Section Seven: Projected Coordinate Systems (AKA “Projections”)

In the last three sections, we looked how reference ellipsoids are combined with geoids via control points to create datums and how geographic grids utilize an angular unit of measure to label intersections of north-south and east-west lines (parallels and meridians) starting at a principal meridian, examining the latitude/longitude system as an example. We then learned that combining a specific datum with a geographic grid creates an accurate way to locate those labeled points utilizing the geographic coordinate system. In this final section, we will look at the final concept that connects geodesy to GIS: projected coordinate systems.

While it is possible to use a geographic coordinate system in the GIS and print a map showing angular units, this system doesn’t make a ton of sense in our minds. If I told you the area of a particular place was 0.00034 degrees squared, how big is that? As big as Rhode Island or as big as Texas? You (I’m going to make a fair assumption here) most likely have no idea. If I gave you directions to my house by saying “travel along Main Street at a heading of 270 for 0.029 degrees latitude and make a left on 9th. You’ll find my house (the bright blue one) at 2 08’59.96” N, 110 50’09.03”W”, you most likely would have a tough time trying to find it without the use of a GPS app or receiver. Overall, we do not perceive the world in angular units, but instead by linear units - the area of a place is 1 mile square or you tell people to 'head 2 miles west on Main and turn right on 9th'.

Which leads us to our first step of making flat maps which utilize linear units of measure: “remove” the crust of the earth and press it flat. Once it’s flat,we can use a wooden ruler and measure in linear units (inches and centimeters, and since a map is defined as a two-dimensional scale model of the surface of the Earth which conveys a message to the user, each inch or centimeter would represent some real world unit such as miles or kilometers).  However, the process is not as simple as removing the crust of the Earth and pressing it flat. Like peeling a tangerine and smashing the peel flat with your hand against a table, if you were to peel off the crust of the Earth and attempt to make a flat map out of it, it will look just the same – no real shape and far from useful.

FIGURE 2.20: Making a Flat Map of the World is So Similar Yet So Different Than Simply Removing the Crust and Smashing It Flat Like a Tangerine Peel

The process, utilizing a projection method to create a 2D projected coordinate system (a flat map from a round globe) is a more complicated then simply peeling off the crust and smashing it flat, and results is a side effect called map distortion, but as a bonus for the complication (as we are about to see) comes a much neater, easier to understand product.   We will spend some time in the next section exploring the idea of projection methods and creating projected coordinate systems, but for now, just understand it is the method we use to get the shapes from the round Earth onto a flat map.

When making projected coordinate systems, because we are taking something round and representing it as something flat, our choices really only are: 1. a funky shape (like the orange peel above) that presents us with zero distortion, or 2. using a projection method, create a more familiar shape, such as a rectangle, but introduce distortion into the map. The problem with the former is the lack of usefulness of the funky shape (how would you measure the area of South America on that orange peel?), and the problem with the latter is there is no specific projection method which reduces or preserves all kinds of distortion. 

Distortion is a by-product of the projection process.  Since we don't want flat maps which look like the orange peel we see above, when we move the land and ocean masses back into place, some measurement becomes distorted.  To preserve the shape of the continents, we need to stretch and move them around, which results in the area no longer being true.  If we try to preserve the area, the shape becomes distorted.  When the continents are moved and the shapes distorted, the distance between them become affected.  If the distance is held true, then the shape and area are going to become distorted. The only way to keep all of the measurements true is to make the flat map in the shape of the smashed orange peel - which takes us back to the beginning where the maps was useless.  

Some projection methods are designed to keep the shapes of the represented areas (specifically “conformal”), while other are designed to preserve distance (specifically “equidistant”), and still others preserve direction, shape, and scale. Listed out, the six major kinds of distortion we are battling when we make projections from geographic coordinate systems are:

  1. Shape (the shape of the world feature vs. the shape drawn on the map)
  2. Area (the measured area of a world feature)
  3. Distance (the measured distance between two world features)
  4. Direction (the cardinal direction between two world features, minus distance information)
  5. Bearing (the cardinal direction measuring from one world feature to any other)
  6. Scale (comparing the size of two world features vs. the same two drawn on a map)

Or as one GIS 101 student once put it, "Map distortion from the projection method is some SADD BS".

As much as I'd like to tear into a plastic globe each semester to illustrate the point, that isn't really practical.  Instead, watch the following video.  It does a great job of explaining projected coordinate systems and distortion.  It will help you understand distortion and the rest of the section.

2.7.1: Creating Projections

To understand how it is the round world is made flat via the projection, imagine yourself as an Earth Explorer, standing in the center of a clear globe.  After high-fiving the guys from Fragglerock, you look out and see all of the continents surrounding you. If you look in the direction of the Washington Monument, you’ve noticed someone has used it to stab the center of a piece of stiff cardboard, much like a lunch server would do with a completed ticket. Let’s call this piece of cardboard a developable surface (when the developable surface is flat, we call this azimuthal). This piece of paper is so large from your point of view, it creates a background for the entirety of North and South America. Since the cardboard is stiff, it touches the Earth only at the Washington Monument (the tangential point) and ‘floats’ above the surface at all other points.

From the center of the globe, you take your handy-dandy Light Gun of Science (LGoS), and begin to point it at each city in North and South America.  As the beam from your LGoS passes through a city and strikes the cardboard, it leaves a labeled mark (it is a Light Gun of Science, after all). Just like a movie projector throws the image from the lens to the wall, your LGoS projects a labeled city point from the edge of the globe to the cardboard.

You notice the distance the light travels from the edge of the globe to the cardboard at Washington DC is zero, since the cardboard touches the globe at that point.  You also note the distance the light travels moving away from DC increases. And because the Washington Monument is not in the exact center of the United States, the distance the light has to travel between New York and the cardboard is shorter then the distance between Los Angeles and the cardboard.

FIGURE 2.21: LGoS vs North and South America
LGoS vs North and South America: Technically An oblique gnomonic projectionThe Mapparium in the Mary Baker Eddy Library in Boston lets you experience the Earth’s geography without as many compromises.

After you’ve marked all the major cities, you then use the LGoS to mark the entire edge of the North and South American continents. Once you take the cardboard off the Washington Monument, you connect the dots to draw the outline of the two landmasses, use a ruler to draw a neat grid over the whole thing, and give it a name which reflects the area it shows and the method used to create it.

Congratulations, you’ve made your first projection.

FIGURE 2.22: LGoS vs North and South America
The result of your projection. Oblique gnomonic projections distort the map in an increasing manner away from the tangential point
The Main Point
Projected coordinate systems are created when a single source of light is used to project an image of land features onto a flat surface in order to transfer the round Earth to a flat map, with it, connecting the labeled points on the GCS to the labeled points on the PCS. This method comes with some consequences in the form of distortions which are presented as shape, area, size, bearing, distance, and direction, noting that the distortion near the place where the developable surface touches the GCS has the least amount of distortion while those places far away have the most distortion. Some projected coordinate systems are designed preserve between one and three factors, but need to give up the others in order to do so.

2.7.2: Projection Methods

As discussed in the first section, the main goal of creating a projected coordinate system is to create a flat representation of a round Earth (a map) which uses linear units of measure (since we best think in linear units) while minimizing distortion. If you look at your Oblique Gnomonic Projection made with your LGoS, there is distortion. The light traveling a longer distance from the surface of the globe at Los Angeles before striking the developable surface vs New York. This longer distance of travel between the 3D and the 2D objects is the source of the previously discussed distortion (along with the need to stretch, bend, and pull the land masses to fit inside the desired developable surface). The Oblique Gnomonic projection method creates a distortion of area and distance, meaning if you were to measure the distance or calculate the area on the resulting map, you'd notice the measured values will vary from the real-world accepted values (as based on geodesy) and that variation will not be constant across the map. Your map will show an increase in area and distance radiating away from the tangential point (Washington Monument).

Figure 2.23: The Three Main Types of Projection Methods - Azimuthal (Planar), Cylindrical, and Conical

While the LGoS example was a silly visualization of how one might create a projected coordinate system, in reality map projections are created using a rather complicated, mathematical process. As this is not a geodesy nor calculus course, we will simplify it into a generalized five-part process.

  1. First, we select a developable surface. In our LGoS example, we selected a developable surface known as azimuthal or planar, which started our flat and ended flat. The other two most common examples of developable surfaces (but in no way the limit) are cylindrical and conical, which start out as a cylinder or cone, respectively. These 3D options are slipped over the Earth, sometimes with the Earth fitting neatly inside the shape, and others where the Earth is a bit too big and bulges outside the shape in a few places.
  2. The second part of creating a projection is to select an aspect. Since the Earth is more or less a sphere, that sphere has no limitations upon how it can fit inside a cylinder or cone, or how many ways you can stick a piece of cardboard to it. If you imagine the toddler game where they are required to place the wooden shapes inside the correctly matched holes, the limit is the game piece and the opening.

    Yet, with a sphere and a cylinder or cone, there are no limits. As long as the sphere is smaller in diameter than the cylinder or widest part of the cone is, it really doesn't matter which way you turn it before placing it inside (In reality, when it comes to making projections, we are also not limited by the diameter, as we just let the Earth bulge out, but then again, projections are all based on math and are a representation of the real world). But back to aspects. Since there is no limitation to how the cylinder, cone, or azimuthal surface will fit over a sphere, we need to define a few common ways to turn the sphere prior to making the projection - normal aspect, transverse aspect, and oblique aspect. These terms vary based on which projection they are referring to, that is to say, normal for one may not be the same normal for another, but in general, normal is the most common way to place the developable surface over the Earth for that projection, transverse is to turn the developable surface 90 degrees from normal, and oblique is all those angles in between. Both of these concepts are explained a little more clearly in the following diagram.

Figure 2.24: A Summary of The Most Common Aspects and Developable Surfaces
  1. The third part of the three-part projection process is to transfer all of the points from the geographic grid on the GCS to the developable surface. We've learned that the term projection comes from the idea of passing light at a single source through the globe and tracing the resulting an image, and thus that is what we are doing. If the image I'm tracing is the geographic grid, I can also transfer the labeled intersections of that grid (remember the whole goal behind creating a GCS is to draw a grid over the surface of the Earth and assign each intersection of the horizontal and vertical lines with an address). After I've transferred the labeled grid from the Earth to the developable surface with the Earth turned at some aspect, I can cut one side of the 3D developable surface and roll it out flat.
    Once I have the geographic grid transferred to the projection and I know the address of each point along the edge of a physical land body, I can mark each point on the projected coordinate system in the proper place, essentially creating a World Map Dot-to-Dot and connect all of the dots on the flat map to draw in the landmasses I wish to represent.
  2. Last, I give the projection a logical name that might represent the developable surface and/or the aspect used, the distortion I'm attempting to preserve, the area in the world it's best fit to use, and since it was a tough road to get to my new projection, maybe my name. All projections will have at least one of these descriptive words, and many will have all.

Conic Projections

Conical projections use a developable surface which start in the shape of a cone. The cone is slipped over the Earth and is either tangent along a single line or secant along two parallel lines around the entire planet. After the projection is completed, the cone is removed and slit up one side. Conical projections reduce distortion closest to the tangent or secant lines, with the distortion increasing as one moves away from these areas.

Conical projections can be equatorial (normal) - meaning the cone is tangent at the equator; transverse - the cone is tangent along a meridian; or oblique - the cone is tangent along another path.

Examples of conic projections include Lambert Conformal Conic, Albers Equal Area Conic, and Equidistant Conic projections 

Figure 2.25: Conical Projection Examples

Cylindrical Projections

Cylindrical projections use a cylinder slipped over the Earth with either a single tangential line or two secant lines. After the map elements are projected onto the cylinder, it is slit and rolled flat. Cylindrical projections are probably the most common, for they result in a rectangular map that does not have distortion like an azimuthal projection. Cylindrical projections, like conical have the least amount of distortion near the tangent or secant lines, then the distortion increases one moves away.

Like conical, cylindrical has the three main kinds projections: equatorial (normal), transverse, oblique. Examples of cylindrical projections include Mercator, Transverse Mercator, Oblique Mercator, Plate Carr, Miller Cylindrical, Cylindrical equal-area, Gall Peters, Hobo Dyer, Behrmann, and Lambert Cylindrical Equal-Area projections.

Figure 2.26: Examples of Cylindrical Projections
Equatoriall (normal) projections are tangent along the Equator, with the cylinder parallel to the poles. Mercator (discussed in the West Wing video) is one of the most popular ways to see this projection.
transverse-mercato01

Transverse projections are tangent along a meridian (most often along the Prime Meridian, but it's not mandatory). Transverse Mercator projections are popular; Universal Transverse Mercator (UTM) is a transverse Mercator projection superimposed with a grid for navigation purposes. UTM is just as popular as latitude and longitude when it comes to the settings on your GPS unit.

Oblique projections are a cylindrical projection along any line that is not the equator or a meridian. Oblique cylindrical projections are used to reduce distortion locally, not just at the Equator or Prime Meridian. 

Azimuthal (Planar) Projections

Azimuthal Planar Projection projections, also known as planar projections, are projections where a rectangular developable surface is tangent at a single point or secant along a path (the developable surface slices through part of the globe) and map elements are projected from a single light source. There are six common locations for the light source, each one used to reduce distortion in a different way (see table).

Azimuthal projections can be normal - which in an azimuthal projection, the developable surface is tangent at either the North or South pole; transverse - the tangential point is somewhere along the equator; or oblique - the tangential point is anywhere else (the projection you made was oblique because the Washington Monument is located neither at a pole or along the equator)

Examples of azimuthal projections include: Azimuthal Equidistant, Lambert Azimuthal Equal-Area, Gnomonic, Stereographic, and Orthographic projections.

Figure 2.27: Example of Light Paths to Create Azimuthal Projections

Orthographic Stereo GnomonicGnomonic
Vertical perspective,near-sideVertical perspective, far-side

Oblique perspective (non-azimuthal)Stereo

2.7.3: Projection Methods to Reduce Specific Distortion

Certain projections are designed to reduce specific error throughout a map. Starting with the developable surfaces, we will look at five of the more common methods used to reduce specific distortion: equal area, conformal, equidistant, true direction, and compromise.

Equal Area

The goal of equal area maps, as the name suggests, is to create a map where each of the land masses represented is given an equal amount of area. Equal area projections are useful where relative size and area accuracy of map features is important (such as displaying countries / continents in world maps), as well as for showing spatial distributions and general thematic mapping such as population, soil and geological maps.

In the image to the right, the map is covered by a series of orange ellipses, called distortion ellipses. Distortion ellipses, known as Tissot’s indicatrix, start a circles placed on the globe. As the projection is created, the distortion ellipses distort in a manner equal to the map’s distortion at the place upon which they are centered. This method allows for a user to visualize the map’s distortion without any measuring equipment. Distortion ellipses do not show up on the final map; they are for visualization purposes only. When we examine the distortion of the ellipses, the shape is distorted, but the area remains constant throughout. This tells us that area is the factor being preserved.

Figure 2.28: Gall-Peters cylindrical equal-area projection

Conformal

Conformal maps serve the purpose of preserving shape, distance, and bearing, at the expense of area and scale. Just like in the West Wing clip and the BuzzFeed “Maps Lie”, it is explained continents away from the Equator are larger in size. When we understand that it is impossible to preserve all six characteristics and conformal maps, such as the Mercator projection, aim to preserve shape and distance, we then understand that Mercator had no intentions of “lying” to anyone, nor did he want to create social inequality. He just wanted a quality map to navigate with. Preservation of shape, distance, and bearing makes conformal map projections suitable for navigation charts, weather maps, topographic mapping, and large scale surveying.

In the image, we see the distortion ellipses as circles. This tells us the shape is preserved, but area is distorted away from the Equator. Looking at this image, is the Mercator projection a tangent or secant developable surface?

Figure 2.29: Mercator - conformal projection

Equidistant

Equidistant, similar but different then conformal projections, aim to preserve distance, but only from the tangential line or lines. This means that when you use an equidistant map, the measured distance from the place where the developable surface came into contact with the globe will be correct, but distances measured between other points will be incorrect. Equidistant projections are used in air and sea navigation charts, as well as radio and seismic mapping. They are also used in atlases and thematic mapping.

In the image, we see the ellipses as circles which are not distorted in shape or size at the equator, but become increasingly so as you move away. When compared to the conformal example, we see the continents becoming distorted in shape, but the distortion ellipses do not stop far from the north and south edges.

Figure 2.30: Equirectangular (equidistant cylindrical) projection

True Direction

Similar to the equidistant projection, which starts with a cylindrical developable surface, true direction starts with an azimuthal developable surface. Much like your oblique gnomonic projection, all directions and bearing away from the Washington Monument are preserved, but if you were to measure between Los Angeles and New York, the measurement will be incorrect. True-direction projections are used in applications where maintaining directional relationships are important, such as aeronautical and sea navigation charts.

Compromise

Compromise projections attempt to balance all of the distortions in one map. This means that none of the six are “perfect”, but each one is is balance with the others, the idea being that no one place is grossly distorted in comparison to any other place on the map. Compromise maps are used to preserve the look of the finished product, a wall or book map, for example. Two common types of compromise maps are the Robinson and Winkel Tripel projection (both of which we will look at in lab).

In the image, which is a Robinson projection, we see none of the ellipses are terribly distorted in size, shape, or distance from each other. But because they are all distorted in all six ways, this map wouldn’t be perfect for navigation, nor preserving area for measurements, nor comparing the shapes to a globe.

Figure 2.31: Robinson Projection

2.7.4: Projection Methods to Projected Coordinate Systems

Up to this point, we've been examining projection methods and not projected coordinate systems.  Just like when we learned geographic coordinate systems, there were steps we took to move from the geoid, to a reference ellipsoid, to a geographic grid, to a datum, and finally to a geographic coordinate system. Projection methods stop being just methods and start to be projected coordinate systems after two more steps have been taken: 1. a linear unit of measure is decided, and 2. a coordinate system origin is established.  All coordinate systems have a single origin point, a point usually labeled zero, zero, however, some systems, like UTM, use a different origin value to prevent negative numbers.  No matter if the origin is labeled zero, zero or something else, the action is the same where each of the intersections along the X and Y axis are labeled, counting up or down from the origin values.

Utilizing a variety of origin points, which come from moving around one secant line or splitting the difference between to tangential lines, allows for the distortion to be the least along either the X or Y axis, depending on the aspect of the projection method.  This movement, in addition to a large selection of reference ellipsoids allows for the maps we create to have the highest accuracy and precision paired with the least amount of distortion while still using linear units.

As stated above, the second factor that a projected coordinate system needs that takes it from a projection method to establish the linear units it's going to use, most commonly, feet, meters, and international feet.  When a geographic grid's intersections are labeled via measuring from either the equator or principal meridian, angles are easier to use since we are dealing with a sphere which later goes through a affine transformation to become an ellipsoid.  When we are making measurements on a flat map - a projection - we then switch to linear units of measure, which are much more convenient and make more sense to us.

To review, a projection method includes selecting a developable surface, such as a cone, cylinder, or flat plane (azimuthal), and an aspect, selecting which direction to turn the Earth within the given developable surface.  It becomes a projected coordinate system when a linear unit and a single secant line or pair of tangential lines have been selected prior to transferring the land and ocean masses to the develoapable  surface, which is then cut and unrolled into a flat map.  The selection of the placement of the secant or tangential lines define the location of the system's origin, as these lines will have the least amount of distortion.  Selecting and moving the secant or tangential lines is made easier with software like ArcGIS, since it allows you to simple type in which line(s) you'd like to use and it makes all of the adjustments necessary.

As you will learn throughout the semester, there are several words in GIS which are used incorrectly or may have a double meaning. This is in no way done to confuse you (really), but it’s just a language that has been used by lots of folks, most of whom came from other sciences.  As GIS is a fairly young science, most of it, including the terminology, comes from other places.  This, in addition to a whole lot of "DIY GIS", has led to a fair number of words which are used either incorrectly, partially correct, or that have two, and sometimes three, meanings.  The term "projection" is one of those words which, in addition to having a couple of meanings (which, luckily, are mostly correct and mostly similar), is used as a rather "all encompassing" term to mean "geographic coordinate system", "projected coordinate system", "projection method", and "convert data from one coordinate system to another".

Technically, the word “projection” or “to project” is in reference to the action of creating a projection and “projected coordinate system (PCS)” is the result of the action, after a linear unit and system origin has been set.  For example, "Lambert Conformal Conic" is   However, you’ll almost always here a projected coordinate system (and geographic coordinate systems, for that fact) referred to as a “projection”, simply because it’s shorter and easier. This text attempts to use the correct terms in the correct places while you’re still learning the process, but later will switch to using “projection” to mean both the action and the product - but by then, you’ll be able to note the difference in context.

We noted earlier that the word projection refers to the method used to create a the first part of a projected coordinate system. This idea will come up again when we begin to explore coordinate systems (both geographic and projected) in the GIS software. For every projected coordinate system, it will have a very specific name that represents where is should be used and the method by which it was created, as well as additional information about the method with which it was created - including the name. That is to say, the name of the PCS is the most specific thing, consisting of where you should use that particular PCS and the method it was created, and not exactly to be redundant, as there are always cases which vary from the norm, you will find the projection method listed, which is slightly less specific. For example, you might see  USA Contiguous Lambert Conformal Conic  as the specific name of a projected coordinate system and Lambert Conformal Conic as the projection method.  The name Lamber Conformal Conic, as we learned earlier, means the projection method used a cone as the developable surface and it is meant to be used with projected coordinate systems which need to preserve shape (conformal) within the contiguous United States.

2.7.5: Practical Projections

So far in this Chapter, we have looked at the idea of geodesy and measuring the Earth, representing the Earth’s odd shape as a geoid and a mathematical ellipsoid, creating measurable locations on the Earth’s surface by combining reference ellipsoids and datums to create geographic coordinate systems, how to use the idea of projections to convert a round Earth to a flat map, and lastly, how different projections are designed to minimize different distortions. Whew! That’s a lot of information - but we are not done quite yet. A solid understanding of how GCS’s and projections are created is essential to understanding how we use them in GIS, so we are going to finish off this chapter with a bit of practical projection use.

Map Scale


Before we look at how to choose a projection, lets first look at the idea of map scale. Map scale is simply the mathematical ratio of the distance between two points represented on a map and the same two points in the real world expressed in one of two ways:

  1. A Statement of Equivalency or Verbal Scale
  2. A Representative Fraction
  3. Scale Bars

Statement of Equivalency or Verbal Scale


statement of equivalency (also known as verbal scale) is when the relative scale is expressly defined on a map: 1 cm = 1 kilometer; 1 inch = 10 miles, 5 nanometers = 12 kilometers (even though that last one would be a bit silly to use).  For every measured unit on the map, one would travel the same distance in the real world.  For the map where one inch equals ten miles, if you were to use a wooden ruler and measure an inch on the map, you can safely assume that the distance between those two points in the real world would be 10 miles.  

The catch with a statement of equivalency is, however, that the statement is only good for those units.  If you were to measure the same map as above in centimeters, you could not assume that the distance between one centimeter is going to be ten miles.  You could, using unit conversion, assume that 2.54 centimeters on the map would equal ten miles, however, but that means you'd be doing a whole lot of math and conversions.  This is why, when you don't know what units your map's audience best thinks in, is to use representative fractions instead.

Representative Fractions


Similar to statements of equivalency, a representative fraction is unitless.  Presented as a ratio such as 1:24,000; 1:100,000; 1:1,000,000; and so on, one unit on the map equals the stated units in the real world.  In other words, a map that has a scale of 1:24,000 means that one inch equals 24,000 inches in the real world OR 1 centimeter equals 24,000 centimeters in the real world.  With representative fractions, you don't need to assume what units your audience thinks best in, but you also assume they know how many inches are in a mile, since the representative fractions only work with a single unit at a time.  

Scale Bars


Lastly, with maps, we often see scale bars drawn on the bottom.  Like a statement of equivalency, the scale bar will be shown in a set unit of measure, and not a representative fraction.  Unlike statement of equivalencies, however, you could potentially use a ruler of a different unit scale to measure the map.  If the map scale bar is shown in inches and miles, you could use a centimeter ruler to find the distance in miles.  You're not exactly limited by the unit of measure of the ruler, but since the scale bar is set up for one kind of units, you can't convert the units back and forth, but you could successfully use a different kind of ruler.

  • One note: When it comes to reproducing maps, the only one of these three which stays constant is a scale bar.  Statement of equivalencies and representative fractions are only true for the size of map the originally refereed to.  If a map is copied at a smaller or larger size, the two ratios will be incorrect while the scale bar remains correct.

Large and Small Scale


In GIS and cartography, we often use the terms large scale and small scale, which are equally as often mixed up. When we look at the representative fraction scale of a map and see a large number in the denominator, we tend to incorrectly conclude the map is the large scale, when in fact a large denominator leads to a small number (a number a long distance from one on the number line)- thus a small scale map. Large scale maps have a small denominator where the number is closer to one on the number line.

The confusion is clear. When you see a map with a representative scale fraction with a large denominator and a larger geographic extent, your first inclination is to assume the map has a large scale. If we keep looking back on the representative fraction, we see that fractions with a smaller denominator are closer to one on the number line (and a scale of 1:1 is the real world) and fractions with larger denominators are further away from the representative size being the same as the real world. To keep it straight, think that smaller denominators lead to larger features (more like the real world) and larger features lead to large scale maps.

Figure 2.32: Small and Large Scale Maps

Projection Selection

One of the most common questions asked by introduction to GIS students after learning about projections is: "What projection is best for my project"?  This is one of those questions which has the unfortunate answer of "Well, that depends".  When you look at how projections are created and what distortions come into play with their use, you will start to realize that being able to select the "proper" projection is just an understanding of the gains and setbacks of each kind of projection. While there are no real “rules” on projection selection, there are a few guidelines to help you choose:

  1. What distortion can you give up in favor of the one(s) you really need?
    • All projected coordinate systems come with some distortion, as we've learned there are no projections which preserve all kinds at one time.  The biggest decision of which projection to use for a project is that of distortion.  If you have a project which deals with solving the the area which has changed, selecting a projection which preserves shape will produce errors in the solved measurements.  On the other hand, when you go to present your work at the end of the project, keeping the projection which preserved area and distorted shape will look quite odd to your reader.
  2. How big is your project area?
    • Now that we understand what scale is in regards to GIS and cartography, we can explore projections in these same terms. As we learned with reference ellipsoids, better accuracy and precision come with better localization, and the same is held true with projections. Selecting a localized projection will decrease distortion close to the tangential point, and that distortion will increase as you move away, so choosing a projection designed for use in Florida and applying it to Oregon is most likely a bad idea. And when you are completing a project which covers the entire United States, choosing a projection designed for either Oregon or Florida (or the whole Earth, for that fact) is also a bad idea.
  3. What unit of measure (feet, inches, degrees...) would you like your map to be measured in?
    • We know that geographic coordinate systems consist of a datum, a reference ellipsoid, and an angular unit of measure (such as degree with the latitude/longitude system) - which means that spatial objects collected in a GCS will be measured in that unit of measure. For example, if you create a GIS layer of roads, and the chosen GCS is WGS84, the length of the roads will be measured in degrees. When we go through the process of creating a projection, moving from the round (measured in angles) to the flat, we can move from an angular unit of measure to a linear one, such as feet and meters. When you move the roads layer you created in WGS84, and move it, or project it, from the GCS WGS84 to a flat map (say WGS84 California (Teale) Albers), you are now capable of measuring in feet, yards, and miles.
  4. What is the end goal of your project?
    • While we will spend quite a bit of time discussing this in Chapter Nine, for now, we can just look at the end goal as how is it the map should look? Will it be displayed on a wall and you need to preserve shape and size or used for navigation where distance and bearing are important. Choosing the correct projection leads to the proper end product.

Examples Of Projected Coordinate Systems

Up to this point, we've spent quite a lot of time examining the process of creating projected coordinate systems: the projection methods, the distortions that are a result of the process, kinds of developable surfaces, and kinds of aspects.  To be a quality GIS technician, it's important to understand the process of creating these coordinate systems which we use every day.  With this understanding comes the ability to make better decisions about maps, products, and solutions.  Within the GIS software, you will find all of the information about how a particular projection was created, which will help you understand it's strength and weaknesses.

In the next section, will look at four common projected coordinate systems and examine how they were made, which - as stated above - will let us in on the benefits of selecting that projection for a specific project.

Normal Mercator and Transverse Mercator

Normal Mercator

The Normal Mercator projection is one of the most common, and most controversial, projections out there. Just like it was stated in the West Wing clip we watched, the Mercator projection is a conformal cylindrical projection, tangent at the Equator, and designed to preserve shape, distance, and bearing. Adjacent to the tangential line, distortion is minimal, but increases rapidly as one moves north or south. The Mercator projection (and it’s cousin, Web Mercator) is used for large areas (small scale) nearest the mid-latitudes.

Transverse Mercator

The Transverse Mercator projections, like we learned in the previous section, turn the tangential line from a parallel to a meridian, which reduces distortion along a North-South line, with that distortion increasing moving east or west.

Figure 2.33: Normal and Transverse Mercator Projected Coordinate Systems
Normal Mercator Projected Coordinate SystemTransverse Mercator Projected Coordinate System

Lambert Conformal Conic

Looking at the name, we immediately know this projection is designed to preserve shape, distance, and bearing (conformal) and is created from a conic developable surface (and was designed by a guy named Lambert). Designed by Johann Heinrich Lambert in 1772, the Lambert Conformal Conic projection is used to map large areas (small scale), especially in the mid-latitudes, where the secant lines are placed at roughly 20°N and 60°S.

Figure 2.34: Lambert Conformal Conic Projected Coordinate System

Universal Transverse Mercator (UTM)

Universal Transverse Mercator is a technically a planar coordinate system - a specially designed method for finding a location on a two-dimensional map, which is based on a projection. Like with the Lambert Conformal Conic projection, we can look to the name - Universal Transverse Mercator - to see the projection is based on the transverse Mercator projection. The “universal” part comes in by altering the transverse methods slightly instead of being tangent at one line and projecting the entire globe, a smaller projection is made by placing the cylinder secant at two lines 6° apart, creating a strip or “zone”. The secant lines are then rotated, and another strip is taken (360° / 6° = 60 zones). By stitching together all 60 zones together, a complete world projection is made with distortion reduced in every zone (vs at a only single meridian). Like the Transverse Mercator projection, however, the distortion is too extreme at the poles (the sweet spots are between 84°N and 80°S) and UTM is not used.

Figure 2.35: Universal Transverse Mercator Projected Coordinate System

Since UTM is a planar coordinate system, it utilizes a Cartesian Coordinate system affixed to an origin and labels XY intersections by measuring the linear distance from that origin. Yet, the UTM system delivers the geographic grid in a unique way - with 60 different origins, one for each zone.

Using the equator as the X axis, each UTM zone is bisected right down the middle creating a central meridian, with 3° on the east and 3° on the west (and the Y axis). North of the equator is the North Zone, denoted with the number of the zone and an N, such as Zone 18N, and an S for south, such as Zone 18S. If the origin was labeled 0,0 where the equator and central meridian intersect in each zone, that means there would have to be negative numbers - those west of the central meridian and south of the equator. To combat the negative numbers, the 0,0 was arbitrarily changed to 500,000 and 10,000,000, meaning coordinates west of the central meridian start counting down from 499,999 meters and south of the equator count down from 9,999,999 meters. Since UTM measurements come from a projection and not a GCS, the terms false easting and false northing are used in place of North and East, like we see with Latitude and Longitude.

Whew! That was a lot of information in two paragraphs. Let’s look at some examples to help understand what we just read.

Figure 2.36: The UTM System Explained
 
 
First, we place the cylinder secant along two lines 6° apart to make a zone.Since our goal is to reduce distortion and create a flat map with planar coordinates, the two curved sections are overlapped with the neighboring zone to make a rectangular strip.

 
The North and South halves of each zone contains a false and a true origin, with the false origin located at 0 mE, 0 mN, as expected, and the false origin located where the central meridian and equator intersect. The North Zone true origin is located at 5000,000 mE, 0 mN while the South Zone true origin is located at 500,000 mE, 10,000,000 mN. The arbitrary numbers assigned to the false origins prevents the use of negative numbers for UTM coordinates. 

Since each zone is labeled the same way in reference to the UTM coordinates, each unique coordinate pair occurs 120 times, thus making the addition of the zone number and North or South designation imperative for the coordinate label. For example, 420235 mE, 3868323 mN occurs 120 times, but UTM 11N 420235 mE, 3868323 mN occurs only once.

State Plane Coordinate System (SPCS)

The UTM planar coordinate system is a great example of a “global” system designed to minimize distortion and locate positions over a large area using series of multiple, smaller projections stitched together. An excellent example of a “local” coordinate system with the same purpose is the State Plane Coordinate System or SPCS.

SPCS breaks the United States into sections by state, and in many cases, breaks states into zones, all with the goal of reducing distortion and labeling locations. For east-west trending states, such as Tennessee and Colorado, utilize a Lambert Conformal Conic projection, while north-south trending states such as Illinois and Mississippi, use a Transverse Mercator projection (The panhandle of Alaska didn’t fall into either category, thus an Oblique Mercator was used). Thinking about what we’ve learned so far, why were the Lambert Conformal Conic and Transverse Mercator projections used?

Figure 2.37: State Plane 
 
The State Plane Coordinate System (SPCS) with zone numbers

In addition to each state being projected independently to reduce distortion, most of the states were actually broken down further, such as Colorado which contains three zones - Colorado North (0501), Colorado Central (0502), and Colorado South (0503). This means the distortion for each section is about as small as it can be. One thing to note is while distortion is reduced greatly for each zone, the inter-zone distortion can be greater then desired, especially when two neighboring states are created from two base projections.

Like UTM, SPCS is a not a projection, but rather a planar coordinate system, utilizing a false origin originally located in Meades Ranch, Kansas (when the system was based on the North American Datum 1927 NAD27), and the idea of counting coordinates with a false easting and a false northing. With an effort to convert angular units of measure (degrees) into a Cartesian Coordinate System for ease of navigation, the SPCS was developed in 1930. Since then, the system has been revised a few times, most notably in 1983 after NAD83 was completed, and more recently, when the origin was moved to the center of the Earth to improve calculations when using the system with GPS units. SPCS can be found in feet, meters, and international feet.

Also like UTM, the idea of using a technique of projecting small sections and merging them together into a large network can be tough for a new GIS student, since the answer “What projection is your data in?” regarding data near Denver would be “SPCS Colorado Central (0502)”. SPCS is often used as a local projection (since each zone is based on a projection), even though it is technically a planar coordinate system. Think about it this way: if you are working in Oregon and would like the data to be as exact as possible, you know that if you use Oregon North, the cone used in the projection was tangent to a line running through the center of the zone, vs a cone which was tangent thought the center of the United States for a continent-based projection. When the projection is designed for a smaller area, distortion is reduced, larger projections have the least amount of distortion along the tangential line, and increasing distortion from there. If your entire area is only a few hundred miles across, the distortion is negligible, and when the area is thousands of miles across, the distortion is noticeable.