1. Models of the Earth sphere, oblate ellipsoid geoid
The earth can be modeled as a
sphere,
oblate ellipsoid
geoid

2. Earth Shape: Sphere and Ellipsoid

3. Definitions: Ellipsoid
Also referred to as Spheroid, although Earth is not a sphere but is bulging at the equator and flattened at the poles
Flattening is about 21.5 km difference between polar radius and equatorial radius
Ellipsoid model necessary for accurate range and bearing calculation over long distances  GPS navigation
Best models represent shape of the earth over a smoothed surface to within 100 meters

4. The Spheroid and Ellipsoid
The sphere is about 40 million meters in circumference.
An ellipsoid is an ellipse rotated in three dimensions about its shorter axis.
The earth's ellipsoid is only 1/297 off from a sphere.
Many ellipsoids have been measured, and maps based on each. Examples are WGS84 and GRS80.

5. Earth as Ellipsoid

6. WGS 84 Geoid defines geoid heights for the entire earth
Geoid: the true 3-D shape of the earth considered as a mean sea level extended continuously through the continents
Approximates mean sea level
WGS 84 Geoid defines geoid heights for the entire earth

7. Earth Models and Datums

8. Definition: Datum
A mathematical model that describes the shape of the ellipsoid
Can be described as a reference mapping surface
Defines the size and shape of the earth and the origin and orientation of the coordinate system used.
There are datums for different parts of the earth based on different measurements
Datums are the basis for coordinate systems
Large diversity of datums due to high precision of GPS
Assigning the wrong datum to a coordinate system may result in errors of hundreds of meters

9. Commonly used datums Datum Spheroid Region of use NAD 27 Clark 1866
Canada, US, Atlantic/Pacific Islands, Central America
NAD 83
GRS 1980
Canada, US, Central America
WGS 84
Worldwide

10. The Datum
An ellipsoid gives the base elevation for mapping, called a datum.
Examples are NAD27 and NAD83.
The geoid is a figure that adjusts the best ellipsoid and the variation of gravity locally.
It is the most accurate, and is used more in geodesy than GIS and cartography.

11. Geoid

12. Map Scale
The ratio of the distance between two points on the map and the real world (earth) distance between the same two points
Map measurement / Real world measurement

13. Map Scale
Map scale is based on the representative fraction, the ratio of a distance on the map to the same distance on the ground.
Most maps in GIS fall between 1:1 million and 1:1000.
A GIS is scaleless because maps can be enlarged and reduced and plotted at many scales other than that of the original data.
To compare or edge-match maps in a GIS, both maps MUST be at the same scale and have the same extent.
The metric system is far easier to use for GIS work.

14. Types of Scale
Graphical
Verbal
Representative Fraction

15. Graphical Scale 10 5
10 miles

16. Verbal Scale
One inch equals one mile, or;
One inch to one mile

17. Representative Fraction
Ratio between map distance and ground distance for equivalent point
Unit free. Ratio is true regardless of units

18. Representative Fraction
Expressed as fraction
1/100,000
Or Ratio
1:100,000

19. Large Scale vs. Small Scale
Small scale = Large area
Small scale = Large Denominator
1:500,000
Large scale = Small area
Large scale = Small denominator
1:50,000

21. Location Reference Systems
Relative
Absolute

22. Relative Location Systems
Real world descriptions
Corner of 34th and Fifth

23. Absolute Location Systems
Absolute description
Uses mathematical coordinates to define the position of grid intersections with respect to a defined (accepted) origin

24. Absolute Location Systems
Common Absolute Location Systems
Global Coordinate System
Cartesian Coordinate Systems
Universal Transverse Mercator (UTM) Coordinate System
State Plate Coordinate System

25. Geographic Coordinates
Geographic coordinates are the earth's latitude and longitude system, ranging from 90 degrees south to 90 degrees north in latitude and 180 degrees west to 180 degrees east in longitude.
A line with a constant latitude running east to west is called a parallel.
A line with constant longitude running from the north pole to the south pole is called a meridian.
The zero-longitude meridian is called the prime meridian and passes through Greenwich, England.
A grid of parallels and meridians shown as lines on a map is called a graticule.

26. The Global Coordinate System
Spherical coordinate system
Unprojected
Expressed in terms of two angles
latitude
longitude
Latitude and longitude are traditionally measured in degrees, minutes, and seconds (DMS).

27. Origin for the Global Coordinate System
Latitude
positive in northern hemisphere
negative in southern hemisphere
Longitude
positive east of Prime Meridian
negative west of Prime Meridian

28. Geographic Coordinates as Data

29. The Global Coordinate System
Longitude
Angle formed by a line going from the intersection of the prime meridian and the equator to the center of the earth, and a second line from the center of the earth to the point in question

30. The Global Coordinate System
Latitude
Angle formed by a line from the equator toward the center of the earth, and a second line perpendicular to the reference ellipsoid at the point in question

31. Cartesian Coordinate Systems
Computationally, it is much simpler to work with Cartesian coordinates than with spherical coordinates
x,y coordinates
referred to as “eastings” & “northings”
defined units, e.g. meters, feet

32. Map Projections
A transformation of the spherical or ellipsoidal earth onto a flat map is called a map projection.
The map projection can be onto a flat surface or a surface that can be made flat by cutting, such as a cylinder or a cone.
If the globe, after scaling, cuts the surface, the projection is called secant. Lines where the cuts take place or where the surface touches the globe have no projection distortion.

33. Map Projections (cont.)
Projections can be based on axes parallel to the earth's rotation axis (equatorial), at 90 degrees to it (transverse), or at any other angle (oblique).
A projection that preserves the shape of features across the map is called conformal.
A projection that preserves the area of a feature across the map is called equal area or equivalent.
No flat map can be both equivalent and conformal. Most fall between the two as compromises.
To compare or edge-match maps in a GIS, both maps MUST be in the same projection.

34. Projection
Method of representing data located on a curved surface onto a flat plane
All projections involve some degree of distortion of:
Distance
Direction
Scale
Area
Shape
Determine which parameter is important
Projections can be used with different datums

35. Projections
The earth is “projected” from an imaginary light source in its center onto a surface, typically a plate, cone, or cylinder.
Planar or azimuthal
Conic
Cylindrical

36. Cylindrical Projections
Secancy: projection surface cuts through globe, this reduces distortion of larger land areas
Tangency: only one point touches surface
Used for entire world
Parallels and meridians form straight lines

37. Example Cylindrical Projections
Shapes and angles within small areas are true (7.5’ Quad)
Distances only true along equator

38. Conic Projections Can only represent one hemisphere
Often used to represent areas with east-west extent (US)

39. Lambert is used in State Plane Coordinate System
Albers is used by USGS for state maps and all US maps of 1:2,500,000 or smaller
Lambert is used in State Plane Coordinate System
Secant at 2 standard parallels
Distorts scale and distance, except along standard parallels
Areas are proportional
Directions are true in limited areas

40. Azimuthal Projections
Often used to show air route distances
Distances measured from center are true
Distortion of other properties increases away from the center point

41. Orthographic:
Used for perspective views of hemispheres
Area and shape are distorted
Distances true along equator and parallels
Lambert:
Specific purpose of maintaining equal area
Useful for areas extending equally in all directions from center (Asia, Atlantic Ocean)
Areas are in true proportion
Direction true only from center point
Scale decreases from center point

42. Other Projections Pseudocylindrical
Unprojected or Geographic projection: Latitude/Longitude
There are over 250 different projections!

43. Pseudocylindrical:
Used for world maps
Straight and parallel latitude lines, equally spaced meridians
Other meridians are curves
Scale only true along standard parallel of 40:44 N and 40:44 S
Robinson is compromise between conformality, equivalence and equidistance

44. Mathematical Relationships
Conformality
Scale is the same in every direction
Parallels and meridians intersect at right angles
Shapes and angles are preserved
Useful for large scale mapping
Examples: Mercator, Lambert Conformal Conic
Equivalence
Map area proportional to area on the earth
Shapes are distorted
Ideal for showing regional distribution of geographic phenomena (population density, per capita income)
Examples: Albers Conic Equal Area, Lambert Azimuthal Equal Area, Peters, Mollweide

45. Mathematical Relationships
Equidistance
Scale is preserved
Parallels are equidistantly placed
Used for measuring bearings and distances and for representing small areas without scale distortion
Little angular distortion
Good compromise between conformality and equivalence
Used in atlases as base for reference maps of countries
Examples: Equidistant Conic, Azimuthal Equidistant
Compromise
Compromise between conformality, equivalence and equidistance
Example: Robinson

47. Local Coordinate Systems
A coordinate system is a standardized method for assigning codes to locations so that locations can be found using the codes alone.
Standardized coordinate systems use absolute locations.
A map captured in the units of the paper sheet on which it is printed is based on relative locations or map millimeters.
In a coordinate system, the x-direction value is the easting and the y-direction value is the northing. Most systems make both values positive.

49. Coordinate Systems for the US
Some standard coordinate systems used in the United States are
geographic coordinates
universal transverse Mercator system
military grid
state plane
To compare or edge-match maps in a GIS, both maps MUST be in the same coordinate system.

50. USA In The UTM Zones

51. The Universal Transverse Mercator Coordinate System
60 zones, each 6° longitude wide
Zones run from 80° S to 84° N
Poles covered by Universal Polar System (UPS)

52. UTM Zone Projection Transverse Mercator Projection
applied to each 6o zone to
minimize distortion

53. UTM Coordinate Parameters
Unit
meters
Zones:
6o longititue
N and S zones
separate coord
X-origin
500,000 m east
of central meridian
Y-origin
equator

54. Advantage of UTM Settings
Zone central meridian Eastings = 500,000 meters
North pole Northings = 10,000,000 meters
allows overlap between zones form mapping purposes
give all eastings positive numbers
tell if we are to the east or west of the central meridian
provide relationship between true north and grid north

55. State Plane Coordinate System
Each state has one or more zones
Zones are either N-S or E-W oriented (except Alaska)
Each zone has separate coordinate system and appropriate projection
Unit: feet
No negative numbers

56. Map Projections for State Plane Coordinate System
E-W zones: Lambert conformal conic projection
N-S zones: Transverse Mercator Projection

57. Pros and Cons of SPCS Advantages:
The system is used primarily for engineering applications
e.g. utility companies, local governments to do accurate
surveying of facilities network (sewers, power lines)
More accurate than UTM.
feet vs. meters
SPCS deals with smaller area
Disadvantages:
Lack of universality cause problems for mapping over large areas such as across zones and states

58. Projections and Datums
Projections and datums are linked
The datum forms the reference for the projection, so...
Maps in the same projection but different datums will not overlay correctly
Tens to hundreds of meters
Maps in the same datum but different projections will not overlay correctly
Hundreds to thousands of meters.

59. Determining datum or projection for existing data
Metadata
Data about data
May be missing
Software
Some allow it, some don’t
Comparison
Overlay may show discrepancies
If locations are approx. 200 m apart N-S and slightly E-W, southern data is in NAD27 and northern in NAD83

60. Selecting Datums and Projections
Consider the following:
Extent: world, continent, region
Location: polar, equatorial
Axis: N-S, E-W
Select Lambert Conformal Conic for conformal accuracy and Albers Equal Area for areal accuracy for E-W axis in temperate zones
Select UTM for conformal accuracy for N-S axis
Select Lambert Azimuthal for areal accuracy for areas with equal extent in all directions
Often the base layer determines your projections

61. GIS Capability A GIS package should be able to move between
map projections,
coordinate systems,
datums, and
ellipsoids.

62. Summary
There are very significant differences between datums, coordinate systems and projections,
The correct datum, coordinate system and projection is especially crucial when matching one spatial dataset with another spatial dataset.