Map Projections and Coordinate
Systems
Surveying 101 for GIS Professionals
2013 Kentucky GIS Conference
Jeremy Gould – Kentucky Transportation Cabinet
Agenda
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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
°
°
°
°
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
CLARKE 1866 Ellipsoid
(NAD’27)
Earth Mass
Center
GEOID
GRS80 Ellipsoid
(NAD’83)
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
(-85.31 °, 37.55 °)
Datum 2 may have a long-lat of
(-85.30 °, 37.54 °)
The same point has different coordinates
because of the shift/rotation of the ellipsoid
w
P
DATUM 1
z
1
1
v
u
2
DATUM 2
2
x
y
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)
Polar Axis
Central Meridian
(SF = k0)
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)
Central Meridian
Polar Axis
North Standard
Parallel (SF = 1)
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
North Zone
State Plane South Zone
Projected Coordinate Systems
KENTUCKY SINGLE ZONE PROJECTION
S in g le Z o n e (a ll co u n tie s)
S in g le Z o n e co ve ra g e a re a (e n tire sta te ).
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
85° 45’ 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
36° 20’ N
False Northing
1,000,000 m
0m
500,000 m
False Easting
1,500,000 m
500,000 m
500,000 m
Linear unit of measure for all zones is the U.S. Survey Foot (USFt)
(1 USFt = .3048006096012 meters)
Projected Coordinate Systems
COORDINATE SPACE COMPARISON
1,500,000 m
4921245 ft
NAD'83 SINGLE ZONE
1,000,000 m
328083 ft
750,000 m
2460623 ft
NAD'83 SOUTH ZONE
500,000 m
1640415 ft
NAD'83 NORTH ZONE
250,000 m
820207 ft
NAD'27 NORTH ZONE
EASTING
2,000,000 m
6561660 ft
1,750,000 m
5741453 ft
1,500,000 m
4921245 ft
1,250,000 m
4101038 ft
1,000,000 m
328083 ft
750,000 m
2460623 ft
500,000 m
1640415 ft
250,000 m
820207 ft
NAD'27 SOUTH ZONE
0m
0 ft
0m
0 ft
NORTHING
1,250,000 m
4101038 ft
Projected Coordinate Systems
S1
E1 < G1 < S1
G1
G2 < E2 < S2
SF = Grid Scale Factor
S2
E1
E2
SF>1
SF=1
Axis of Rotation
E = Distance on ellipsoid
G = Distance on grid
S = Distance on surface
 = Geodetic latitude
Ellipsoid
G2
Topographic Surface
(Ground)
SF<1
Projection Grid
E3 < S3 < G3
SF=1
SF>1
G3
S3
E3
Equatorial Plane
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
1/1.000059148
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
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Geographic Coordinate Systems
Ellipsoids
Geoid
Horizontal Datums
Projected Coordinate Systems
Project Datum Factors
References
• http://kartoweb.itc.nl/geometrics/index.html
• http://resources.arcgis.com/en/help/main/10.1/index.html#//003r0000000
1000000
• http://training.esri.com/gateway/index.cfm?fa=catalog.webCourseDetail&
courseid=24
• http://transportation.ky.gov/Highway-Design/Pages/SurveyCoordination.aspx
• http://www.agc.army.mil/Missions/Corpscon.aspx
• Basic GIS Coordinates, Second Edition: Jan Van Sickle
• http://www.esri.com/news/arcuser/0703/geoid1of3.html
Questions?
Hopefully after this talk your project won’t look like this.