Projections

There are many systems for projecting a spherical surface or portion of it onto a flat plane. An ideal projection would accurately represent all the spatial characteristics of the projected features (shapes, areas, relative positions and distances) using a consistent scale, but no such projection system exists. Actual projection systems are based on various trade-offs between accuracies of position, shape and size.  There are several categories of projection system:

• Equidistant projections preserve correct relative distances of all features from the center of the map.
• Equal-area projections preserve correct areas.
• Conformal projections preserve correct shape.
• Azimuthal projections preserve correct direction or orientation of all features relative to the center of the map.

• A cylindrical projection would result from wrapping a cylinder of paper tangent to the equator, and recording the projection from a filament between the poles.  The polar points project as lines at the top and bottom of the map, and meridian (longitude)  lines which actually converge at the poles would appear parallel.
• A conic projection would result from placing a paper cone over one hemisphere, tangent to some latitude, and recording the projection from a point light source at the center of the globe.  Parallels (latutude lines) are shown as arcs radiating from the pole; meridian  lines are shown as straight lines converging at the pole.
• An azimuthal projection results from placing a flat piece of paper tangent to a single point on the globe.  An azimuthal projection may be lit from the center of the globe (a "gnomonic" projection) or from an infinite distance through the other side of the globe opposite the tangency point (an "orthographic" projection).  All straight lines drawn on a gnomonic projection will correspond to great circles (sections through the center of the Earth).

Coordinate Systems

While most coordinate systems are flat grids, the lat-long coordinate system references a global surface directly, and is not (strictly speaking) a projection at all.  Its origin is the intersection of the prime meridian (0 degrees longitude) passing through Greenwich, England with the equator (0 degrees latitude). The Earth's 360 meridian lines are about 69 miles apart at the equator and converge at the poles. The 180 parallels are equidistant, with about 69 miles separating each degree. A cylindrical projection with parallel meridians such as the lat-long "plate carre" map below will impart a horizontal stretch to map features near the poles; this is why Greenland looks bigger than Australia even though it's actually much smaller.

All coordinate systems except lat-long are based on specific projections.  The universal transverse mercator (or UTM) system divides the earth by meridian into 60 6-degree grid zones. Coordinates are expressed in meters. Each zone's nominal origin is its westernmost point on the equator. Coordinates are referenced in meters to the zone's midpoint on the equator, which is assigned the coordinates (500000,0)

For example, Zone 18 (below) extends from 72 degrees west to 78 degrees west, and covers much of the Northeast US including Delaware. The center of the zone is 75 degrees west longitude line which lies just east of Cape Henlopen. The map below includes a portion of the UTM grid system in black. Note that these grid lines diverge significantly from true north (indicated by the bounding longitude lines at 72 and 78 degrees west).

Delaware uses two different state plane coordinate systems.  The older system is in feet, based on the 1927 North American Datum (NAD).  The newer system is in meters,  based on the 1983 NAD.   Older 1:24,000 USGS quad sheets for Delaware use the 1927 datum, the newer ones use the 1983 datum, and include alternative reference ticks in the lower left corner for the older system.

A datum defines a reference surface for a portion of the Earth, or perhaps the whole Earth. Why the different datums? Map projection problems are complicated by the fact that the Earth is not a perfect sphere at all: its poles are somewhat flattened, its equator is somewhat bulged, and it has a slight pear shape. As surveying techniques have improved, geodesists (people who study the shape of the earth) have refined their models of the Earth's surface. Each datum is based on a spheroid ("ellipsoid"), which is a mathematical representation of a curved surface on which x,y,z surface reference coordinates and a center-of-the-earth reference point are defined. Like datums, some spheroids are local, or defined for specific regions of the Earth; others are global.

NAD 1927 is based on the 1866 Clarke spheroid, while NAD 1983 is based on the 1980 Geodetic Reference System spheroid.  In Delaware, a point referenced in NAD83 will be about 208 meters north and 1 meter east of the same point referenced in NAD27.

The WGS datums of 1972 and 1984, widely used by the Department of Defense, are based on the World Geodetic System 1972 and 1984 spheroids, respectively.

Coordinate Systems in ArcMap

You should make it a habit to specify an appropriate coordinate system in the data frame's Properties menu at the start of any project, and to check that the coordinate systems of the various map layers you are using are correctly specified in the layer Properties "Source" tab.   Once the data frame's coordinate system is specified, the program is generally pretty smart about reconciling geodata specified in other coordinate systems in your map. But problems often arise when you fail to specify the coordinate system of the data frame, and/or load data referenced in different coordinate systems that are not specified.

Some of the geodata layers that you download for your class projects will not have their coordinate systems specified, so you will have to specify these with Arc Toolbox's Data Management--Projections--Define Projection tool, or by editing the layer's Properties in Arc Catalog. Since most of your work is with Delaware data, you should learn to recognize local coordinates in the most commonly-used systems: lat-lon, UTM and State Plane. Here are examples of typical northing-easting coordinates for the Newark area:

• Geographic--Spheroid-Based--GRS 1980 (lat-lon): 39.79452, -75.76523 --specified in decimal degrees referenced north from the equator and east (negative means west) from the Greenwich (England) meridian.
• Projected--UTM--NAD 1983--Zone 18N: 4392195.83, 433440.61 --specified in meters, referenced north from the equator and east or west of the center meridian which is specified as 500,000 meters; as shown in the image above, the center meridian for UTM zone 18 is 75 degrees west. UTM coordinates for Delaware mostly begin with 4's, northings are in millions, eastings are in hundreds of thousands.
• Projected--State Plane--NAD 1983 (meters)--Delaware: 184146.35, 170155.49 --specified in meters, referenced north and east from an origin located in the northern end of Hamilton County, Virginia. Within Delaware, DE State Plane (NAD83 meters) easting coordinates range from 160000 to 235000 meters; northing coordinates range from 50000 to 204000 meters.

If you need to specify the coordinate system for a geodata layer, always specify the actual system the coordinates are in. Students sometimes think they can re-project a geodata layer simply by specifying a different coordinate system for it. The layer winds up in La-La-Land and the student winds up in hell.

To convert an unprojected (lat-lon) or differently-projected layer to the coordinate system specified for the data frame, you can use Data--Export Data to create a copy that will be in the coordinate system of the data frame.  When reprojecting geodata, Arc uses its library of transformations to translate the coordinates of each feature in the source data into the corresponding coordinates for the alternative projection.  ArcToolbox's Data Management tools include formal vector feature and raster Project utilities. 

Use Toolbox's Data Management--Projections utilities to specify a layer's coordinate system or reproject geodata from one coordinate system to another. You can use Arc Catalog to define the layer's coordinate system in the Properties window.

You may encounter situations where you are tempted to create a custom coordinate systems, specifying the projection center, false easting and northing values (to shift the nominal origin), etc. Do not succumb to this temptation! Data generated in a customized projection may be impossible to reconcile with data in a standard projection.  For example, if you were analyzing an area that spans UTM zones 17 and 18, so you might concoct a UTM zone 17.5.  You would be better advised to use some other standard coordinate system.

Later on in this course you will be expected to geo-reference some raw data, such as a scan of a map or a satellite image. Raw satellite images typically have lat-lon header information that the GIS may use to position them approximately, but you can use the Georeferencing toolbar to get them positioned much more accurately.  (You may have to use Arc Catalog to clear the raw layer's coordinate system specification before you can georeference it. The Georeferending process involves using the mouse to define links between points on the raw layer and one or more geo-referenced layers. Once you get a series of links that fit the raw layer to the geo-referenced layers with minimal error, you can create a "rectified" version of the raw layer. The Georeferencing utility uses the links to calculate a transformation equation.  Reprojecting an image or raster involves resampling: the program creates a blank raster in the target projection and populates its cells with values extracted from corresponding cells in the source raster, as identified by the transformation equation.

Raster cells are supposed to have uniform ground size. Be aware that cells in lat-lon rasters don't. 

Format conversions for unprojected (lat-lon) data are pretty straightforward.  Unprojected data coordinates are either decimal degrees dd.dddddd, or degrees and decimal minutes (an idiotic hybrid!) dd mm.mmmm, or the traditional degrees, minutes and (decimal) seconds: dd mm ss.ss. Since there are 60 minutes per degree and 60 seconds per minute, the conversion from degrees-minutes-seconds to decimal degrees is: dd.dddddd = dd + mm/60 + ss.ss/3600, and the conversion from decimal degrees D to degrees-minutes-seconds is: dd  mm  ss.ss = int(D)  int((D - int(D))*60)  (D - int(D) - int((D - int(D))*60))*60

"Proper" maps

This is a GIS class, not a computer cartography class, so we generally don't bother with all the cartographic niceties.  But a "proper" map should have...

• a title and supporting text annotations that clearly explain what it is all about,
• a legend that explains the symbology used. 
• a scale, either a horizontal bar representing specified distance units, or a scaling ratio. The 1:24,000 scale ratio used in many USGS quad maps means that one inch on the map equals 2,000 feet (=24,000 inches).  As noted above, however, projections imply scale inconsistencies, which are trivial for maps of small areas where projection from sphere to plane involves minimal distortions, but possibly significant in maps of large areas.  Scale bars ae generally preferable to scale ratios if there is any chance someone will resize (e.g., photocopy) your map. On a resized map the scale bar will still be correct but the scale ratio will be wrong.
  
• a north arrow.  This is a convention for maps oriented so north is at the top of the page, but essential for maps where north is not the top of the page.  To rotate a north arrow in ArcMap's layout, change its Calibration Angle in the Properties menu.  Planar projections imply a north angle that varies slightly from one side of the map to the other.  For example, USGS 7.5-minute quad maps in UTM projection are actually trapezoids (see the UTM Zone18 map above): the top and bottom are parallel, but the sides converge noticeably toward the north.  The same is true of the UTM zone 18 map shows above.   Keep in mind that true north and south refer to the Earth's rotational axes. The Earth's magnetic north is currently located about 73^oN,100^oW beneath Prince of Wales Island in northern Canada, so that as you move east or west from 100^oW the declination angle correcting your compass reading to true north will increase. (Magnetic north keeps shifting rapidly enough that declination angles may change significantly over just a few years; the Earth has actually reversed its magnetic polarity many times in the past billion years. True north moves much more slowly as the Earth wobbles a bit over millennia.) Some maps, including USGS quad maps, indicate the declination angle between magnetic north and true north.
  
• projection information.  This is often omitted from published maps since most people don't know or care much about projections, but serious cartographers do care.
  
• optional elements: a neatline around the border of the map, supporting graphics, data tables or other elements.