International Ellipsoid, from 1924, used in most of the rest
of the world (but developed in the US from US data!).
2. Methods of Projection
Many projections can be visualized as literally projecting a
light source through a transparent globe onto a surface. The light
source can be any number of places -- at the center of the globe,
at the opposite side of Earth, or out in space, for instance (see
figure below for examples). The map surface onto which the
projection is made can be various shapes, and can also be at
various places. In all projections, the map surface touches the
globe at at least one point. This is because any map is most
accurate where it touches the globe; there is no distortion here.
The contact point between globe and map is called the point of
tangency; if it is a line, it's a line of tangency, standard line,
or (if it is a parallel) standard parallel. Away from the tangent
locations, the map surface gets further from the globe, and hence
more distorted. Most all projections nowadays are done by computer
using equations that relate lat/long to x/y coordinates on the
map.
Projections may be made onto three basic shapes, with three
types of projections resulting:
a. Planar
Also called azimuthal. In this case, the globe is
projected onto a flat surface. The "light source" can
be from several locations. Usually, the flat surface
touches the globe at a single point. Most often planar
projections are used for Polar regions, and the
tangent point is the North or South Pole.
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b. Cylindrical
Here a cylinder is wrapped around the globe, usually with the
map surface touching the globe at a circle (a great circle, to be
exact--a circle whose center coincides with the center of Earth).
Cylindrical projections are the only one of the three main types
that can show the entire globe, and so most world maps are
cylindrical.
The most famous cylindrical projection is the one named for
Gerhardus Mercator, who developed it in 1569. It was valuable for
early navigators, since straight lines on a Mercator map are also
compass headings. Unfortunately, it greatly distorts the sizes of
areas near the Poles (see section 3 below), so it should not be
used as a general-purpose world map!
c. Conical
The third type of projection is made onto a cone. Usually this
means contacting the globe along one of the parallels (lines of
latitude), i.e., a small circle. Although we cannot use conic
projections for a world map, they are excellent for
continent-sized areas in the mid-latitudes. Most maps of the
United States are made with conic projections.
Conic projections are usually made more accurate by "sinking"
the cone part way into the globe (remember, this is all done with
computers, not literally!). Then we have two lines of tangency, or
two standard parallels, along which the map is extremely accurate.
This two-line approach is also called the secant case, as opposed
to the simple tangent case. You may see maps of the US with a
statement on the bottom like: "Lambert Conformal Conic Projection,
48° and 33° Standard Parallels." The projection was
developed by J.H. Lambert (1728-1777), an important figure in
cartography.
d. Other
Some projections are not based on any of the above three
shapes, and cannot be visualized as literally being projected.
Instead, they simply have equations that tell where to plot each
latitude/longitude coordinate from the globe. Some examples are
the Sinusoidal and van der Grinten projections. World maps are
often made with these projections, since they may have less
distortion than cylindrical projections.
3. Qualities of Projections
The other major factor you need to know about a projection is
the qualities about the globe that the projection either preserves
or distorts. Most projections can preserve one or more of the
following qualities, but none can retain all of them. Note that
the projection method (planar, cylindrical, or conical) does not
necessarily mean any of these qualities below are preserved or
distorted. It all depends on how the projection is done.
a. Equal-Area. Some projections show all areas in true
proportion to their real areas on the globe. For example, a dime
placed on the map would cover the same area regardless of where
placed. To show areas truly, a map must distort most of the other
qualities below, at least subtlely. But if you need a
general-purpose map of the world or continental area, an
equal-area map, or a map that is very close to equal-area, is your
best bet. Some examples: Sinusoidal, Albers Conic Equal-Area, and
Lambert Azimuthal Equal-Area.
b. True Shape, or Conformality. Another important
characteristic of the globe that can be distorted on a map is the
shape of areas. This distortion problem is obvious on a
cylindrical map that is equal-area, because the higher latitudes
near the Poles have to be distorted to preserve areas. The
Mercator projection is conformal, but at the expense of area. The
Mercator shows Greenland almost as large as South America, when in
reality it is about 1/8 the continent's size. Some people have
accused developed nations (which are mostly in the higher northern
latitudes) of intentionally portraying their lands as larger than
developing countries (which are mostly in lower, tropical
latitudes). One projection, known as the Peters projection, has
been promoted as the "true" world map, since it shows countries
with true areas. Peters is indeed equal-area, but does a number on
shape--as one person put it, it makes the world look like it was
hung on a laundry line. Many other equal-area projections are
available that do a better job with shape.
c. True Scale. In no map can you use one scale
accurately for the whole map. Some distortion occurs, although it
is slight in many maps. Some projections can preserve true scale
and distance along one or more lines. These are may be called
equidistant projections. A popular planar projection for polar
areas is known as the Azimuthal Equidistant, which has true scale
from the central tangent point--the Pole--to any other point on
the map. You could also use an Azimuthal Equidistant map centered
on your location to measure distances accurately to any other
place on the globe. Some map software can draw such a map for
you.
d. True Direction. The last major quality of maps is
direction. Maps that preserve it are called azimuthal. Most planar
projections preserve true direction away from the center of the
map (usually the Pole) and so azimuthal is nearly synonymous with
planar projection.
e. Other Qualities. Some projections are designed to
have specialized qualities. The Mercator projection is one: all
constant compass headings (rhumb lines, or loxodromes) are
straight lines. The Gnomonic projection is another: all great
circle routes are straight lines. As you may know, great circle
routes are the shortest distances between points on the globe. For
example, when you fly from San Francisco to London, you don't fly
along a parallel of latitude, but over the polar route; this is
along a great circle. If you're flying or sailing, then, you can
combine the gnomonic and Mercator maps for navigating. First you
draw your route on the gnomonic map (a straight line connecting
the two places), then transfer the route to the Mercator map as a
series of straight segments that approximate the gnomonic line.
This way, you can follow the straight segments on the Mercator map
with a compass, and turn only when you need to follow the next
segment. You may notice this when you're flying and the pilot
periodically turns to follow these segments.
B. Coordinate Systems
How can we describe locations on Earth? If someone asks you,
"where is Hawaii?", what do you tell them? You can give them
directions relative to your position ("swim 2000 miles
south-southwest"). Other ways are also possible, but what if you
needed to pinpoint a location for people coming from many
directions? Or if you wanted to record a location for later
reference? Or if you had no landmarks to guide you? This is the
purpose of coordinate systems. They are ways of describing
locations on Earth in reference to an established grid. You have
probably been exposed to the most common method,
latitude/longitude, but there are many other methods in use.
1. Latitude & Longitude
The latitude and longitude system is also called the
geographical grid. This grid exploits the fact that Earth is
nearly a sphere, and that it spins on an axis. Looking down on the
globe from above the North Pole, we can fit a circle to the
rotating Earth. We could assign each location along any circle
that surrounds the Pole a measurement in degrees. A circle has 360
degrees. We could use this range of numbers, going from 0° to
360°. Alas, early map-makers didn't do this, exactly. They
wanted low numbers on both sides of the Prime Meridian (the 0
line). As a result, the globe is divided into hemispheres, each
assigned longitude between 0° and 180°, with the
addition of East or West to differentiate the halves. The lines of
longitude are meridians.
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To complement the east-west measurement, a north-south
measurement is necessary, so that we may pinpoint
locations. Since we only need to measure along one
meridian, we only need to assign measures to a
half-circle, or 180 degrees. Once again, it's more
complicated than necessary. Rather than go from 0°
at the North Pole to 180° at the South Pole (or
vice-versa), the system starts with 0° at the
halfway point (the Equator), and measures north and south
to 90° at the Poles. Each line of latitude is a
circle; these lines are called parallels (sensible, since
they are parallel to one another).
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With this system we can pinpoint any location on Earth. Since a
degree of latitude spans about 111 km, each degree can be broken
down to get more exact. A degree is composed of 60 minutes (60'),
and a minute is composed of 60 seconds (60")--just like a clock.
Based on this system, SSU lies at 38° 20' 46" N, 122°
40' 30" W. If you're uncomfortable with this system, you should
practice looking up locations on a globe or atlas.
Lat/long is cumbersome to use for at least two reasons. First,
notice that meridians converge at the Poles. A degree of longitude
decreases from about 111 km at the Equator to 0 at the Poles;
1° is about 88 km in Sonoma County. Convergence makes
lat/long poor for use as a rectangular grid, where we want simple
x,y coordinates for locations. Second, lat/long is not a decimal
system. How far is it from 114° 34' 54" to 116° 14' 33"?
Not very far, but you'd have trouble giving me the distance even
in terms of degrees/minutes/seconds. For these reasons, lat/long
is usually replaced by other coordinate systems, especially at the
local level, for most descriptions of location. Most of these
systems, including those below, use a projection of the globe onto
a flat surface, onto which we can then draw an x/y grid.
2. State Plane Coordinates (SPC)
The National Geodetic Survey developed the SPC system beginning
in 1933. Eventually every state was covered, with coordinates
identified both on maps and on the ground, so that surveyors and
cartographers could accurately identify and measure locations. The
key to this system is that rather than having one coordinate
system for the entire US, separate systems were assigned to
smaller zones. Each zone used its very own projection and
coordinate center and system. 120 zones cover the US. Within each
zone, you are never far from the standard line. This way, the
coordinates would be extremely accurate within each zone (less
than 1 foot per 10,000 feet of measurement, in fact). The problem,
of course, is that coordinates between zones don't match up, so
the SPC system is not useful for small-scale (large-area) maps
that include more than one zone.
Nearly all states have multiple zones, but zones never cross
county lines. California has 7 zones, most extending as east-west
bands; Sonoma County's zone extends to Lake Tahoe. Los Angeles
County has its own zone (naturally). Each state uses either the
Lambert Conformal Conic or the Transverse Mercator projection
(California uses the first).
Within each zone, locations are identified by x,y coordinates
in feet. Any x,y coordinate system needs an origin, that is, where
the coordinates are (0,0). In order to keep all SPC numbers
positive, the origin for each zone is placed off to the southwest
of the actual zone covered. This origin is not the actual center
of the projection (that is, where the globe "touches" the sheet
projected onto). That actual center is in the middle of each SPC
zone, so that coordinates are most accurate there. In short, the
actual center is assigned an arbitrarily large coordinate (such as
2,000,000 feet East, 400,000 feet North), and all other
coordinates are measured from there. This puts the "false origin"
off to the southwest.
SPC coordinates are shown on all USGS topographic maps. Usually
tick marks on the margins of the map show regular spacing of the
grid, and selected marks have the actual coordinates in feet. By
examining the topographic map for Cotati, we can find that the
SPCs for SSU are 1,806,500' E, 246,200' N. As mentioned above, the
SPC system is used widely in conducting local land surveying and
public works. It can be used by the cartographer and geographer
not only to identify coordinates of places, but to calculate
distances between locations by use of the Pythagorean Theorem, as
described in the next section.
3. Universal Transverse Mercator (UTM)
The UTM grid is similar to the SPC system, at least regarding
how you use it at the local level and in being marked on all USGS
topographic maps. The principal differences are that the
coordinates are given in meters, not feet, and that the zones are
much larger. UTM zones extend north-south, practically from Pole
to Pole. The UTM grid system covers the entire globe (well, almost
-- except for very near the Poles).
You encountered the Mercator projection before. In the standard
Mercator, the cylinder is "wrapped" around the Equator, and areas
become very distorted toward the Poles. A transverse Mercator
projection turns the cylinder, so that the circle of contact with
the globe is around a pair of meridians. This way, the projection
is very accurate on a north-south zone near the standard line. Of
course, once again it distorts severely at large distances away
from the meridian.
The Universal Transverse Mercator grid gets around the
distortion problem by the same method as the SPC system. The UTM
has many zones, each with its own projection centered on a
meridian. There are 60 zones to be exact, each 6° wide (which
covers Earth, 60 x 6° = 360° around). Within each zone,
then, the grid is very accurate in matching true Earth distance
and direction. As with the SPC system, going across zones is
difficult, so the UTM is meant primarily for local and regional
measurement.
The UTM was adopted and thus popularized by the Army in 1947.
The Army included the UTM grid on its topographic maps; later the
USGS added UTM coordinates to most of its maps and photoquads. The
Army numbered each 6°-wide zone around the globe from 1 to
60, starting at 180° W and going east; northern California is
in zone 10. They also lettered north-south segments of each zone
from A (south) to Z (north). The north-south segments aren't
necessary, and so are rarely used outside the military.
Within each UTM zone, x,y coordinates can be given in meters.
Like SPCs, an origin is needed, and is placed outside the zone off
the southwest corner. The north-south center of the zone is
arbitrarily designated as 500,000 meters east (that is, east of a
false origin off to the west). "Eastings" (x-coordinates) for
locations east of the center are higher than this, up to about
850,000 m E; westward the coordinates decrease, down to about
150,000 m E; the zone doesn't extend all the way to the false
origin. The "northing," or north-south (y) coordinate, depends on
which hemisphere you're in. For the Northern Hemisphere part of
each zone, the measurement starts at the Equator with 0 and
measures the number of meters north (up to about 8,800,000 m N at
80° N). In the Southern Hemisphere, the Equator is designated
arbitrarily as 10,000,000 m N, and coordinates decrease as you go
south toward the South Pole.
Examples: A location with coordinates 334,400 m E, 4,203,600 m
N would be 334,400 meters east of the false origin, or (500,000 -
334,400 =) 165,600 meters west of the central line. It would be
4,203,600 meters, or 4,203.6 km, north of the Equator. The UTM
coordinates for SSU are: 4,243,540 m E, 528,390 m N (these are
actually close to the coordinates for Stevenson 3065).
The UTM grid is shown on all recent USGS topographic maps. The
latest topographic maps draw in the grid as thin black lines. All
topos with the UTM have tick marks along the margin, along with
values for eastings or northings next to most ticks. Except for a
few values near the corners, the easting or northing value is
abbreviated . For example, instead of printing "3,445,000 m N",
the tick would be labeled 3445, with the thousands and
meters-north assumed from the context.
The UTM grid, even if drawn in on the map, may not give us the
exact coordinates for a given location. Even on 7 1/2-minute
quads, the grid is only every 1,000 meters (1 km). How can you
determine coordinates more precisely? The answer is called a
roamer. This is simply a sheet of paper, plastic or other material
that has finer distance intervals marked off that match the scale
of the map. Starting from the nearest grid lines, you can measure
over to the location and come close (at least within 100 m) to the
actual easting and northing coordinates.
Another big advantage of the UTM (or SPC) grid is that once you
have coordinates for two locations within the same zone,
calculating the distance between them is simple. Just apply the
Pythagorean Theorem. If the two locations are at the coordinates
(x1, y1) and (x2, y2), then the distance (D) between them is:
For example, say you find coordinates for two cities:
Springfield at 294,100 m E, 3,428,900 m N, and Garden City at
292,400 m E, 3,428,100 m N. The distance between them is then:
that is, the distance is 1880 meters, or 1.88 kilometers.
As you can see, the UTM grid is a very useful system for
tracking Earth locations. It is used extensively in remote
sensing, computerized mapping, and geographic information systems.
It is worth your while to familiarize yourself with it.
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