Map Projections
What does the world look like?
AITOFF
AZIMUTHAL EQUIDISTANT
BEHRMANN EQUAL AREA CYLINDRICAL
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CYLINDRICAL EQUAL AREA
BONNE
CRASTER PARABOLIC
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ECKERT I
ECKERT III
ECKERT V
There are many more …
Why does the world have so many different faces? Because the earth is a globe, but the
map is a flat plane, to draw locations on the surface of a sphere on a flat paper, schemes
are designed to transfer the round surface to flat plane. The mathematical transformation
of the earth’s round surface to a flat map sheet is referred as map projection.
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Sphere, Ellipsoid, Sea level, Datum, and Geoid
Before we go further into projections, we need to take a look at our earth.
How big is the earth?
Distance around the equator: 40,075,452.7 meters.
Distance around the poles: 39,939,593.9 meters.
The earth is not a perfect sphere; it is an ellipsoid (or spheroid), an ellipsoid that is
squeezed in the axis direction. To precisely describe an ellipsoid, we need to measure its
“flattening”.
Semi-major axis (a)
Semi-minor axis (b)
For the World Geodetic Reference System of 1984 (WGS84)
a = 6,378,137.0 meters
b = 6,356,752.3 meters
So flattening f = (a – b) / a = 1/298.257
How height is measured? A point on the earth surface (usually the sea level of major river
estuary) was used as a reference of surface elevation measurement. This reference point
is not very good as a reference for large areas such as continents.
An estimate of the ellipsoid was designed to allow calculation of elevation of every point
on earth, including sea level. This ellipsoid is often called a datum. In 1983 a new
datum was adopted for the United States, called the North American Datum of 1983
(NAD83), based on measurements taken in 1980 and accepted internationally as the
geodetic reference system (GRS80).
The U.S. military has also adopted the GRS80 ellipsoid but refined the values slightly in
1984 to make the world geodetic system (WGS84).
With different reference systems or datum, elevation of a location could be significant
different. See figure below:
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Height
Topography
Geoid
Sea level
Ellipsoid
Sphere
Geoid: the equipotential surface of the Earth's gravity field which best fits, in a least
squares sense, global mean sea level (source: NGS).
Map Projections
Given that the earth can be approximated by a shape like the sphere of the ellipsoid, how
can we go about converting data in latitude and longitude into a flat map, with x and y
axes? The simplest way is to ignore the fact that latitude and longitude are angles at the
center of the earth, and just pretend that they are x and y values.
This map will range from 90 degrees north to 90 degrees south, and from 180 degrees
east to 180 degrees west. The corresponding (x, y) values are from (-180, -90) to (+180,
+90). This map is now a map projection, because the earth’s geographical (latitude,
longitude) coordinates have been “mapped” or projected onto a flat surface.
We can “project” the sphere (or ellipsoid) onto any of three flat surfaces and then unfold
them to make the map: plane, cylinder, or the cone. Projections onto these three surfaces
are called azimuthal (planar), cylindrical, and conic, respectively.
We can also choose the way the surfaces contact the sphere or ellipsoid surface. If the
projection surface simply touches the global, we refer the project as tangent. A planar
projection is tangential to the globe at one point. Tangential cones and cylinders touch the
globe along a line. If the projection surface intersects global surface, we refer the
projection as secant. Secant cones and cylinders touch the globe along two lines.
Whether the contact is tangent or secant, the contact points or lines are significant
because they define locations of zero distortion. Lines of true scale are often referred to
as standard lines. In general, distortion increases with the distance from the point of
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contact. If a standard line coincides with a parallel of latitude, it is called a standard
parallel.
Azimuthal projections
Polar
Equatorial
Oblique
Transverse
Oblique
Cylindrical projections
Equatorial
Conic projections
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Cartographers have devised thousands of different map projections. Fortunately, they all
fall into a set of “types” that are quite easily understood. The simplest way to evaluate a
projection is by how it distorts the earth’s surface during the transformation from a sphere
or ellipsoid to a flat map. Some projections preserve the property of local shape, so that
the outline of a small area like a state or a part of a coastline is correct. These are called
conformal projections. They are easily identified, because on a conformal projection the
lines of latitude and longitude grid (graticule) meet at right angles, although not all rightangle graticules mean conformal projections. Conformal projections are employed
mostly for maps that must be used for measuring directions, because they preserve
directions around any given point. Examples are the Lambert Conformal Conic and the
Mercator projections.
Lambert Conformal Conic Projection
Mercator Projection
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The second category of projections preserves the property of area. Projections that
preserve area are called equal area or equivalent. Many GIS packages compute and use
area in all sort of analyses, and as such must have area mapped evenly across the surface.
Examples of equal area projections are Albers equal area and the sinusoidal projections.
Albers Equal Area Projection
Sinusoidal Projection
The third category of projections preserves distance but only along one or a few lines
between places on the map. They are called equidistant projections. The simple conic
and the azimuthal equidistant projections are examples. These projections are not
frequently used in GIS.
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A final category is that of the miscellaneous projections. These are often a compromise,
in that they are neither conformal nor equivalent, and sometimes are interrupted or
broken to minimize distortion. Examples of miscellaneous are Goode’s homolosine
(made by patching different projections together) and Robinson projection.
Goode’s Homolosine projection
Robinson projection
In the United States, the bulk of the 1:24,000 topographic map series of the USGS uses a
polyconic projections, the Clarke 1866 ellipsoid, and the NAD27 datum. The recent
movement to the NAD83 datum and its corresponding ellipsoid, the GRS80, have
“moved” features by as much as 300 meters on the ground or 12.5 millimeters (0.49”) on
the 1:24,000 map. While integrating spatial data sets, make sure they have the same
datum and projection.
The transformation of the ellipsoid shape of the earth to a two-dimensional surface
cannot be accomplished without some element of distortion, through shearing, tearing, or
compression. Equal-area (or equivalent) maps, for example, preserve area relationships,
but tend to lose conformality (preservation of shape). Conformal projections, on the
other hand, maintain shape over small areas but produce areal distortion. In thematic
mapping, it is frequently more important to maintain correct area properties, in that area
is often an important element of what is being shown by the map. Therefore, shape is at
times compromised through the choice of an equivalent projection. For small scale maps,
in fact, conformality cannot be maintained over the entire area; rather, the projection may
preserve shape best along a standard line, with shape distortion increasing with distance
from the line.
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