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)

Datum

Spheroid

Prime

MeridianProjection

Parameters

Projection

Projected

Coordinate

System

Geographic

Coordinate

System

Coordinate System

Angular

Unit

Linear

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] ]

PROJCS A2

GEOGCS A

PROJCS A1

Projection

Conversion Pathways

(lon, lat)

(x, y)

(λ, φ)

PROJCS A2

GEOGCS A

PROJCS A1

GEOGCS B

PROJCS B1

Projection

Geographic Transformation

(lon, lat)

(x, y)

(λ, φ)

Conversion Pathways

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

10Figure 1.2

Geographic Coordinate Systems

11Figure 1.3

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”

SpheroidSphere

Figure 1.5 15

Background geometry

Circle: all axes are the same length

Ellipse: 2 axes

f = (a – b)/a (flattening)

e2 = (a2 – b2)/a2 (ellipticity

squared)

Se

mim

ino

r

ax

is (

b)

Semimajor axis (a)

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)

Earth’s SurfaceEarth-centered datum (WGS84)Local datum (NAD27)

Local datumcoordinate system

Earth-centered datumcoordinate system

What is a datum?

Classical geodesy (before 1960) – 5 quantitiesLatitude 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 constantsThree 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 DatumsDefined for countries, regions,

or the world

World: WGS84, WGS72

Regional:

ED50 (European Datum 1950)

Arc 1950 (Africa)

Countries:

GDA 1994 (Australia)

Tokyo

(gcs, geogcs)

Name (European Datum 1950)

Datum (European Datum 1950)

Spheroid (International 1924)

Prime Meridian (Greenwich)

Angular unit of measure (Degrees)

Geographic coordinate systems

Geographic transformations

“datum” transformations

Convert between GCS

Includes unit, prime meridian, and spheroid

changes

Defined in a particular direction

All are reversible

dXdY

X

dZY

Z

(0,0,0)

(145,-39,6)

Relationship between two datums

X Y

Z

Rotations

Transformation methods

Equation-based

Molodensky, Abridged Molodensky, Geocentric

Translation

Coordinate Frame, Position Vector, Molodensky-

Badekas

Longitude Rotation, 2D lat / lon offsets

File-based

NADCON, HARN, NTv2

Transformation exampleWGS 1984 (WGS 1984)

a = 6378137.0 meters

f = 1 / 298.257223563

e2 = 0.0066943799901…

European 1950 (International 1924)

a = 6378388.0

f = 1 / 297.0

e2 = 0.006722670022…

40 different geographic transformations

Geocentric Translation

Position Vector, Coordinate Frame

NTv2

Why so many?

Areas of use

Accuracy

Method Accuracies

NADCON 15 cm

HARN/HPGN 5 cm

CNT (NTv1) 10 cm

Seven parameter 1-2 m

Three parameter 4-5 m

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]]

Example of GT in WKT format

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

X -

Y +

X +

Y +

X -

Y -

X +

Y -

X

Data

usually here

Y

Visualize a light shining through the

Earth onto a surface

plane

This much earthsurface has to fitonto this muchmap surface . . .

therefore, much of the Earth surfacehas to be represented smaller thanthe nominal scale.

projection

Fitting sphere to plane causes stretching or

shrinking of features

Projecting Earth‟s surface always involves

distortion:

shape

area

distance

direction

More on projections

Projection properties

Conformalmaintains shape

Equal-areamaintains area

Equidistantmaintains distance

Directionmaintains 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

Tangent

projection surface touches spheroid

Secant

surface cuts through spheroid

No distortion at contact points

Increases away from contact points

More on

projection surfaces

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

X

Y

Central

Meridian:

X = 0

Latitude of

Origin:

Y = 0

false easting =

500,000

false northing =

10,000,000

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

GCS

PCS

Spatial Domain

12,000,000

4,000,000

10,000

90

180 10,000 4,000,000

UTM

(meters)

UTM

(feet)

Decimal

Degrees

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?