Projections Tutorial

This is the introductory topic on geographic projections for users new to GIS and mapping. It is a good refresher for experienced users wishing to understand how Manifold deals with projections.

Geographic projections are a way of showing the curved surface of the Earth on a flat surface like a piece of paper or a computer monitor.

For over two thousand years educated people have known that the Earth is round and have realized as a matter of elementary geometry that any flat map showing the surface of our curved Earth will in some way change the shape of what it portrays. For over two thousand years geographers have been inventing ways of using flat maps to show the curved surface of the Earth in ways that minimize such distortions.

Imagine we make a globe out of a flexible material with a world map painted onto the material. Now, let’s cut the material and try to flatten it. If you’ve ever tried to flatten a deflated ball you know this is not possible to do without stretching the flexible material in some areas and compressing it in others. If we deform the peeled "skin" of our atlas globe in this way to make it flat we will end up changing the shape of continents and other items illustrated in our world map.

There is no one way of projecting the curved surface of the Earth onto a flat sheet that does not cause some distortion, and there is no one projection that is suitable for all purposes for which people use maps. However, for virtually every usage there are projections that minimize distortions of importance for that task. For example, if one needs to measure areas in a flat map there are projections that will guarantee the area contained by various shapes is correct even if the shapes of the objects shown appear quite differently than they do on a globe. Other projections do a good job of showing continents in a shape similar to that seen on a globe even though they do not allow for accurate measurement of areas.

Note: We refer to the Earth as a sphere in this topic even though it is a slightly flattened ellipsoid.

How Projections Work

Any shape that is curved in only one direction can be unrolled into a flat map without distorting the appearance of objects drawn on it. For example if we take a cylinder and cut it lengthwise we can unroll the cylinder into a flat map. The trick to any elementary projection is to place the Earth within such a shape and to "project" lines out from the Earth onto the shape to show where to draw the projected outlines of items on the globe. We can then "unroll" the shape and see our projected map on a flat surface.

The first and most obvious such projection to use is a simple cylindrical projection. Place the Earth inside a big cylinder that touches the Equator and then transfer points on the globe to the cylinder. The simplest way to do this is to imagine the cylinder is graph paper with 360 boxes in circumference and 180 boxes up and down.

If we use the longitude and latitude coordinates in degrees of a place on the sphere and transfer it to our cylindrical roll of graph paper we end up with a map like the above. Since one point needs to be the center of the unrolled cylinder, nearly universal usage is to use the intersection of the Equator with the zero Meridian running through Greenwich, England. We can then count degrees plus and minus 180 degrees in longitude and plus and minus 90 degrees in latitude.

The above presentation is called the Geographic or Latitude / Longitude projection. We can think of it as our default projection. It produces a good effect in areas near the Equator, but results in immense distortion close to the poles.

There are other ways of transferring points from the surface of the globe onto an enclosing cylinder. Most of these, such as the Mercator projection, use some mathematical formula to alter the ratio between degrees of latitude on the globe and vertical measurements on the cylinder. What they all have in common is that accuracy is good near the Equator where the cylinder is very close to the globe. To a greater or lesser degree all cylindrical projections centered on the Equator fall off in accuracy as distance from the Equator increases.

If we wish to make maps of places along the Equator we could use a cylindrical projection and just show those regions. What would be shown in those maps would be relatively free of distortion. One problem with this is that the Equator for the most part lies over water whereas the greatest demand for maps is in populated zones. A quick glance at a world map shows that most populated zones occur in a North-South direction.

Turning the cylinder so that it is tangent to the Earth along a meridian (longitude line) instead of tangent to the Equator results in what is called a transverse cylindrical projection. We can now make local maps anywhere along the darker, North-South line of tangency and if the maps are not too big they will be relatively free of distortion. However, this only works along the line of tangency. If we pick a North-South line running through Athens we can make maps all the way from Scandinavia down the length of Africa, but any maps using this projection in North and South America would be hopelessly distorted.

One possible solution is to use not one projection, but many transverse cylindrical projections with the cylinder rotated slightly along the Equator. In fact, one scheme of mapping the Earth called the Universal Transverse Mercator (UTM) plan does just this. UTM maps the Earth with a transverse cylinder projection using 60 different lines, each of which is a standard "UTM Zone". By rotating the cylinder in 60 steps (six degrees per step) UTM assures that all spots on the Earth will be within 3 degrees of the center, tangent line of one of the 60 cylindrical projections. (The Gauss Kruger system is a European system akin to UTM that also uses a transverse cylinder rotated in six degree steps).

To map any spot on Earth in UTM, one picks the UTM Zone centerline that is closest to it and then makes a map using that cylindrical projection.

The illustration above shows a small section of the earth near the tangent line projected onto the cylinder, and then the cylinder being unrolled into a flat sheet. If we want to save the X,Y locations of points on our flat sheet we can now measure them as though the flat sheet were graph paper and use the resulting coordinates in a digital, flat map.

The above illustration shows a key concept that often proves confusing to GIS newcomers: although "unprojected' data about locations on the Earth are specified in degrees, all projected maps specify the coordinates of the objects on them using X,Y coordinates using meters, feet or other linear measures. These coordinates are computed relative to some origin on the flat sheet established by the projection in use.

Computer files that contain projected maps therefore contain coordinates like

44030976,38403088
44030984,38403080
44030900,38403077

and not longitude,latitude coordinate numbers such as

-110.3484, 44.2856
-110.3463, 44.2889
-110.3511, 44.2902

Latitude,longitude coordinates are normally in decimal degrees as above, while the coordinate numbers in projected files are most often meters in X and Y directions from some origin known to the projection. It is as if the green sheet in the illustration above were an enormous piece of graph paper on which the map is drawn "full size" and then measured off in meters.

In a well run GIS system the internal coordinates of projected maps may be hidden from the user because the GIS software will automatically translate the internal map drawing coordinates into Latitude/Longitude values on the fly. Manifold, for example, will show cursor position in a projected map view using Latitude and Longitude values. What is going on is that Manifold is automatically translating internal projected coordinates like 44030984,38403080 into the equivalent Longitude and Latitude values.

Conic Projections

The main problem with cylindrical projections is that they do a poor job of minimizing distortion except for very close to the line of tangency. They are a poor choice for mapping large countries (such as the US or Russia) that have great East-West extents.

A better choice for mapping such regions is a conic projection, which projects shapes from the Earth’s sphere onto a cone. Cones, of course, can be unrolled into a flat sheet without any deformation. Locations near the line where the cone is tangent to the Earth will be relatively free of distortion. By using taller cones we can move the line of tangency nearer to the Equator and by using fatter, more open cones we can move the line of tangency closer to the pole.

We can see the practical effect of a conic projection by considering a map of North America shown in the Latitude / Longitude projection. This is an "unprojection" that simply takes each coordinate in degrees and plots it using equal sized X and Y degrees at all locations:

The geographic cylindrical projection greatly overstates the size of northern regions.

Using a conic projection, we can transfer the shape of North America to the cone (in the region marked in red on the cone) and then unroll the cone to make a flat map. That flat map can then be used as "graph paper" to measure off coordinate locations with which we could build a flat, digital map.

The resulting flat map provides a much better impression of the true shape of North America. It is interesting to note that since most schoolchildren are taught geography from maps using cylindrical projections that greatly distort Northern regions, the average person thinks Alaska and Greenland are many times larger than they really are. The above conic projection uses a tangent line cutting through the "lower 48" US states and so optimizes their appearance while understating the apparent size of Alaska.

When both are viewed in Lambert Conformal Conic projection using parameters midway between the "lower 48" and Alaska and Alaska is moved over the "lower 48" US and rotated to preserve apparent meridian angles, it's clear that Alaska is very large, but not as large as is commonly thought.

Azimuthal Projections

Azimuthal projections show one hemisphere of the Earth at a time by projecting lines upward from the globe onto a flat disk tangent to the globe at one point.

By centering the disk over any particular point on the Earth, one can achieve a view of the Earth as it appears from space from high over that point. The Orthographic projection is the classic "view from space" azimuthal projection of the Earth.

Projections and Projection Parameters

Virtually all projections in common use fall into one of the above three categories. They are either cylindrical (regular or transverse), conic or azimuthal projections as customized by slightly different projection parameters. Projection parameters are options in how the projection is arranged.

For example, the Orthographic projection can be centered on any point on Earth by specifying the latitude and longitude of the desired central point. Conic projections may be customized by specifying the parallel of latitude at which the cone should be tangent.

Specifying a projection together with various optional parameters will drive the mathematical conversion of longitude,latitude degree coordinates into the numbers used within the projected coordinate system. When we encounter a computer file with projected data numbers such as…

44030976,38403088
44030984,38403080
44030900,38403077

…we will not be able to make geographic sense of these number unless we known in which projection with which optional parameters they are intended to be used.

Some GIS formats are "smart" and automatically save the projection parameters in use together with the data. During import of drawings from such formats, Manifold will fetch all necessary parameters from such "smart" formats automatically and will load the coordinates properties for that drawing with the correct parameters necessary to use the data.

When importing projected drawings from "dumb" GIS formats that do not save the projection information with the data we will need to know what projection and parameters should be used with that drawing. We will then have to enter this information manually into that drawing's coordinate properties so Manifold can use the data as intended.

Projections dialogs in Manifold are set up so they automatically present available options for the projection in use. Some specialized projections allow specification of an elaborate set of optional parameters.

False Easting and False Northing

Once a map is constructed using a given projection, the map is a flat surface. Distances on that flat surface may be measured as X and Y rectangular coordinates, with the X coordinate being the distance to the right of the vertical line passing through the origin or the center of a projection. A negative X coordinate represents distance to the left. In practise a false X or false easting is frequently added to all values of X to eliminate negative numbers.

Likewise, the Y rectangular coordinate is the distance above the horizontal line passing through the origin or center of a projection, with negative Y being the distance below. In practise, a false Y or false northing is frequently added to all values of Y to eliminate negative numbers.

The use of false easting and false northing is a relic of days when map projection computations were done by hand, so that computation with negative numbers was less convenient. In modern times we let computers do all the computational drudgework so false easting and northing are no longer essential. However, they continue to live on within projected digital maps created using older methods. Manifold allows use of false easting and false northing with many projections.

A Historical Note on the Round Earth

School children are often wrongly taught that Columbus sailed Westward to China to prove that the Earth is round. Once launched on his journey Columbus is often portrayed as heroically pressing on despite the opposition of his sailors, who feared their little fleet would fall off the edge of a flat Earth. That is almost the exact opposite of the truth.

Most educated people in Columbus's day knew the Earth was round. In fact, they not only knew the Earth was round they knew the size of the Earth as well. Almost everyone except Columbus accepted the estimate for the radius of the round Earth computed by Eratosthenes of Cyrene (276-195 B.C.). Eratosthenes figured the Earth's radius to be about 6267 kilometers, a figure remarkably close to the modern mean of about 6371 kilometers. In the 1490's educated people had known for over one thousand five hundred years the actual size of the round Earth. Since ancient days cartographers had even created projections to deal with the representation of a round earth on flat maps.

Even many uneducated people knew the Earth was round. Among uneducated people sailors especially believed the Earth to be round because of the frequent observation at sea that tall points such as mountains come into view above the horizon as the distance to an objective becomes closer. Many "round Earth" visual effects incompatible with a flat Earth are easily seen by the human eye at sea.

Columbus met much opposition at Court to his plan precisely because people knew the Earth was a very large sphere. The ships of Columbus's day were so slow that they could not be loaded with enough food and water to voyage directly to China westward from Europe. Without the then-unknown continents of North and South America to use as re-supply points the direct voyage would be so long that the crew would die before making landfall.

Columbus based his plans for his voyage on the argument that the Earth is smaller than it truly is. Educated people were unimpressed with what they regarded as his chain of wishful-thinking assumptions that "proved" Eratosthenes was wrong and that a Westward voyage was just barely feasible. When Columbus launched across the Atlantic his sailors were fearful that in the event his estimate of the Earth's size was wrong and everyone else was right they would expire of thirst and starvation.

As it turns out Eratosthenes was right and Columbus was wrong about the size of the Earth. Columbus simply had the good fortune of rediscovering a New World before he and his crew died proving the true size of the round Earth. In all fairness it should be pointed out that despite his flawed belief in a small world Columbus was a master admiral of unparalleled skill, intelligence and personal courage. A failed and quarrelsome administrator on land, Columbus is indisputably one of the greatest leaders who ever took to sea. He is alone among the early voyagers in executing and surviving four successful voyages to the New World.

Christopher Columbus by Sebastiano del Piombo

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