1. Map Projections & Coordinate Systems
How does a cartographer deal with the translation of a round-ish Earth to a flat map/screen?
How and where are 2-dimensional coordinates assigned?

2. Datums - Spheroids - Ellipsoids - Geoids…
Turns out Columbus was wrong…
The Earth’s shape is not truly spherical: There is a slight bulging at the equator and flattening at the poles due to the centrifugal force generated by the Earth’s rotation
The closest mathematical approximation of Earth’s shape is an oblate spheroid or an ellipsoid
Better is a Geoid (not a mathematical model but a model of mean sea level based on survey measurements taken across the planet)
Not a problem for small scale maps of the Earth - a sphere is sufficient
In order to be accurate, larger scale maps must use an ellipsoid (or geoid) as a base (earth model)
Datums are built upon an ellipsoid (or a geoid) in conjunction with local/regional survey control points (Ex: North American Datum 1927 (NAD27); Kertau 1948 )
An example of an ellipsoid

3. Map Projections
Methods for flattening (unrolling) a roundish earth onto a flat surface
Based on a Datum (which is based on an ellipsoid)
Ellipsoid (earth model) changes, the datum changes – (Clarke 1866 = NAD 1927) v. (GRS 1980 = NAD 1983)
The shapes of the earth land features are ‘projected’ onto a flat surface – as if a light were aimed at the planet casting a shadow

4. Datum Shift 4789 541 4790 542 4788 543 Datum corner NAD27 1000m 700m
When there is a change in datums, coordinate values change for all points on the ground.
“The grid moves, but the ground does not move.”
Switching from one datum to another results in different coordinate values for the same location. This is because the point of origin for each datum may be slightly different.
The graphic above uses Universal Transverse Mercator (UTM) coordinate system (described later in this lesson), which is represented by the blue grid, to demonstrate datum shift. Each UTM grid square is one kilometer (1,000 meters) on a side, and its location on the map is given by the black numbers along the left and bottom edges of the map. Assume, for the purposes of this example, that the school is 700 meters to the east of the UTM gridline closest to it on the left, and 275 meters north of the first UTM gridline below the school (as shown by the orange arrows and numbers).
To understand datum shift, it’s important to remember that ground features, and map features, remain stationary when a change in datum occurs on a map, or on the ground. However, the coordinate system (represented here by the blue UTM grid) does shift with a change from one datum to another.
It is the grid, not the map features or points on the ground, that moves when a switch in datums occurs.
Introducing a new datum to the map gives every feature on that map (including Highlands School in the example) a new set of coordinates. Using the school as an example, and NAD27 as the map’s datum shown above, the school’s coordinates are:
easting meters, and northing meters.
The next slide shows how those coordinates for the school change when datum NAD83 is used on the map instead of NAD27.

5. Datum Shift Datum corner NAD83 4789 541 4790 542 4788 543 1000m 600m
In the above graphic, the physical features on the map remain in place. But switching the UTM grid from NAD27 to NAD83 has shifted the grid rather dramatically on the map. The map, and the physical earth it represents, has remained stationary. Only the UTM grid has moved. Which means that the Highlands School now has a new set of coordinates in datum NAD83: easting meters and northing meters.
Compare that to the NAD27 coordinate from the previous slide: easting meters and northing meters.
Since UTM is based on the metric system, the amount of offset between NAD27 and NAD83 can be easily measured on this example map. For easting, the difference between NAD27 and NAD83 is 100 meters, or 328 feet ( – ). For northing, it’s 75 meters, or about 230 feet ( – ). A person navigating by GPS using NAD27 coordinates programmed into a receiver that has its datum set to NAD83 will end up at a point 100 meters east and 75 meters south of his or her desired destination.
On an incident, datum shift can create a potentially dangerous situation if the coordinates refer to a safety zone, pick up point, medivac point, helicopter drop, or other such operation. To ensure correct datum matching, the GPS Specialist should always verify the datum for any coordinates (regardless of the coordinate format, such as lat/long or UTM) when those coordinates are provided by another person, derived from a map, or supplied by another GPS receiver.
It’s best if all GPS users on an incident should operate with one datum and coordinate system standard on the ground (air operations may have to use another datum).
To alleviate potential datum errors, the GPS Specialist should follow these guidelines:
When receiving coordinates provided by another person, always obtain the datum used to derive those coordinates, and use that datum.
When plotting coordinates from any map, determine the datum used on the map from the map’s legend.
When passing on coordinates to other GPS users, provide them with the datum used to derive those coordinates.
Determine what datum the incident GIS Specialist, or the incident agency, wants GPS data collected in. If no preference is given, it’s best to choose WGS84.
Or, upon arriving on a new incident, the GPS Specialist should find out what datum other GPS users are using on the incident.
Always set a GPS receiver to the correct datum before entering coordinates into a GPS receiver.

6. Basic Types of Map Projection
Common Developable Surfaces
Plane
Cone
Cylinder

7. Basic Types of Map Projection
Common Developable Surfaces
Plane
Cone
Cylinder

8. Map Projections - Cylindrical
Tend to be Conformal
Globe is projected onto a cylinder tangent at equator (typically)
Low distortion at equator
Higher distortion approaching poles
A good choice for use in equatorial and tropical regions, e.g., Ecuador, Kenya, Malaysia

9. Mercator Projection A cylindrical projection
Invented by Gerhardus Mercator - Flemish cartographer - in 1569
A special purpose projection intended as a navigational tool
A straight line between two points gives a navigator a constant compass bearing to the destination - not necessarily the fastest route

10. Map Projections - Conic
Tend to be Equal Area
Surface of globe projected onto cone tangent at standard parallel
Distorts N & S of standard parallel(s)
Normally shows just one semi-hemisphere in middle latitudes

11. Map Projections – Planar or Polar
Planar or Polar Projection -- Conformal
Conformal
Surface of globe is projected onto a plane tangent at only one point (frequently N or S pole)
Usually only one hemisphere shown (often centered on N or S pole)
Works well to highlight an area
Sometimes used by airports
Shows true bearing and distance to other points from center/point of tangency

12. Map Projections – Distortion
Conformal vs Equal-area
(The Great Debate)
Preserve true shapes
Preserve angles
Exaggerate areas
Graticules perpendicular
Show true size (area)
Distorts shapes, angles and/or scale
(squish/stretch shapes)
Graticules not perpendicular OR

13. Distortion: direction and distance
Conformal vs Equal-area
Northwest
North
Northeast

14. Conformal

15. Equal Area

16. Map Projections - families & examples
Elliptical/Pseudocylindrical (football) Projection
Tend to be equivalent (equal-area)
Not bad for world maps
Mollweide projection

17. Map Projections - families & examples
Goode’s Homosoline Interrupted Elliptical Projection
Equivalent/equal area
Good for climate, soils, landcover - latitude and area comparisons
Mild distortion of shapes
Interrupts areas - oceans, Greenland, Antarctica - sometimes reversed

18. Map Projections - families & examples
Waterman Polyhedron “Butterfly” Projection
Good approximation of continents’:
size
shape
position

19. Johann Heinrich Lambert
Map Projections
Two to Remember
Lambert invented two of the most important and popular projections in use today
1. Conformal Conic
A conic with two standard parallels (used for some State Plane systems)
2. Transverse Mercator A rotated cylindrical with the tangent circle N-S instead of along the Equator (used for UTM & some State Plane systems)
Johann Heinrich Lambert
( )
Lambert Conformal Conic
Transverse Mercator

20. Map Projections Coordinate Systems
For a spatial database to be useful, all parts must be registered to a common coordinate system.
Coordinate Systems (other than latitude-longitude) use a particular Projection, as well as a particular Datum (which is based upon a particular Ellipsoid or Geoid)…

21. (a spherical coordinate system)
Coordinate Systems
Three to Remember
1. State Plane
2. Universal Transverse
Mercator (UTM)
3. Latitude-Longitude
(a spherical coordinate system)

22. Coordinate Systems: State Plane Coordinate System
Created in the 1930’s, zones follow state/county boundaries
Each zone uses a projection:
Lambert’s Conformal Conic (E-W zones)
Transverse Mercator (N-S zones)
Each zone has a centrally located origin, a central meridian and a false origin established to the W and S
Don’t have to deal with negative numbers
Uses planar coordinates (instead of Lat./Long. spherical coordinates)
Square grid with constant scale - distortion over small areas is minimal
USA only
False origin for WA. N. zone
Zones of the SPCS for the
contiguous US

23. Coordinate Systems: State Plane Coordinate System

24. Coordinate Systems: Universal Transverse Mercator
Convenience of a plane rectangular grid on a global level
Popular in scientific research
A section from a transverse Mercator projection is used to develop separate grids for each of 60 zones
Low distortion along the tangent central meridian, increasing E & W
Works great for large scale data sets and satellite image rectification though some areas cross zones (WA, TN, etc.)
Beginning at 180o, Transverse Mercator projections are obtained every 6 degrees of longitude along a central meridian

25. Coordinate Systems: Universal Transverse Mercator
60 N-S zones each spanning 6o of longitude (0.5o overlap each side) from 84o N - 80o S
In polar regions the Universal Polar Stereographic grid system (UPS) is used
Each zone has an origin, central meridian, and false origin, just as with SPCC
Coordinates read similar to SPCC but in meters:
UTM zones (10-19 North) covering the lower 48 states

26. Coordinate Systems: Lat./Long. (Geographic Coordinates)
Works for a sphere or spheroid
Lines of latitude begin at the equator and increase N and S toward the poles from 0o to 90o
Degrees of Latitude are constant
Lines of Longitude begin at some great circle (prime meridian) passing through some arbitrary point
1o of Longitude = 1o of Latitude only at the Equator. Degrees of Longitude get smaller (converge) towards poles
Technically, Unprojected (a spherical coordinate system)
NOT Projected
“Geographic Coordinate System”

27. So why does this matter?
GIS programs must know what projection was used for data creation and where the Coordinate System’s point of origin is…
Map data or satellite images in different projections, coordinate systems, or referenced to different datums may not overlay properly…

28. So why does this matter?
Things work best in ArcMap if the Data and the Data Frame use the same coordinate system, projection and datum…
ArcMap can project data (using one coordinate system) to another, different coordinate system (e.g., that of the Data Frame) if the coordinate systems of the data and the data frame are properly defined
If the Data and Data Frame use different Datums, a Datum Transformation must be chosen

29. Summary
A round-ish earth must be ‘projected’ onto a developable surface in order to make a flat map. Common developable surfaces are:
Plane
Cylinder
Cone
Two common Projections used in the USA are:
Lambert’s Conformal Conic
Transverse Mercator (Cylindrical, a variant of the classic Mercator projection)
Coordinate Systems assign a unit of measurement and a point of origin. These require a projection as well as a datum (earth model).
Three common Coordinate Systems used in the USA are:
Latitude-Longitude (which is technically unprojected (or ‘geographic’), but still requires a datum, and uses speherical coordinates as opposed to planar coordinates)
UTM (Universal Transverse Mercator)
State Plane