1. Map projections and datums

2. Maps are flat
flat
Earth is curved

3. Map Distortion No map is as good as a globe.
A map can show some of these features
True direction or azimuth
True angle
True distance
True area
True shape
But not all of them!

5. common frame of reference
Coordinate System
common frame of reference
for all data on a map

6. GIS needs Coordinate Systems to:
perform calculations
relate one feature to another
specify position in terms of distances and directions from fixed points, lines, and surfaces

7. Coordinate Systems
Cartesian coordinate systems: perpendicular distances and directions from fixed axes define positions
Polar coordinate systems: distance from a point of origin and an angle define positions

8. Each coordinate system uses a different model to map the Earth’s surface to a plane

9. GCS Geographic Coordinate Systems
Degrees of latitude and longitude
Spherical polar coordinate system
“Real” distance varies

10. Spherical Coordinates
Any point uniquely defined by angles passing through the center of the sphere
Meridian 
Equator 

11. The Graticule Map grid (lines of latitude and longitude)
A transformation of Earth’s surface to a plane, cylinder or cone that is unfolded to a flat surface

12. Decimal Degrees 30º 30' 0" = 30.5º 42º 49' 50" = 42.83º
35º 45' 15" = ?

13. Standard Geographic Features
Parallels of latitude are drawn relative to the Equator and meridians of longitude are drawn relative to the central meridian.

14. Parallels of Latitude
Equator
Slicing the Earth into pieces

15. Measuring Parallels
Give the slices values

16. Lines of Longitude Antimeridian A Meridian A
Meridians of longitude slice the Earth relative the pole.
Establish a way of slicing the Earth from pole to pole

17. Prime Meridian
Establishes an orthogonal way of slicing the earth

18. Longitude
North America
Values of pole-to-pole slices

19. Earth Grid Comparing the parallels and the lines
In some GISs, the longitude west of the Prime Meridian and the latitudes South of the Equator are negative.
Comparing the parallels and the lines

20. Latitude and Longitude
Combining the parallels and the lines

21. Grid for US
What is wrong with this map?
Parallels and Lines for US

22. Sphere vs. Ellipsoid Globes versus Earth
The earth is almost a perfect sphere. The ellipticity of a sphere is 0.0, whereas that of the earth is Flattening of the earth occurs at the poles. And bulging occurs at the equator.
The south pole is closer to the equator than the north pole.

23. Shape of the Earth Approximated by an ellipsoid
Rotate an ellipse about its minor axis = earth’s axis of rotation
Semi-major axis a = 6378 km
Semi-minor axis b = 6356 km NP b a SP

24. Ellipsoids and Geoids
The rotation of the earth generates a centrifugal force that causes the surface of the oceans to protrude more at the equator than at the poles.
This causes the shape of the earth to be an ellipsoid or a spheroid, and not a sphere.
The nonuniformity of the earth’s shape is described by the term geoid. The geoid is essentially an ellipsoid with a highly irregular surface; a geoid resembles a potato or pear.

25. The Ellipsoid
The ellipsoid is an approximation of the Earth’s shape that does not account for variations caused by non-uniform density of the Earth.
Examples
Clarke 1866
Clarke 1880
GRS80
WGS60
WGS66
WGS72
WGS84
Danish

26. Satellite measurements have led to the use of geodetic datums WGS-84 (World Geodetic System) and GRS-1980 (Geodetic Reference System) as the best ellipsoids for the entire geoid.

27. The Geoid
The maximum discrepancy between the geoid and the WGS-84 ellipsoid is 60 meters above and 100 meters below.
Because the Earth’s radius is about 6,000,000 meters (~6350 km), the maximum error is one part in 100,000.

28. Geodetic Datums

29. Geodetic Datum
Defined by the reference ellipsoid to which the geographic coordinate system is linked
The degree of flattening f (or ellipticity, ablateness, or compression, or squashedness)
f = (a - b)/a
f = 1/294 to 1/300

30. Geodetic Datums A datum is a mathematical model
Provide a smooth approximation of the Earth’s surface.
Some Geodetic Datums
WGS60
WGS66
Puerto Rico
Indian 1975
Potsdam
South American 1956
Tokyo
Old Hawaiian
European 1979
Bermuda 1957

31. Common U S Datums NAD27 North American Datum 1927
WGS84 World Geodetic System 1984 (based on NAD83)

32. Map Projections
A map projection is a systematic arrangement of grids on a flat surface (map).
Map projections are required because the earth is spheroid.
One easy way to understand what map projections are is to visualize a light shining through the earth onto a surface, called a projection surface.
Any representation of the earth will have some distortion, in shape, area, distance, or direction.
The four basic properties of map projections are:
Shape
Area
Distance
Direction
Maps are flat.
Different projections produce different distortions.
The characteristics of each projection make them useful for some purpose.

33. Making a Map Concept of the Light Source
The light source can be taken as the center of the earth, producing a gnomonic projection. The point of tangency is a single point on the surface of the earth.
Shape distortion increases with distance from the center.
Directional measurements are accurate from the center.
No line distances are accurate.
Distortion of area measurements increase with distance from the center.
The stereographic projection views the surface from one surface to the opposite surface.
Local shape is accurate.
True scale is at the center and so distortion of areas increases with distance from the center.
From the center, directions are accurate. Local angles are accurate everywhere.
Distance errors increase with distance from the center.
The orthographic projection views the earth from an infinite point in space.
Shapes are minimally distorted near the center, maximum distortion occurs near the edge.
Area get smaller away from the center
Directions are only true from the central point.
Distances decrease away from the center.
Concept of the Light Source

34. Projection Families
Although there are many projections, they can be grouped into one of three families according to the projection surface.
Universal Transverse Mercator is a system in which the earth is divided into 60 zones, each spanning six degrees of longitude. Each zone has its own central meridian. Therefore each zone spans 3 degrees west and 3 degrees east of its central meridian.
The origin for each zone is the Equator and its central meridian. To eliminate negative coordinates, the projection alters the coordinate system at the origin. The value given to the central meridian has a false easting of 500,000, and a false northing of 0. For locations in the southern hemisphere, the origin is assigned a false easting of 500,000 and false northing of 10,000,000.
The UTM system has accurate representation for small shapes and minimum distortion of larger shapes within each zone.
Areas have minimal distortion within each zone.
Directional angles in UTM are locally true.
Distances along central meridians in UTM are constant, but at a scale factor of to reduce lateral distortion.
Errors and distortion increase beyond one UTM zone.

35. Types of Projection Families
An infinite number of projections can be produced by changing the orientation of the projection surface. There three basic types of projections: azimuthal, cylindrical, and conic.
The normal orientation for cylindrical projections places the cylinder parallel to the polar axis. The cylinder is upright and tangent to the globe at the equator
The normal orientation for conic projections has the cone tangent to parallel of interest and the axis of the cone is in line with the axis of the globe.
Transverse projections have the projection surfaces at 90 degree angles to the normal projections.
The surfaces onto which the earth’s surface can be projected can have an infinite number of orientations to the earth itself.
Any surface at non-polar or non-transverse positions is called an oblique projection.

36. Standard Point/Line for Projection
For regular or normal azimuthal projections, the orientation is tangent to the poles of the earth.
A projection has either a point of tangency or a line of tangency.
Distortion increases as you move away from the point or line of tangency. The only place where the map is true to scale is at the point or line of tangency.

37. Regular Azimuthal
Azimuthal projections are the easiest to understand. In them, the grid network is projected onto a plane

38. Azimuthal Projections
In a the azimuthal projection the scale changes dramatically across the United States. This is not a good choice for a map of the US.

39. Azimuthal Projections
Shapes are distorted everywhere except at the center
Distortion increases from center
True directions can be plotted from the center outward
Distances are accurate from the center point

40. Polyconic Projections
A series of conic projections stacked together
Have curved rather than straight meridians
Not good choice for tiles across large areas

41. Albers Conic Equal Area Projections
Good choice for mid-latitude regions of greater east-west than north-south extent
Scale factor along two standard parallels is
Scale is reduced between the two standard parallels and increased north or south of the two standard parallels

42. Equal Area Projections
Projections that preserve area are called equivalent or equal area.
Equal area projections are good for small scale maps (large areas)
Examples: Mollweide and Goode
Equal-area projections distort the shape of objects

43. Conformal Map Projections
Projections that maintain local angles are called conformal.
Conformal maps preserve angles
Conformal maps show small features accurately but distort the shapes and areas of large regions
Examples: Mercator, Lambert Conformal Conic

44. Conformal Map Projections
The area of Greenland is approximately 1/8 that of South America. However on a Mercator map, Greenland and South America appear to have the same area.
Greenland’s shape is distorted.

45. Map Projections
For a tall area, extended in north-south direction, such as Idaho, you want longitude lines to show the least distortion.
You may want to use a coordinate system based on the Transverse Mercator projection.

46. Map Projections
For wide areas, extending in the east-west direction, such as Nebraska, you want latitude lines to show the least distortion.
Use a coordinate system based on the Lambert Conformal Conic projection.

47. Map Projections
For a large area that includes both hemispheres, such as North and South America, choose a projection like Mercator.
For an area that is circular, use a normal planar (azimuthal) projection

48. The UTM System

49. Universal Transverse Mercator
1940s, US Army
120 zones (coordinate systems) to cover the whole world
Based on the Transverse Mercator Projection
Sixty zones, each six degrees wide

50. UTM Zones Zone 1 Longitude Start and End 180 W to 174 W
Linear Units Meter
False Easting ,000
False Northing 0
Central Meridian W
Latitude of Origin Equator
Scale of Central Meridian

51. UTM Zones Zone 2 Longitude Start and End 174 W to 168 W
Linear Unit Meter
False Easting ,000
False Northing 0
Central Meridian W
Latitude of Origin Equator
Scale of Central Meridian

52. UTM Zones Zone 13, Colorado, Nebraska Panhandle, etc.
Longitude Start and End W to 102 W
Linear Unit Meter
False Easting ,000
False Northing 0
Central Meridian W
Latitude of Origin Equator