Section Seven: Projection Methods

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 the selected units for measuring angles.  Choices include degree and radians.  to label intersections of north-south and east-west lines (parallels and meridians) starting at a principal meridian the north-south line from which the labeling begins.  East-west lines have a very obvious start point: the equator.  North-south lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. , examining the latitude also known as 'parallels' the east-west portion of a geographic grid measured with angles between 0 and 90° /longitude system as an example. We then learned that combining a specific datum with a geographic grid the result of using an established angular unit of measure to label the intersections of north-south and east-west lines on the surface of the Earth starting the labels at a principal meridian the north-south line from which the labeling begins.  East-west lines have a very obvious start point: the equator.  North-south lines must start somewhere, so when it is established for a particular geographic grid, it can be considered the principal meridian. 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 the science of measuring and monitoring the size and shape of the Earth and the location of points on its surface 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 measurement of plane angle, representing 1⁄360 of a full rotation (circle). In full, a degree of arc or arc degree. Usually denoted by ° 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 measurement of plane angle, representing 1⁄360 of a full rotation (circle). In full, a degree of arc or arc degree. Usually denoted by ° latitude also known as 'parallels' the east-west portion of a geographic grid measured with angles between 0 and 90° 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 projected coordinate system (a flat map from a round globe) is amore complicated then simply peeling off the crust and smashing it flat, and results is a side effect called map distortion In GIS, the unavoidable inaccuracies which occur when transferring features from a geographic coordinate system to a developable surface.  Comes in six flavors: Shape: the shape of the geographic feature vs. the shape drawn on the map Area: the measured area of a world feature Distance: the measured distance between two world features Direction: the cardinal direction between two world features, minus distance information Bearing: the cardinal direction measuring from one world feature to any other Scale: comparing the size of two world features vs. the same two drawn on a map , 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 technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected 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 technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected 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 technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected method which reduces or preserves all kinds of distortion.

Distortion is a by-product of the projection technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected 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 technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected methods are designed to keep the shapes of the represented areas (specifically “ conformal serve the purpose of preserving shape, distance, and bearing, at the expense of area and scale ”), 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 student once put it, "Map distortion from the projection technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected 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 a projection technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected , 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 (similar to what you see when you're standing on the bridge of the Mapparium (Figure 2.21b). 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 a geometric shape which will not be distorted when flattened.  Used as the base shape to transfer features during projections.  Most often a cone, cylinder, or plane (azimuthal) (when the developable surface a geometric shape which will not be distorted when flattened.  Used as the base shape to transfer features during projections.  Most often a cone, cylinder, or plane (azimuthal) 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 between the globe's surface and the cardboard increases as you move& away from Washington DC. 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
A: LGoS vs North and South America: Technically An oblique gnomonic projection technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected	B: The Mapparium in the Mary Baker Eddy Library in Boston lets you experience the Earth’s geography without distortion as you look from the inside out.

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. (Figure 2.21a) 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 creation method (Figure 2.22)

Congratulations, you’ve made your first projection technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected .

FIGURE 2.22: LGoS vs North and South America
The result of your projection technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected . Oblique gnomonic projections distort the map in an increasing manner away from the tangential point

2.7.1.1: Geographic Coordinate Systems - The Beginning of Projections

This whole section is all about creating projected coordinate systems which have the advantage of flat maps and linear units, and like the sections on Geographic Coordinate Systems, it will seen like we are building "the" Projected Coordinate System or "the three" Projected Coordinate Systems- which isn't true. There are thousands of coordinate systems. In fact, at the time of writing - 2022 - There are currently 6,709 coordinate systems (795 Geographic and 5,914 Projected) and supported by Esri software - and that not even counting the 882 depreciated coordinate systems and 286 active/depreciated coordinate systems which don't fall into the Geographic or Projected Coordinate System categories (vertical and engineering) which are listed in the EPSG database electronic storage container with a top-down structure in which the items contained are related to each other and that relationship allows for the data to be quickly and efficiently queried and retrieved for use. ! Below, you can see a list of all the Esri supported Geographic and Projected Coordinate Systems.

The process of creating a projected coordinate system all begins with some Geographic Coordinate System - which one? - The one that is appropriate to create the Projected Coordinate System with the least amount of distortion for the area being mapped. This is because every Projected Coordinate System is a projection technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected of a Geographic Coordinate System. Earlier, in the LGoS example, you were standing inside the Mapparium and shooting the laser through the glass wall to the azimuthal plane (the cardboard stabbed on the Washington Monument). In that case, the GCS used to create the PCS was the Mapparium. The Mapparium was the 3D Geographic Coordinate System from which the land masses were being projected on to the azimuthal plane to create the Projected Coordinate System. When a Projected Coordinate System is being built, it starts with a selected Geographic Coordinate System from which to make the projection technically: the result of using one of variety of methods to transfer the geographic locations of features from a geographic coordinate system to a developable surface everyday use: any coordinate system, geographic or projected on to the developable surface a geometric shape which will not be distorted when flattened.  Used as the base shape to transfer features during projections.  Most often a cone, cylinder, or plane (azimuthal) . This is a very important piece of information to keep in mind for the remainder of this section. Any time an ellipsoid is mentioned, that ellipsoid is coming from the Geographic Coordinate System. It's not just a random ellipsoid that was selected to make a Projected Coordinate System.

The Main Point
All Projections are based on a selected Geographic Coordinate System. The selection for the GCS is made based on the area the PCS will be created. For example, the Projected Coordinate System specifically named "USA Contiguous Lambert Conformal Conic" is based on the Geographic Coordinate System specifically named "GCS North American Datum 1983"

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