Map projections and distance on Earth

The representation of Earth on a flat surface -- a map -- whether a sheet of paper or computer screen, inevitably results in distortion. A sphere or spheroid is the rough geometric equivalent of Earth; and the basic inequality of form between a sphere and a plane lies at the heart of this distortion. More precisely, distortion is a direct result of the mathematical transference of three-dimensional spherical coordinates onto a two-dimensional planar field, or, what is referred to as a projection (Christopherson 2006:25; G. & C. Merriam 1969:xxvi). While the spheroidal globe is near to a true representation of Earth, it’s limited by scale and presentation, i.e. only half of a globe is visible at any one time. Projections employed constructing large scale maps depicting relatively small areas in great detail, have less significance than projections employed for small scale maps depicting the entire Earth or a continent (Birdseye 1929:1; Alpha and Snyder 1982).

Numerous projections have been developed over the past two millennia (Snyder 1993:1) but the number of projections is potentially unlimited (Robinson 1960:66). Further, all projections preserve or display well one representational quality at the sacrifice of others. Such qualities include shape (form), area, distance, and direction (Greenhood 1964:114). Map projections preserving the shape or form of land masses are said to be conformal. (Less commonly these are referred to as orthomorphic projections (Greenhood 1964:115).) Map projections preserving the measured area of map features are said to be equal area, equivalent or homolographic maps (Greenhood 1964:115). Map projections preserving the distance between mapped features are said to be equidistant (Robinson 1960:58). Maps projections preserving the direction between mapped features from a central point of tangency are described as azimuthal (Robinson 1960:59). Due to widespread misconceptions about Earth engendered by the use of some projections, geographers and cartographers have at times advocated the discontinuation and use of certain categories of map projection (Committee on Map Projections 1989:222).

To illustrate the variation of distance as it is represented by maps made using different projections, six world maps were constructed in ESRI’s ArcMap program using six projections: two of the maps were made using a conformal map projection; two were made using an equal area map projection; two were made using an equidistant map projection. Conformal projections used were the Mercator and Gall Stereographic (Figure 1); equal area projections used were Goode’s Homolosine and Bonne (Figure 2); equidistant projections employed were the Sinusoidal and Equidistant Conic (Figure 3).


Figure 1.

Following construction of the six maps in ArcMap, I employed ArcMap’s digital measuring tool to measure the distance between two select cites located on different continents: Washington, DC (38° 53' 42" N, 77° 02' 12" W) and Kabul, Afghanistan (34° 31' 00" N, 69° 10' 59" E). The correct measured distance between Washington, DC and Kabul, Afghanistan is approximately 6919 miles (Travel Distance Calculator Between Cities 2010). The results of measuring the distance between the two world cities are listed below in Table 1.

No map shows scale correctly across its entire area. In addition to area and shape, map scale also distorts in parts of a map other than its point or line of tangency (Snyder 1982: 6). This feature is the reason distance on the various projections employed for this exercise is so varied. Mercator projection maps of the world represent equatorial regions well, but they’re greatly distorted toward the poles, a condition that’s the result of representing the poles (points) as lines equivalent in length to the equator (Alpha and Snyder 1982).


Figure 2.

A stereographic projection exaggerates area. Representation of area -- and by extension scale -- increases from the map’s central point of tangency (Robinson 1960:82). Goode’s homolosine -- an interrupted projection utilizing several standard meridians -- combines elements of homolographic (nee equal-area) and sinusoidal projections with variable scale across the map (Snyder 1982:221). Scale on both sinusoidal projections and the heart-shaped Bonne projection are true only along their central meridian and parallels (Snyder 1993: 50, 62), whereas scale on an Equidistant Conic projection map is true only along meridians (Snyder 1993:122).

Figure 3.

Although none of the six projection’s Washington-to-Kabul measurement was correct, these inaccuracies may be due to coarse resolution of each map and the limitations of precision in the ArcMap measuring tool.

Table 1. Projection category, type of projection and measured distance in miles between Washington DC and Kabul, Afghanistan using six separate projections.

Conformal maps:
Mercator projection 10,119
Gall Stereographic projection 7,135

Equal Area maps:
Goode’s Homolosine projection 9,986
Bonne projection 6,753

Equidistant maps:
Sinusoidal projection 8,103
Equidistant Conic projection 6,975

Analog comparison

As a comparison, I measured the distance between Washington and Kabul using an old analog method. The tools required are a globe and a string. The measurement is simple as follows:

STEP 1. Stretch a length of string on the globe from Washington, DC to Kabul, Afghanistan as close as is practicable approximating the path along a great circle (Figure 4).

Figure 4.













STEP 2. Removing the string from the two points of the measurement, placing one end of the string at 0° lat, 0° lon, stretching it along the equator to obtain its length in degrees of longitude (Figure 5).

Figure 5.












STEP 3. Multiply the number of degrees measured by 69 miles.

For this operation, the distance the string stretched along the equator was approximately 100° of longitude. Multiplying 100 by the distance between 1° along a great circle (69 miles), the distance between Washington, DC and Kabul, Afghanistan is about 6900 miles (100 X 69 = 6900). This measurement is closer to the actual value between the two cities (6919 miles) than the measurement obtained from any of the projected maps, illustrating the dictum that the globe remains the truest representation of Earth.

ADDENDUM:

Of course, none of these provide the real method by which one would obtain the shortest distance to Kabul from Washington. For that there are these three steps:

(1) Have the CIA secretly train and arm thousands of Afghan fundamentalist jihadists (including Osama bin Laden) to defeat an occupying army of Russians during the 1980s.

(2) Allow the remnants of these jihadists to evolve into the ultra fundamentalist Taliban government during the 1990s.

(3) Feign surprise when these products of CIA training provide haven for terrorists with a deep abiding hatred for the products of U.S. foreign policy.

Follow these fool-proof steps and you’ll be in Kabul in no time.

Peace.

References

Alpha, Tau Rho and John P. Snyder 1982. The Properties and Uses of Selected Map Projections. Miscellaneous Investigations Map I-1402, United States Geological Survey. United States Government Printing Office, Washington, DC.

Birdseye, C. H. 1929. Formulas and Tables for the Construction of Polyconic Projections. Bulletin 809, United States Geological Survey. United States Government Printing Office, Washington, DC.

Christopherson, Robert W. 2006 Geosystems: An Introduction to Geography, Sixth Edition. Pearson/Prentice Hall, Upper Saddle River, New Jersey.

Committee on Map Projections, American Cartographic Association 1989 'Geographers and Cartographers Urge End to Popular Use of Rectangular Maps'. The American Cartographer. Vol 16, No. 3, pp. 222-223.

Greenhood, David 1964. Mapping. The University of Chicago Press, Chicago, IL

Merriam, G. & C. Co. 1969. Webster's Geographical Dictionary. G. & C. Mirriam Co., Springfield, MA

Robinson, Arthur H. 1960. Elements of Cartography, 2nd Edition. John Wiley and Sons, New York.

Snyder, John P. 1982. Map Projections Used by the U.S. Geological Survey. Bulletin 1532, United States Geological Survey. United States Government Printing Office, Washington, DC.

Snyder, John P. 1993. Flattening the Earth: Two Thousand Years of Map Projections. The University of Chicago Press, Chicago, IL

TRAVEL DISTANCE CALCULATOR BETWEEN CITIES www.mapcrow.info. Data retrieved 7 July 2010.