Maps and Legends
7 TECHNICAL DETAILS
It has been said that the making of a good map requires
a compromise between all sorts of distortions so that in the
end none is too objectionable. Some maps keep area true at
the expense of shape while others keep local directions exact
but alter dimensions. The projections available in "Maps and
Legends: The Cartographer" cover both of these kinds of maps
and more. By studying the differences between the
projections that you can make with this program you will see
how these compromises can be handled to produce maps that are
both effective and pleasing to view.
In this section we present some background information
on each of the projections available in Maps and Legends.
For those who are mathematically inclined we have also
provided the formulae that describe each projection.
Cylindrical Projection
The cylindrical projection provided in "Maps and
Legends" is the rectangular even-spaced projection. It is
one of the simplest projections to express mathematically.
As its name implies, this projection results from projecting
surface features outward onto a cylinder which is wrapped
about the earth and aligned north and south. Parallels and
meridians form straight horizontal and vertical lines, and a
given change in latitude or longitude corresponds to the same
horizontal or vertical distance anywhere on the map.
The formulae which relate latitude and longitude to
horizontal (x) and vertical (y) position on the map are:
x = a * (longitude) - b
y = c * (latitude) - d
In these expressions (and those which follow) a, b, c,
and d are constants which are determined from your
specification of the map coordinates to be displayed on the
screen. With the "Cylindrical" projection option of Maps and
Legends you specify the region to be mapped in both latitude
and longitude. As a result, you control the relative scales
in these two directions (a and c). You can choose any region
of the globe to appear within the screen. You can also use
this feature to distort your map by stretching it in latitude
or longitude. Some interesting effects can be produced this
way. The disadvantage of this method is that you must use
care if you want your map to show the same distance scale in
both dimensions. Mercator Projection: The Mercator
Projection is named for its developer, the great Dutch
cartographer Gerard Mercator (1512 - 1594). In Maps and
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Legends this projection is available as one of the "World
Map" projections.
The Mercator projection is closely related to the
cylindrical projection, but it differs in a very important
way. In the Mercator projection a given angular distance in
latitude corresponds to a greater distance on the map closer
to either pole. As a result, areas nearer the poles appear
larger than they really are. The great advantage of this
projection is that straight lines on the map show true
compass direction. Thus to sail a ship from one point to
another, one had only to sail to the compass course indicated
by a straight line on Mercator's map. The path that resulted
may not be the shortest path, but it would get you to your
destination. This is why Mercator's projection was so
significant to the world navigators of his time. The great
disadvantage of the Mercator projection is the distorted
picture that it presents of the relative areas of various
regions of the world. Accustomed to the Mercator projection
from maps studied in school, few people can believe that
South America is really more than nine times larger in area
than Greenland.
Mathematically, the Mercator projection is expressed
as:
x = a * (longitude) - b
y = c - a * log{ tan[(pi/2 - latitude) / 2] }
The constants a, b, and c are determined in Maps and
Legends to produce horizontal and vertical scales in the
proper ratio and to center the map at the longitude which you
enter in the dialog. Because of the way the Mercator
projection is constructed, a finite map can never show the
poles. In Maps and Legends the overall scale is defined to
show the major land areas of the world well. If the scale
were decreased to display Antarctica, for example, the rest
of the world would appear quite small.
Conic Projections:
To understand how a conic projection is defined, imagine
the earth inside an ice cream cone (for southern hemisphere
projections), or wearing a dunce cap (for maps in the
northern hemisphere). Features are projected onto the map
from the center of the earth, and the map is displayed by
cutting the cone and unrolling it. The conic projection was
first published by Ptolemy in the 2nd century A.D.
Conic projections are at their best for showing regions
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of the world that span a wide range of longitudes in the mid-
latitudes, like the continental US. A conic projection is
exact on the parallel (or parallels) where the cone
intersects the earth. (The cone can be made a bit smaller,
to cut the earth along two parallels.) These parallels are
called the standard parallels of the map. The latitude of
the standard parallels determines the angle of the cone.
Conic projections encompass a large family of map
projections, some of which also go by other names. If, for
example, the standard parallel of the conic projection is the
equator, the cone becomes a cylinder and a cylindrical
projection results. If the standard parallel is at the pole,
the cone flattens into a sheet and the projection is the same
as the azimuthal equidistant projection described below. If
you request a conic projection covering a large longitude
range near a pole, the cone on which the map is projected,
when unrolled, will not fill the display. This is not a
problem with the program, it's just the way the conic
projection works.
Maps drawn with the conic projection can be quite
accurate in scale. However, the distortions in direction
produced by the conic projection are responsible for the fact
that many Americans think Washington and Maine are farther
north than Minnesota.
In this program, standard parallels are chosen that
split the center meridian 1/4 from the top and the bottom of
the region that you select. (Unless, of course, you have
chosen a map centered on the equator or the north or south
pole).
The formulae for this map projection are:
x = (a + pi/2 - latitude) * cos( b * longitude)
y = (a + pi/2 - latitude) * sin( b * longitude)
The constant b is the sine of the angle of the cone
(alpha), and a is determined from the latitudes of the
standard parallels and distance between them. If the
standard parallels are at latitudes Lt1 and Lt2 then:
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b = sin(alpha) = cos(X) sin(d)/d
a = d tan(X)/tan(d) - X
where
X = pi/2 - (Lt1 + Lt2)/2 and
d = (Lt1 - Lt2)/2
in other words, X is the average colatitude of the standard
parallels, and d is half the distance between the standard
parallels.
Azimuthal Equidistant Projection
The azimuthal equidistant projection is the only
projection in which every point is shown at both the correct
distance and the correct direction (azimuth) from the center
point. All other distances and directions are distorted. This
projection is used to determine great circle directions. If
you place your home town at the center of the map, you can
easily determine the azimuth for pointing shortwave antennae
for reception of distant radio stations. Straight lines away
from your home at the center of the map will be the great
circle routes flown by airplanes on international flights
from an airport near your home.
A point half way around the world from the center of the
map is 20,000 kilometers away. To generate a map of the whole
world, enter 20000. Do not put any commas in the number. To
generate a map to a smaller great-circle distance, enter a
number less than 20000.
If the center point is at latitude Lt0 and longitude
Lg0, then another point at Lt, Lg will appear on the map at
an x,y position:
x = A cos(Lt) sin(Lg-Lg0)
y = A [sin(Lt) cos(Lg0) - cos(Lt) sin(Lt0) cos(Lg-Lg0)]
where
A = c / sin(c) and
cos(c) = sin(Lt) sin(Lt0) + cos(Lt) cos(Lt0)
cos(Lg-Lg0).
Perspective Projection
Prospective projections are those which show the Earth
exactly as it appears when viewed from some point in space
above the surface. This is one of the best projections to use
for games and space war simulations.
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If you enter a very large number here you will get an
orthographic projection. In practice a good height to use for
an orthographic projection is 999999 miles.
You might enjoy putting in a low height to see a "birds
eye" view of the area around your home town. A good number to
use is 250 kilometers. That is about the altitude of the
Space Shuttle orbit. If you make a low altitude perspective
projection over the Unites States, it's fun to include the
state boundaries. The view can seem very distorted from low
altitudes. Your state will take up much of the field of
view.
If Lg is the longitude at the center of the map then
x = r sin[Lg cos(b)]
y = r cos[Lg sin(b)]
where r = a [ cos(Lt) / cos(b) ] / { h cos(b) + cos(Lt - b)}
This is a lot like a conic projection except cos(b) = n
is the sine of the angle of the cone. b is the colatitude of
the central parallel of the map. The orthographic projection
is the limit where h goes to infinity. Otherwise h is
between 1 and a large number.
Orthographic Projection
In orthographic projections the point of view is at
infinity. Orthographic projections are a special case of
perspective projections. The orthographic projection is the
limit where h, in the calculation for the perspective
projection, goes to infinity.
Other World Map Projections:
We have included a selection of different world map
projections in Maps and Legends. Some of these projections
are useful, some are important in the history of cartography,
others are included just because they are interesting to make
and view.
The Flamsteed, Sanson-Flamsteed, or Sinusoidal Projection
This projection is an equal area projection in which
parallels of latitude are straight and equally spaced. "Equal
area" means just that. A square mile, or hundred square
mile, etc. region of the world will have the same area on the
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map no matter where that region is. (Contrast this to the
Mercator projection.) Equal area, however, does not mean
constant shape. That "square" mile will not be square
everywhere on the map.
In the Sanson-Flamsteed projection all parallels are at
their true distances from the equator. In addition, the
scale along each parallel is exact. The shapes of all
meridians and the outline of the map follow a cosine curve so
that:
x = a (longitude) cos(latitude) + b
y = a (latitude) + c
Sanson was an important early cartographer and he was
the first to publish this projection. Flamsteed used this
projection in his famous star atlas and it was due to him
that this projection became popular.
The Foucault Projection
The Foucault projection is closely related to the
Sanson- Flamsteed projection. This is also an equal area
projection, named after the french mathematician who derived
it. It is interesting to compare the Foucault and the
Sanson-Flamsteed projections. Both projections have straight
parallels of latitude, but in the Foucault projection, they
are not evenly spaced. Notice how these two maps each bring
out different aspects of the regions that they display, and
how they differ in where the map distortions become too
distracting. The Foucault projection can be expressed as:
x = a (longitude) cos(latitude) cos2(latitude/2)
y = a pi tan(latitude/2) / 2
where a is the map scale factor.
The Mollweide Projection
One of the most beautiful world map projections is the
Mollweide projection. It is also an equal area projection
with straight parallels of latitude. Much of the appeal of
the Mollweide projection is due to the way that it solves the
polar distortion problems that are so distracting in the
Sanson-Flamsteed, and especially the Foucault, projections.
In this projection, the outline of the map is an ellipse with
a ratio of 2/1 in the lengths of the major and minor axes.
The Mollweide projection is one of the more difficult to
calculate, because the projection requires the solution of
the transcendental equation:
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2t + sin(2t) = pi sin(latitude)
the resulting value for t then determines the x and y
position on the map via:
x = a (longitude) cos(t)
y = a pi sin(t) / 2
Maps and Legends doesn't solve this equation for every
coordinate point because that would take too long. Instead,
the equation is solved for a number of points in advance, and
the solution for any point of the map is looked up in this
table.
The Werner Projection
Certainly the most remarkable of the equal area
projections is the Werner projection. In this projection the
outline of the world takes the shape of a heart. It is this
unusual shape which justifies its presence in Maps and
Legends. Here, finally, is an equal area projection where the
parallels are neither straight nor parallel. Johann Werner
was a young contemporary of Columbus. While he knew nothing
about the polar regions, he knew that the earth was round and
he had the mathematical genius to develop this map which has
been called "the cartographer's valentine".
Mathematically, the Werner projection resembles the
conic projection, except that n, the angle of the cone,
depends on the latitude:
x = a h sin[B (longitude)] - c
y = -a h cos[B (longitude)] - d
where B = sin(h)/h and the colatitude h = pi/2 - latitude.
The Polyconic Projection
Finally we come to the projection which may be the
ultimate map compromise. The polyconic projection is neither
equal area nor does it preserve directions, distances, or
angles exactly, but it is never far off, either. The
polyconic projection was developed in 1820 by the first
Superintendent of the Coast and Geodetic Survey of the
then-young United States, Ferdinand Hassler. At that time
the US stretched mostly north and south along the Atlantic.
Hassler derived his projection to overcome the errors
associated with the conic projection, when it is applied to
regions that are larger north-south than east-west.
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Conceptually, the polyconic projection is a series of
narrow conic projection strips cut out from the region
surrounding the conic's standard parallel and pasted
together. This projection is ideal for a series of
contiguous maps of a large region. In fact, the topological
maps of the US Geological Survey are all based on the
polyconic projection.
The formulae which express the polyconic projection
are:
x = (Lg) cos(Lt) / H
y = (Lt) + (Lg)(Lg) sin(Lt) cos(Lt) / (2 H)
where H = 1 + {(Lg) sin(Lt) / 2}2
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