# Map Projections and Coordinate Systems

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Map Projections and Coordinate Systems
Surveying 101 for GIS Professionals 2013 Kentucky GIS Conference Jeremy Gould – Kentucky Transportation Cabinet 1 1

Agenda Geographic Coordinate Systems Ellipsoids Geoid
Horizontal Datums Projected Coordinate Systems Project Datum Factors Vertical Datums

Geographic Coordinate Systems
Geographic Coordinates Systems use radial coordinates to locate a point on a specifically defined sphere (ellipse). These are called spherical coordinates. Cartesian point P can also be represented in spherical coordinates (λ,φ, γ) where: λ= +/- degrees longitude Φ= +/-degrees latitude γ= + radial distance from center

Geographic Coordinate Systems
Kentucky (-85.3 °,37.5 °) (85.3 ° W, 37.5 ° N) Equator 0° latitude Prime Meridian 0° longitude

Ellipsoids Ellipsoids are flattened spheroids that when referenced to the earth can be rotated and/or shifted to best fit the earth (geoid) either in part or in whole

Geoid The geoid is an equipotential gravimetric surface resulting in an irregular and non-mathematical approximation of the earth’s size and shape relative to a base of reference that best fits global mean sea level in a least squares sense The geoid is a 3 dimensional surface along which the pull of gravity is a specified constant The geoid is a measured and interpolated surface and not a mathematically defined surface Differences in the density of the Earth cause variation in the strength of the gravitational pull, in turn causing regions to dip or bulge above or below the reference ellipsoid

Geoid Gravity Recovery And Climate Experiment (GRACE) Gravimeters

Ellipsoids There are Global Ellipsoids and Regional (local) Ellipsoids
Two Global ellipsoids are GRS80 and WGS84

Horizontal Datums A datum is a reference surface
A geodetic datum consists of two major components Ellipsoid with a spherical coordinate system and origin Set of points and lines that have been surveyed A geodetic datum is a three dimensional Euclidian reference frame defined relative to an associated ellipsoid oriented to achieve a best fit statistical approximation of the geoid either in part or in whole. The North American Datum (NAD) has been defined by two different ellipsoids, the Clarke ellipsoid of 1866, which was oriented to best fit the North American continent and is the basis of NAD27, and the Global Reference System ellipsoid of 1980 (GRS80) which is a globally defined ellipsoid and the basis of NAD83.

Ellipsoid, Geoid, and Datum
GRS80 Ellipsoid (NAD’83) CLARKE 1866 Ellipsoid (NAD’27) GEOID Earth Mass Center Approximately 236 meters

Horizontal Datums Lat and Long locations of given benchmarks in the NAD27 datum will likely be different from the lat and long of that same benchmark in the NAD83 or WGS84 datum's. The monumented points do not move This is described as a datum shift Shift in coordinate locations from WGS84 to NAD83 is often less than 1 meter Datum shifts between NAD27 and NAD83 are often 100’s of meters

Horizontal Datums Geographic Position (Lat-Long)
(variations between datums for same position) Example: Datum 1 may have a long-lat of ( °, °) Datum 2 may have a long-lat of ( °, °) The same point has different coordinates because of the shift/rotation of the ellipsoid

Projected Coordinate Systems
A mapping projection is a geometric tool that allows a portion of a spherical surface to be represented on a two dimensional surface such as a flat sheet of paper or computer screen in a spatially consistent manner. A State Plane Coordinate System is a specialized mapping projection that allows direct conversion between spherical geographic coordinates of latitude () and longitude (), and rectangular Cartesian coordinates of northing (y) and easting (x).

Projected Coordinate Systems
So how do we get from our Geographic Coordinates to a Projected Coordinate System?

Projected Coordinate Systems
Cylindrical Conical Planar

Projected Coordinate Systems
Transverse Mercator Projection SF < 1 Practical Limit of Projection (SF  k0) SF > 1 Axis of Cylinder Grid Origin Intersection of Ellipsoid and Projection Cylinder (SF = 1) Central Meridian (SF = k0) Polar Axis

Universal Transverse Mercator Coordinate System

Projected Coordinate Systems
KENTUCKY PROJECTIONS UTM Zones 16 & 17 Transverse Mercator (Secant Cylinder) UTM Zone 16 UTM Zone 17

Projected Coordinate Systems
Lambert Conic Projection (Northern Hemisphere) Polar Axis North Standard Parallel (SF = 1) Central Meridian South Standard Parallel (SF = 1) Parallel of Grid Origin (Base Parallel)

State Plane Coordinate Systems
State Plane zones are sometimes identified by the Federal Information Processing System (FIPS) Codes as shown below

Projected Coordinate Systems
KENTUCKY PROJECTIONS North and South State Plane Lambert Conformal Conic (Secant Cone) State Plane South Zone State Plane North Zone

Projected Coordinate Systems
KENTUCKY SINGLE ZONE PROJECTION

Projected Coordinate Systems
KENTUCKY SPCS – NORTH AND SOUTH ZONES NORTH ZONE SOUTH ZONE

Projected Coordinate Systems
Kentucky Projections NAD83 State Plane Coordinate System (Lambert Conformal Conic) Parameter Single Zone North Zone South Zone Central Meridian 85° 45’ W 84° 15’ W North Std Parallel 38° 40’ N 38° 58’ N 37° 56’ N South Std Parallel 37° 05’ N 37° 58’ N 36° 44’ N Base Parallel 36° 20’ N 37° 30’ N False Northing 1,000,000 m 0 m 500,000 m False Easting 1,500,000 m Linear unit of measure for all zones is the U.S. Survey Foot (USFt) (1 USFt = meters)

Projected Coordinate Systems
COORDINATE SPACE COMPARISON 1,500,000 m ft 1,250,000 m ft NAD'83 SINGLE ZONE 1,000,000 m ft NORTHING 750,000 m ft NAD'83 SOUTH ZONE 500,000 m ft NAD'83 NORTH ZONE 250,000 m ft NAD'27 NORTH ZONE NAD'27 SOUTH ZONE 0 m 0 ft 0 m 0 ft 250,000 m ft 500,000 m ft 750,000 m ft 1,000,000 m ft 1,250,000 m ft 1,500,000 m ft 1,750,000 m ft 2,000,000 m ft EASTING

Projected Coordinate Systems
Equatorial Plane Topographic Surface (Ground) Ellipsoid E1 Axis of Rotation S1 G1 G2 E2 S2 G3 E3 S3 E1 < G1 < S1 G2 < E2 < S2 E3 < S3 < G3 Projection Grid North Standard Parallel South Standard Parallel E = Distance on ellipsoid G = Distance on grid S = Distance on surface SF = Grid Scale Factor  = Geodetic latitude SF=1 SF<1 SF>1

Project Datum Factor A Project Datum Factor (PDF) converts grid distances (state plane coordinates) to ground/surface distances. If you were to use a total station to measure distance between two points on the ground and then used GPS to measure the location of the same two points and calculate the distance between those two points on the state plane grid, the two distances would be close but not exactly the same.  This is due to the curvature of the earth combined with the elevation above sea level of the project location.  The grid (state plane projection) is trying to represent the elevated, curved surface of the earth on a flat plane at sea level. The PDF was more prevalent before GPS became popular because total stations were the primary tools used for surveying. Projects were designed using the PDF. This allowed surveyors in the field to measure directly from the designed plans, without having to apply the PDF on the fly in the field.

Project Datum Factor Example

Project Datum Factor Example
0’s Inverse of PDF 1/

Project Datum Factor Example

Conversion Among Coordinate Systems
Exact or approximate mathematical formulas have been developed to convert to and from geographic coordinates (lat and long) to all commonly used coordinate projections Care must be taken when converting among projections that use different datums A datum transformation must be used to convert from one geographic coordinate system to another

Conversion Among Coordinate Systems
Inverse of PDF Inverse of PDF

Vertical Datums Many Vertical Datums GPS provides Elipsoid height

Summary Geographic Coordinate Systems Ellipsoids Geoid
Horizontal Datums Projected Coordinate Systems Project Datum Factors

References http://kartoweb.itc.nl/geometrics/index.html
Basic GIS Coordinates, Second Edition: Jan Van Sickle

Hopefully after this talk your project won’t look like this.
Questions? Hopefully after this talk your project won’t look like this.

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