeters, etc, such that the map fits on the page or screen. It is so simple that folks often forget that there is any projection going on at all – they might even suggest that the coordinates you see are unprojected, but they'd be wrong.

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