1. Basic Coordinate Systems
Grid System
RG 620
April 20, 2016
Institute of Space Technology, Karachi

2. Coordinates are not certain !!
Even if their figures are precise

3. Example
In 1937 the United States Coast and Geodetic Survey set Youghall at latitude 40º 25' "N and longitude 108º 45' "W
In November of 1997 Youghall suddenly got a new coordinate, º 25' "N and 108º 45' "W
Had Youghall actually moved at all?
Of course it did no such thing, the station is right where it has always been
Its datum changed !!!!!!
Originally, in1937 latitude and longitude for Youghall was based on the North American Datum 1927 
In 1997 the basis of the coordinate of Youghall became the North American Datum 1983 (NAD83)

4. Coordinates without a specified datum, are vague…..

5. Reference: David Conner National Geodetic Survey, 2003

7. Coordinate Systems
After projection, it is necessary to set up a coordinate system on the map that will allow a point to be described in X-Y space (or northing and easting)
To describe a location in a universally understandable manner a grid system is necessary
For a useful grid it is necessary for it to define an origin and a uniform grid spacing
There are several types of Coordinate System to represent the Earth’s surface
Uniform Grid Spacing means the distance between grid lines should remain constant.
The problem is often illustrated by trying to flatten part of an orange peel. The orange peel stands in for the surface of the Earth. A small part, say a square a quarter of an inch
on the side, can be pushed flat without much noticeable deformation. But when the portion gets larger problems appear.
A portion of the Earth can be projected directly onto the flat plane. In fact this is the typical method for establishing an independent local coordinate system. 7

8. Coordinate Systems
The method of projection, onto a simple flat plane, is based on the idea that a small section of the Earth, as with a small section of the orange, conforms so nearly to a plane that distortion on such a system is negligible
Mapping a considerable portion of the Earth using a large number of small individual planes
Offers the convenience of working in plane Cartesian coordinates and still keep distortion at manageable levels
Note: when these planes are brought together they cannot be edge-matched accurately
A well-designed map projection can offer the convenience of working in plane Cartesian coordinates and still keep distortion at manageable levels.
They cannot be joined properly along their borders.

9. Coordinate Systems Latitude and Longitudes are used
Some commonly used Coordinate Systems are:
Geographic
Latitude and Longitudes are used
UTM
Shape is preserved and precise measurements in meter
State Plane
Local surveying (with minimum distortion)
Geographic: One of the most common CS in use. 9

10. Geographic Coordinate System (WGS84 datum)
Scale, distance, area, and shape are all distorted with the distortion increasing toward the poles.

11. State Plane Coordinate (SPC) Systems
Standard set of projections for the United States developed in 1930’s
Specifies positions in Cartesian coordinate systems for each state
Used for local surveying and engineering applications
Points are projected from their geodetic latitudes and longitudes to x and y coordinates in the State Plane systems
Conformal mapping system for US with a maximum scale distortion of 1 part in 10,000
Used by state and local governments. All states adopt their own specialized coordinate systems.
Larger states require multiple zones. California has 6 and Alaska has 10 zones.
set of 124 geographic zones. There are 110 zones in the continental US, with 10 more in Alaska, 5 in Hawaii, and one for Puerto Rico and US Virgin Islands.
system is highly accurate within each zone. Outside a specific state plane zone accuracy rapidly declines, thus the system is not useful for regional or national mapping.
It managed to keep the distortion of the scale ratio under 1 part in 10,000 and preserved conformity. It did not disturb the familiar system of ordered pairs of Cartesian coordinates and it covered each state with as few zones as possible whose boundaries were constructed to follow portions of county lines as much as possible, with some exceptions. The idea was that those relying on State Plane Coordinates could work in one zone throughout a jurisdiction. 11

12. State Plane Coordinate Systems
Zones have different projections
Lambert Conformal Conic: for states that are longer east–west, such as Tennessee, Kentucky, North Carolina, Virginia, etc.
Transverse Mercator projection: for states that are longer north–south, such as Illinois, Arizona, New Hampshire, etc.
The Oblique Mercator projection: for the panhandle of Alaska (AK zone 1) because it lays at an angle
Idaho uses a transverse Mercator projection and is divided into three zones - east, central and west.
In Florida both LCC and TM are used
FIPS: Federal Information Processing System
Most state plane zones are based on either a transverse Mercator projection or a Lambert conformal conic projection. The choice between the two map projections is based on the shape of the state and its zones. States that are long in the east-west direction are typically divided into zones that are also long east-west. These zones use the Lambert conformal conic projection, because it is good at maintaining accuracy along an east-west axis. Zones that are long in the north-south direction use the Transverse Mercator projection because it is better at maintaining accuracy along a north-south axis. The panhandle of Alaska, whose maximum dimension is on a diagonal, uses an Oblique Mercator projection, which minimizes the combined error in the X and Y directions 12

13. False Easting and Northing?
SPCS: Please note that the central meridian is not the y-axis. If it were the y-axis negative coordinates would result. To avoid them the actual y-axis is moved far to the west of the zone itself.
Not only were angles preserved on the final product, but also there were minimal differences between the length of a measured line on the Earth’s surface and the length of the same line on the map projection, minimal for the measurement technology of the day. In other words, the scale of the distortion was pretty small.
The distortion was held to 1 part in 10,000. A maximum distortion in the lengths of lines of 1 part in 10,000 means that the difference between the length of a 2-mile line on the ellipsoid and its representation on the map would only be about 1 foot at the most.

14. In the old SPCS27 arrangement the y-axis was 2,000,000 feet west from the central meridian in the Lambert Conic projection and 500,000 feet in the Transverse Mercator projection. In the SPCS83 design those constants have been changed. The most common values are 600,000 meters for the Lambert Conic and 200,000 meters for the Transverse Mercator.
However, there is a good deal of variation in these numbers from state to state and zone to zone. In all cases however, the y-axis is still far to the west of the zone and there are no negative State Plane Coordinates. No negative coordinates, because the x-axis, also known as the baseline, is far to the south of the zone. Where the x-axis and y- axis intersect is the origin of the zone and that is always south and west of the zone itself. This configuration of the axes ensures that all State Plane Coordinates occur in the first quadrant and are, therefore, always positive.

15. State Plane Coordinate Systems
Large states are divided into zones to limit distortion error and maintain said accuracy
One or more zones in each state with slightly different projection in each zone
Boundaries of zones follow state and county lines
The number of zones in each state is determined by the area the state covers
The number of zones ranges from 1 to 10 (in Alaska)
Each zone has a unique central meridian
The easting origin for each zone is always placed an arbitrary number of feet/meter west of the western boundary of the zone, eliminating the need for negative easting values.  The northing origin, however, is not at the equator as in UTM, but rather it is placed at an arbitrary number of feet/meter south of the state border.
Property survey, construction projects. 15

16. State Plane 1927 vs. 1983
Originally based on the North American Datum of and the measurement unit was feet
Now being converted to North American Datum of (NAD83) (will use meters as unit of measure)
Due to datum change some zones are redefined
Datum change: some of the states are redefining zones and primary coordinates e.g. California changed from 7 zones to 6, Montana changed from 3 zones to 1, SC & Nebraska changed from 2 zones to 1.
No negative coordinates.
GRS80 spheroid used by NAD83.
The foundation of the original State Plane Coordinate System, SPCS27 was NAD27 and its reference ellipsoid Clarke 1866. 16

17. Source: http://www.pdhcenter.com/courses/l117/l117content.pdf

18. Scale Factor
Where K is the scale factor for a line, K1 is the scale factor at one end of the line and
K2 is the scale factor at the other end of the line.
In Figure 4.9 a typical 158-mile State Plane Coordinate zone is represented by a grid plane of projection cutting through the ellipsoid of reference. As mentioned earlier between the intersections of the standard lines, the grid is under the ellipsoid. There, a distance from one point to another is longer on the ellipsoid than on the grid. This means that right in the middle of a SPCS zone the scale factor is at its minimum. In the middle a typical minimum SPCS scale factor is not less than , though there are exceptions. Outside of the intersections the grid is above the ellipsoid where a distance from one point to another is shorter on the ellipsoid than it is on the grid. There at the edge of the zone a maximum typical SPCS scale factor is generally not more than

19. Scale Factor
Scale factor varies with the latitude in the Lambert projection.
Distortion lessens and the scale factor approaches 1 as a line nears a ______??
standard parallel

20. Universal Transverse Mercator Coordinate System
Global coordinate system
Globe is divided into narrow longitude zones
Best used for north-south oriented areas (little distortion in this direction)
Successive swaths of relatively undistorted regions created by changing the orientation of the cylinder slightly
These swaths are called UTM zones
Each zone is six degrees of longitude wide
Total _?_____zones
Error is less than 0.04%
Total 60 zones.
Used in USGS topo maps and DEM.
UTM coordinates are in meters, making it easy to make accurate calculations of short distances between points (error is less than 0.04%)
Zone boundaries follow arbitrary lines of longitude rather than boundaries between jurisdictions.
In terms of scale it is four times less accurate than typical State Plane Coordinate systems. Yet the ease of using UTM and its worldwide coverage makes it very attractive for work that would otherwise have to cross many different SPCS zones. 22

21. Universal Transverse Mercator Coordinate System
These zones are numbered from west to east
Zone 1 begins at the International Date Line (1800 W), Zone 2 at 174°W and extends to 168°W
Each Zone is further divided into Eastern and Western halves by drawing a center line called Central Meridian
Zones are further split north and south of the equator
Zone 60: starts at 174 degrees E and extends to International Date Line
10-19: UTM zones for the lower 48 contiguous states of the United States of America 23

22. Universal Transverse Mercator Coordinate System
At equator a zone is about 40,000/60 = 667 Km wide
Any point can be described by ‘Easting’ and ‘Northing’ values
Northing is the distance to the equator, while easting is the distance to the "false easting", which is uniquely defined in each UTM zone
The equator is used as the northing origin for all north zones (northing value of zero)
South zones have a false northing value added to ensure all coordinates within a zone are positive
For UTM south zones, the northing values at the equator are set to equal 10,000,000 meters
40,000 = circumference of earth 24

23. UTM – Easting and Northing25

24. The UTM secant projection gives approximately 180 kilometers between the lines of exact scale where the cylinder intersects the ellipsoid (total = 360 km). The scale factor grows from along the central meridian of a UTM zone to at 180 km to the east and west.
In state plane coordinates, the scale factor is usually no more than 1 part in 10,000. In UTM coordinates it can be as large as 1 part in 2,500.

25. Source: Text book

26. Universal Transverse Mercator Coordinate System
Important thing to remember
Coordinate values are discontinuous across
UTM zone boundaries, therefore, analyses are
difficult across zonal boundaries

27. Horizontal Zoning
Latitudes are divided into zones lettered from A at the South Pole to Z at the North Pole
Spacing is not regular throughout
A and B zones are within the south circle of 80 degrees
Zones Y and Z cover the north polar region north of 84
Rest of the zones extend from 80 degrees south latitude to 84 degrees north latitude degrees
Zone X is 12 degrees wide (from 72 to 84 degrees North)
I and O not used
Rest of the zones are 8 degree wide
Zone M and N are just South and North of Equator respectively

28. UTM Zones 30

29. UTM Zones 31

30. UTM Zones - Pakistan

31. UTM – Finding Grid Zone Finding Grid Zone for any Latitude
In calculation take west longitude as (-) negative and east longitude as (+) positive
Add 180 and divide by 6
Round off the resultant value to the next higher number
For example, Denver, Colorado is near 105° W. Longitude, -105°.
-105° + 180° = 75°
75°/ 6 = 12.50
Round up to 13
Example 2: Greenwich Prime Meridian is at …….. Longitude?
Z=30/31 33

32. Quiz
E = 500, ,000 = 600,000
N = 10,000,000 – 1,000,000 = 9,000,000
156 W
153 W
150 W

33. Measuring Distance Distortion
Comparing map distance with the Great Circle Distance
Remember the Example from Text Book where the Great Circle Distance between two point A and B was = KM
Identify coordinates of the equivalent points on UTM grid
Calculate the distance between these points
Negative scale distortion when features are compresses or reduced in size
Positive scale distortion when features are expanded
Calculate distance using Pythagorean formula

34. Grid Distance

35. Variation between Datums
Federally produced datum transformation software is available free of cost at NOAA’s National Geodetic Survey (
Reference: David Corner

36. Conversion Among Coordinate Systems

37. References http://www.ncgia.ucsb.edu/giscc/units/u013/u013_f.html
Images:
Peter H. Dana, Department of Geography, The University of Texas at Austin
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