Coordinate Systems, Projections,
and Transformations
An Overview
Outline
Acronyms, Terminology
Coordinate System Components
Conversion between Coordinate
Systems
The PEAL (Projection Engine Acronym List)
PE (Projection Engine)
GCS, GEOGCS (Geographic Coordinate System)
PCS, PROJCS (Projected Coordinate System)
VCS, VERTCS (Vertical Coordinate System)
GT, GEOGTRAN (Geographic Transformation)
VT, VERTTRAN (Vertical Transformation)
WKT (Well-Known Text String)
EPSG (European Petroleum Survey Group)
(www.epsg.org)
Coordinate System
Projected
Coordinate
System
Projection
Projection
Parameters
Geographic
Coordinate
System
Datum
Linear
Unit
Spheroid
Prime
Meridian
Angular
Unit
Well-Known Text String
PROJCS["Test",GEOGCS["GCS_WGS_1984",DATUM[
"D_WGS_1984",SPHEROID["WGS_1984",6378137,29
8.257223]],PRIMEM["Greenwich",0.0],UNIT["Degree",0.
0174532925199433]],PROJECTION["Mercator"],PARA
METER["Central_Meridian",120.0],PARAMETER["Stan
dard_Parallel_1",0.0],PARAMETER["False_Easting",10
00000.0],PARAMETER["False_Northing",0.0],UNIT["Fo
ot",0.3048]]
Well-Known Text String
PROJCS[ "Test",
GEOGCS[ "GCS_WGS_1984",
DATUM[ "D_WGS_1984",
SPHEROID[ "WGS_1984", 6378137.0, 298.257223563] ],
PRIMEM[ "Greenwich", 0.0],
UNIT[ "Degree", 0.0174532925199433] ],
PROJECTION[ "Mercator" ],
PARAMETER[ “Central_Meridian“, -120.0],
PARAMETER[ “Standard_Parallel_1”, 0.0],
PARAMETER[ “False_Easting”, 1000000.0],
PARAMETER[ “False_Northing”, 0.0],
UNIT[ "Foot", 0.3048] ]
Conversion Pathways
PROJCS A1
PROJCS A2
(x, y)
Projection
GEOGCS A
(lon, lat)
(λ, φ)
Conversion Pathways
PROJCS A1
PROJCS A2
PROJCS B1
(x, y)
Projection
GEOGCS A
GEOGCS B
Geographic Transformation
(lon, lat)
(λ, φ)
Units, Spheroids, Prime Meridians
Angular - UNIT["Degree", 0.0174532925199433]
UNIT[“Grad”, 0.01570796326794897]
The value is Radians / Unit
Linear - UNIT["Foot", 0.3048]
The value is Meters / Unit
SPHEROID[ "WGS_1984", 6378137.0, 298.257223563]
The values are Semi-Major axis length in Meters,
Inverse Flattening (1 / f)
Prime Meridian – PRIMEM[“Paris”, 2.337229166666667]
PRIMEM[“Greenwich”, 0.0]
The value is Decimal Degrees based on Greenwich
Geographic Coordinate Systems
Figure 1.2
10
Figure 1.3
11
Distances and Angular Units
Figure 1.4
13
More background geometry
Rotating a circle or ellipse creates a sphere or
spheroid (oblate ellipsoid of revolution)
Defines the size and shape of the Earth “model”
Sphere
Spheroid
Figure 1.5
15
Circle: all axes are the same length
Ellipse: 2 axes
f = (a – b)/a (flattening)
e2 = (a2 – b2)/a2 (ellipticity
squared)
Semimajor axis (a)
Semiminor
axis (b)
Background geometry
Figure 1.7
17
Earth as sphere
simplifies math
small-scale maps
(less than 1:5,000,000)
Earth as spheroid
maintains accuracy for larger-scale maps (>
1:1,000,000)
Datums
Reference frame for locating points on Earth‟s
surface
Defines origin & orientation of latitude/longitude
lines
Defined by spheroid and spheroid‟s position
relative to Earth‟s center
Creating a Datum
Pick a spheroid
Pick a point on the Earth‟s surface
All other control points are located relative to the
origin point
The datum‟s center may not coincide with the
Earth‟s center
Two types of datums
Earth-centered
(WGS84, NAD83)
Local
(NAD27, ED50)
Local datum
coordinate system
Earth-centered datum
coordinate system
Earth’s Surface
Earth-centered datum (WGS84)
Local datum (NAD27)
What is a datum?
Classical geodesy (before 1960) – 5 quantities
Latitude of an initial point
Longitude of an initial point
Azimuth of a line from this point
Semi-major axis length and flattening of ellipsoid
Satellite geodesy (after 1960) – 8 constants
Three to specify the origin of the coordinate system
Three to specify the orientation of the coordinate system
Semi-major axis length and flattening of ellipsoid
Why so many datums?
Many estimates of Earth‟s size and shape
Improved accuracy
Designed for local regions
North American Datums
NAD27
Clarke 1866 spheroid
Meades Ranch, KS
1880‟s
NAD83
GRS80 spheroid
Earth-centered datum
GPS-compatible
North American Datums
HPGN / HARN
GPS readjustment of NAD83 in the US
Also known as „NAD91‟ or „NAD93‟
27 states & 2 territories (42 states in PE)
NAD27 (1976) & CGQ77
Redefinitions for Ontario and Quebec
NAD83 (CSRS98) – GPS readjustment
International Datums
Defined for countries, regions,
or the world
World: WGS84, WGS72
Regional:
ED50 (European Datum 1950)
Arc 1950 (Africa)
Countries:
GDA 1994 (Australia)
Tokyo
Geographic coordinate systems
(gcs, geogcs)
Name (European Datum 1950)
Datum (European Datum 1950)
Spheroid (International 1924)
Prime Meridian (Greenwich)
Angular unit of measure (Degrees)
Geographic transformations
“datum” transformations
Convert between GCS
Includes unit, prime meridian, and spheroid
changes
Defined in a particular direction
All are reversible
Relationship between two datums
Z
(145,-39,6)
dZ
(0,0,0)
dX
dY
X
Y
Rotations
Z
X
Y
Transformation methods
Equation-based
Molodensky, Abridged Molodensky, Geocentric
Translation
Coordinate Frame, Position Vector, MolodenskyBadekas
Longitude Rotation, 2D lat / lon offsets
File-based
NADCON, HARN, NTv2
Transformation example
European 1950 (International 1924)
a = 6378388.0
f = 1 / 297.0
e2 = 0.006722670022…
WGS 1984 (WGS 1984)
a = 6378137.0 meters
f = 1 / 298.257223563
e2 = 0.0066943799901…
40 different geographic transformations
Geocentric Translation
Position Vector, Coordinate Frame
NTv2
Why so many?
Areas of use
Accuracy
Method Accuracies
NADCON
HARN/HPGN
CNT (NTv1)
Seven parameter
Three parameter
15 cm
5 cm
10 cm
1-2 m
4-5 m
Example of GT in WKT format
GEOGTRAN["ED_1950_To_WGS_1984_23",
GEOGCS["GCS_European_1950",
DATUM["D_European_1950",
SPHEROID["International_1924",6378388.0,297.0]],
PRIMEM["Greenwich",0.0],
UNIT["Degree",0.0174532925199433]],
GEOGCS["GCS_WGS_1984",
DATUM["D_WGS_1984",
SPHEROID["WGS_1984",6378137.0,298.257223563]],
PRIMEM["Greenwich",0.0],
UNIT["Degree",0.0174532925199433]],
METHOD["Position_Vector"],
PARAMETER["X_Axis_Translation",-116.641],
PARAMETER["Y_Axis_Translation",-56.931],
PARAMETER["Z_Axis_Translation",-110.559],
PARAMETER["X_Axis_Rotation",0.893],
PARAMETER["Y_Axis_Rotation",0.921],
PARAMETER["Z_Axis_Rotation",-0.917],
PARAMETER["Scale_Difference",-3.52]]
ED50 versus WGS84
Figure 1.17
37
Figure 1.18
38
Map Projections
mathematical conversion of
3-D Earth to a 2-D surface
Longitude / Latitude to X, Y
(l, j)
(x, y)
Projected coordinate system
Linear units
Shape, area, and
distance may be
distorted
Y
Data
XY+
X+
Y+
usually here
X
XY-
X+
Y-
Visualize a light shining through the
Earth onto a surface
Fitting sphere to plane causes stretching or
shrinking of features
This much earth
surface has to fit
onto this much
map surface . . .
projection
plane
therefore, much of the Earth surface
has to be represented smaller than
the nominal scale.
More on projections
Projecting Earth‟s surface always involves
distortion:
shape
area
distance
direction
Projection properties
Conformal
maintains shape
Equal-area
maintains area
Equidistant
maintains distance
Direction
maintains some directions
Projection surfaces
Cones, Cylinders, Planes
Can be flattened without distortion
A point or line of contact is created when surface
is combined with a sphere or spheroid
More on
projection surfaces
Tangent

projection surface touches spheroid
Secant

surface cuts through spheroid
No distortion at contact points
Increases away from contact points
Conic Projections
Best for mid-latitudes with
an East-West orientation.
Tangent or secant along 1
or 2 lines of latitude known
as „standard parallels‟.
Cylindrical projections
Best for equatorial or
low latitudes
Rotate cylinder to
reduce distortion
along a line
Planar projections
Best for polar or circular regions
Direction always true from center
Shortest distance from center to another
point is a straight line
Also called azimuthal or zenithal
Can be any aspect
Projection parameters
Central meridian
Longitude of origin
Longitude of center
Latitude of origin, Latitude of center
Standard parallel
Scale factor
False easting, False northing
Y
false easting =
500,000
false northing =
10,000,000
X
Latitude of
Origin:
Y=0
Central
Meridian:
X=0
Choosing a
coordinate system
Often mandated by organization
Thematic = equal-area
Presentation = conformal (also
equal-area)
Navigation = Mercator, true
direction or equidistant
More on choosing
a coordinate system
Extent
Location
Predominant extent
Projection supports
spheroids/datums?
Web Mercator
Online mapping services use a sphere-only
Mercator
Two ways to emulate it
Sphere-based GCS
Projection that can force sphere equations
Mathematically EQUAL
PROJCS["WGS_1984_Web_Mercator",
GEOGCS["GCS_WGS_1984_Major_Auxiliary_Sphere",
DATUM["D_WGS_1984_Major_Auxiliary_Sphere",
SPHEROID["WGS_1984_Major_Auxiliary_Sphere",
6378137.0, 0.0]],
PRIMEM["Greenwich", 0.0],
UNIT["Degree", 0.0174532925199433]],
PROJECTION["Mercator"],
PARAMETER["False_Easting", 0.0]
PARAMETER["False_Northing", 0.0],
PARAMETER["Central_Meridian", 0.0],
PARAMETER["Standard_Parallel_1", 0.0],
UNIT["Meter", 1.0]]
# 102113
PROJCS["WGS_1984_Web_Mercator_Auxiliary_Sphere",
GEOGCS["GCS_WGS_1984",
DATUM["D_WGS_1984",
SPHEROID["WGS_1984",6378137.0, 298.257223563]],
PRIMEM["Greenwich", 0.0],
UNIT["Degree", 0.0174532925199433]],
PROJECTION["Mercator_Auxiliary_Sphere"],
PARAMETER["False_Easting", 0.0],
PARAMETER["False_Northing", 0.0],
PARAMETER["Central_Meridian", 0.0],
PARAMETER["Standard_Parallel_1", 0.0],
PARAMETER["Auxiliary_Sphere_Type", 0.0],
UNIT["Meter", 1.0]]
#3857
(old #102100)
Mercator Equations (std parallel at equator)
Sphere
x = R(λ – λ0)
y = ln(tan(π/4 + φ/2))
Spheroid
x = a(λ – λ0)
y = ln(tan(π/4 + χ/2) where conformal latitude
χ = 2 arctan{tan(π/4 + φ/2)[(1-e sin φ)/(1+e sin φ)]e/2} - π/2
UTM
Universal Transverse Mercator
60 zones, 6° wide
Transverse Mercator
zone 1, central meridian = -177
scale factor = 0.9996
false easting = 500,000 m
In Southern Hemisphere, false northing =
10,000,000 m
SPCS
State Plane Coordinate System
States have 1 or more zones
Uses either NAD27 or NAD83 datums
Uses Lambert Conic, Transverse Mercator, and
Oblique Mercator
Horizons
PCS
GCS
Spatial Domain
12,000,000
UTM
(feet)
4,000,000
UTM
(meters)
10,000
Decimal
Degrees
90
180
10,000
4,000,000
When is a foot not a foot?
esriUnits is limited
ArcMap, “Foot” is US survey foot
US survey foot, 1 ft = 0.3048006096012192 m
Int‟l foot, 1 ft = 0.3048 m
PE, “Foot” is Int‟l foot; “Foot_US” is US survey
foot
9002 UNIT["Foot", 0.3048]
9003 UNIT["Foot_US", 0.3048006096012192]
9005 UNIT["Foot_Clarke", 0.304797265]
9041 UNIT["Foot_Sears", 0.3047994715386762]
9051 UNIT["Foot_Benoit_1895_A", 0.3047997333333333]
9061 UNIT["Foot_Benoit_1895_B", 0.3047997347632708]
9070 UNIT["Foot_1865", 0.3048008333333334]
9080 UNIT["Foot_Indian", 0.3047995102481469]
9081 UNIT["Foot_Indian_1937", 0.30479841]
9082 UNIT["Foot_Indian_1962", 0.3047996]
9083 UNIT["Foot_Indian_1975", 0.3047995]
9094 UNIT["Foot_Gold_Coast", 0.3047997101815088]
9095 UNIT["Foot_British_1936", 0.3048007491]
9300 UNIT["Foot_Sears_1922_Truncated", 0.3047993333333334]
Closing
We‟ve only scratched the surface!
GCS != PCS
Geographic transformations are vital
Measurement in degrees in meaningless
Questions?