1. Chapter 2 - Map Projection
Week 2
Spring 2007

2. Introduction
Same coordinate system is used on a same Data Frame in ArcMap
Projection - converting digital map from longitude/latitude to two-dimension coordinate system.
Re-projection - converting from one coordinate system to another
Certain map projections are better than others for representing different parts of the earth’s land area. Such as, the Albers conical equal-area projection is uniquely suited for portraying the United States.

3. Measuring Earth’s Circumference
Pythagoras(6 B.C.) and Aristotle(4.B.C.) decided the earth was round.
Eratosthenes ( B.C.)-calculated the equatorial circumference to be 40,233 km and today’s measurement is 40,072 km. Error = 161/40072 = 0.4%

4. Size and Shape of the Earth
Shape of the Earth is called “geoid” Newton and Huygens stated earth was flattened at the poles and bulged at the equator.
The sciences of earth measurement is called “Geodesy”
“ellipsoid” - reference to the Earth shape.
b=semiminor axis (polar radius)
f = (a-b)/a - flattening
1/ for GRS1980, and 1/ for Clarke 1866
a = semimajor axis (equatorial radius)
The geoid bulges at the North Pole and is depressed at the South Pole

5. Geoid (shape of the earth)
Ellipsoid – used for showing earth shape
Clark ellipsoid of 1866, used until recently. (NAD27)
NAD (North American Datum) uses Geodetic Reference System (GRS80) reference ellipsoid.

6. Geographic Grid
The location reference system for spatial features on the Earth’s surface, consisting of Meridians and Parallels.
Meridians - lines of longitude for E-W direction from Greenwich (Prime Meridian)
Parallels - line of latitude for N-S direction
North and East are positive for lat. and long. such as Cookeville is in (-85.51, 36.17).

7. Scale Map Scale = map distance / earth distance
RF (representative fraction) – such as 1:25,000, 1:50,000…
Compute the scale with 10-in radius globe
Scale Bar, Verbal Scale (1 in = 2 miles)
Determine scale of “1 inch to 4 miles
Scale problem – distance between two points is 5 mile, what is the scale of a map on which the points is inches apart?

8. DMS and DD (sexagesimal scale)
Longitude/Latitude can be measured in DMS or DD,
For example in downtown Cookeville, a point with (-85.51, 36.17) which is in DD. To convert DD to DMS, we will have to do several steps: for example, to convert to DMS,
0.51 * 60 = 30.6, this add 30 to minute and leave 0.6.
0.6 * 60 = 36, this add 36 to seconds. Thus, the longitude is (-85o30’36”)

9. Exercise - convert New York City’s DMS to DD
New York City’s La Guardia Airport is located at (73o54’,40o46’). Convert this DMS to DD.

10. Exercise - convert New York City’s DMS to DD
New York City’s La Guardia Airport is located at (73o54’,40o46’). Convert this DMS to DD.
54/60 = 0.9 and 46/60 = 0.77
(73.90, 40.77) is the answer.

11. GPS (Global Positional System)

12. Applications of GPS
Location Identification – find out where you are in a big city or highway
Travel – find out distance to the next exit
Survey – streets, building, school, etc.
Forest – where are you now?
Boating/Fishing – locations
Aviation – landing, navigation
Defense Industry – where is the target?

13. At least 3 satellites are required

14. SVs (Space Vehicles)
GPS Constellation consists of 24 satellites that orbit the earth in 12 hours. There are more than 24 operational satellites as new ones are launched to replace older.
Six orbital planes (4 sat. each), 20,200 km altitude, equally spaced (60o apart), and inclined at about 55o with respect to the equatorial plane, which provides 5-8 SVs visible from any point on the earth

15. Exercise Copy Jan19P1.xls to your folder from G:\4210\Data\GPS
Start ArcMap
Use Tools|Add XY Data….
Select Jan19P1.xls from your folder.
Latitude and Longitude will be automatically shown on the X and Y field.
You may now confirm your coordinate system. (undefined coordinate system…Click –Edit – Select – Geographic Coordinate Systems – World – WGS 1984.prj)

16. You may simply ignore this message by clicking OK
You may simply ignore this message by clicking OK. Now you have chance to view the points in DMS. You can change your dataframe coordinate system to UTM16 to view the point in meters.
However, the GPS points are still in WGS1984.

17. Length of Parallels/Angular/Great Circle
Length of parallels = cos() * length of equator
Meridians and parallels intersect at right angles.
Loxodrome – meridians, parallels and equator all have constant compass bearing.
Great circle arc – shortest distance between 2 points on earth, formed by passing a plane through the center of the sphere.
All meridians and equator are great circle. (Is Loxodrome a Great Circle?)
Small Circle – circles on the grid are not great circle. Parallels of latitude of small circle (except equator).
Travel along N-S is the shortest, but not E-W (except along equator)
Azimuth – angle between great circle and meridian (fig 2.10). The azimuth from A to B is the angle made between the meridian passing through A and the great circle arc passing from A to B. Measuring clockwise from geographic north, although possible from the south.

18. Measure Distance on a Spherical Surface
cos D = sin a * sin b + cos a * cos b * cos c
where D is the distance between A and B in degrees
a is the latitude of A, b is the latitude of B and c is the difference in longitude between A and B.
Multiply D by by the length of one degree at the equator,which is miles. For example:
Between Cookeville and New York City, we have a = 36.17, b=40.77, and c = (-73.90) =
cos D = sin36.17 * sin cos * cos * cos (-11.61) = 0.988, cos = 8.885
Distance = * = 615 miles

19. Exercise Add another Dataframe to your current ArcMap Project.
Add layers (cities/contiguous-states) from G:\4210\Data\ESRIDATA\USA to the current dataframe.
Find Cookeville and New York (use Selection | Select By Attribute to select ‘New York’) and find their distance and you will get decimal degrees. So, you need to change the measuring unit from the little tool icon shown in the measuring window.

20. Projection – to represent the earth as a reduced model of reality
Transformation of the spherical surface to a plane surface. Graticule – meridians and parallels on a plane surface.
Projection Process (fig 2.12)
Best fit (earth geoid)
Find a “Reference ellipsoid” and reduce to
“Generating globe” then
Map projection (2D surface)

21. Map projections
Surface Transformation and Distortion caused by tearing, shearing and compression from 3D to 2D.
For a large scale map, distortion is not a major problem. However, the mapped is larger, then distortion will occur.
Conformal - preserves local shapes
Equivalent - preserves size
Equidistant - maintain consistency of scale for certain distance
Azimuthal - retains accurate direction
Conformal and Equivalent - mutually exclusive, otherwise a map projection can have more than one preserved property

22. Projections
Equal-Area Mapping - distort Shape, but important in thematic mapping, such as in population density map.
Conformal Mapping – shapes of small areas are preserved, meridian intersect parallels at right angles. Shapes for large areas are distorted.
Equidistance Mapping – preserve great circle distances. True from one point to all other points, but not from all points to all points.
Azimuthal Mapping – true directions are shown from a central point to other points, not from other points to other points. This projection is not exclusive, it can occur with equivalency, conformality and equidistance.

23. Measuring Distortion
Overlay shapes on maps (fig 2-14) (angular distortion value: 2ω

24. Distortion Tissot’s indicatrix (fig 2-15)
S=max. areal distortion, if S= 1.0, no area distortion
a=b conformal projection. S varies
ab not conformal

25. Simple Case
Secant Case
Conic
Cylindrical
Azimuthal

26. Standard line - the line of tangency between the projection surface and the reference globe
Simple case has one standard line where secant case has two standard lines.
Scale Factor(SF) - the ratio of the local scale to the scale of the reference globe
SF =1 in standard line.
Central line - the center (origin) of a map projection
To avoid having negative coordinates , false easting and false northing are used in GIS. Move origin of map to SW corner of the map.

27. Planes of deformation darker areas represent greater distortion
source of data: Dent, 1999

28. Commonly used map projections
Transverse Mercator - use standard meridians, required parameters: central meridian, latitude of origin (central parallel) false easting, and false northing.
Lambert Conformal Conic - good choice for mid-latitude area of greater east-west than north-south extent (U.S. Tn,,,,). Parameters required: first/second standard parallels, central meridian, latitude of projection’s origin, false easting/northing.
Albers Equal-Area Conic - requires same parameters as Lambert Conformal, with standard parelles at 29.5o and 45.5o is a good choice for mapping the US
Equidistant Conic - preserves distance property along all meridians and one or two standard parallels.

29. Spheroid or ellipsoid- a model that approximate the Earth - datum is used to define the relationship between the Earth and the ellipsoid.
Clarke was the standard for mapping the U.S. NAD 27 is based on this spheroid, centered at Meades Ranch, Kansas.
WGS84 spheroid (almost identical to GRS80 spheroid) - from satellite orbital data. More accurate and it is tied into a global network and GPS. NAD 83 is based on this datum.
Horizontal shift between NAD 27 and NAD can be large..
HARN: High Accuracy Reference Network being developed

30. Coordinate Systems
Plane coordinate systems are used in large-scale mapping such as at a scale of 1:24,000.
Accuracy in a feature’s absolute position and its relative position to other features is more important than the preserved property of a map projection.
Most commonly used coordinate systems: UTM, UPS, SPC and PLSS

31. UTM
Divide the world into 60 zones with 6o of longitude each,covering surface between 84oN and 80oS.
Use Transverse Mercator projection with scale factor of at the central meridian. The standard meridian are 180 km east and west of the central meridian.
N Hemisphere: false origin at the equator and 500,000 meters west of the central meridian:
S Hemmisphere: 10,000,000 m south of the equator and 500,000 m west of the central meridian.
Maintain the accuracy of at least one part in (within one meter accuracy in a 2500 m line)

32. The SPC System Developed in 1930.
To maintain required accuracy of one in 10,000, state may have two ore more SPC zones.
Transvers Mercator is used for N-S shapes, Lambert conformal conic for E-W direction.
Points in zone are measured in feet origianlly.
State Plane 27 and 83 are two systems.

33. PLSS
Divide state into 6x6 mile squares or townships. Each township was further partitioned into 36 square-mile parcels of 640 acres, called sections

34. TN Physiological Map Spatial_Reference_Information:
Horizontal_Coordinate_System_Definition:
Planar:
Map_Projection:
Map_Projection_Name: ALBERS conical equal area
Albers_Conical_Equal_Area:
ALBERS conical equal area
Standard_Parallel: 29.5
Standard_Parallel: 45.5
Longitude_of_Central_Meridian: -96
Latitude_of_Projection_Origin: 23
False_Easting:
False_Northing:
Planar_Coordinate_Information:
Planar_Coordinate_Encoding_Method: coordinate pair
Coordinate_Representation:
Abscissa_Resolution: 1.0
Ordinate_Resolution: 1.0
Planar_Distance_Units: METERS
Geodetic_Model:
Horizontal_Datum_Name: North American Datum of 1983
Ellipsoid_Name: GRS1980
Semi-major_Axis:
Denominator_of_Flattening_Ratio:

35. Exercise
Let’s change the coordinate system of the layer of United States from NAD1983 to Albers Equal Area -Conic
Predefined
Projected Coordinated Systems
North America
North America Albers Equal Area Conic
North_America_Albers_Equal_Area_Conic
Projection: Albers
False_Easting:
False_Northing:
Central_Meridian:
Standard_Parallel_1:
Standard_Parallel_2:
Latitude_Of_Origin:
Linear Unit: Meter