Map Projections and Coordinate Systems

Surveying 101 for GIS Professionals2013 Kentucky GIS ConferenceJeremy Gould – Kentucky Transportation Cabinet

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 longitudeΦ= +/-degrees latitudeγ= + radial distance from center

Geographic Coordinate Systems

Equator0° latitude

Prime Meridian0° longitude

• 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

Ellipsoids

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)Gravity Recovery And Climate Experiment (GRACE)GravimetersGravimeters

Ellipsoids

There are Global Ellipsoids and Regional (local) EllipsoidsTwo 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 MassCenter

Approximately236 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)

z

DATUM 2

x

u

2 y

1

2

1

v

DATUM 1 P

w 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

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

Central Meridian(SF = k0)

Intersection of Ellipsoidand Projection Cylinder

(SF = 1)

Practical Limitof Projection

(SF k0)

SF > 1

SF < 1

Grid Origin

Axis ofCylinder

Polar Axis

Universal Transverse Mercator Coordinate System

Projected Coordinate Systems

KENTUCKY PROJECTIONSKENTUCKY PROJECTIONSUTM Zones 16 & 17UTM Zones 16 & 17Transverse Mercator (Secant Cylinder)Transverse Mercator (Secant Cylinder)

UTM Zone 16 UTM Zone 17

Projected Coordinate Systems

North StandardParallel (SF = 1)

South StandardParallel (SF = 1)

Parallel ofGrid Origin

(Base Parallel)

Central MeridianPolar Axis

• Lambert Conic Projection (Northern Hemisphere)

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 PROJECTIONSKENTUCKY PROJECTIONSNorth and South State PlaneNorth and South State PlaneLambert Conformal Conic (Secant Cone)Lambert Conformal Conic (Secant Cone)

State Plane South Zone

State PlaneNorth Zone

Projected Coordinate Systems

KENTUCKY SINGLE ZONE PROJECTIONKENTUCKY SINGLE ZONE PROJECTION

Projected Coordinate Systems

KENTUCKY SPCS – NORTH AND SOUTH ZONESKENTUCKY SPCS – NORTH AND SOUTH ZONES

SOUTH ZONE

NORTH ZONE

Projected Coordinate Systems

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 0 m 500,000 m

False Easting 1,500,000 m 500,000 m 500,000 m

Kentucky ProjectionsKentucky ProjectionsNAD83 State Plane Coordinate SystemNAD83 State Plane Coordinate System

(Lambert Conformal Conic)(Lambert Conformal Conic)

Linear unit of measure for all zones is the U.S. Survey Foot (USFt)(1 USFt = .3048006096012 meters)

Projected Coordinate Systems

COORDINATE SPACE COMPARISONCOORDINATE SPACE COMPARISON

0 m

250,000 m

500,000 m

750,000 m

1,000,000 m

1,250,000 m

1,500,000 m25

0,00

0 m

0 m

500,

000

m

750,

000

m

1,00

0,00

0 m

1,25

0,00

0 m

1,50

0,00

0 m

1,75

0,00

0 m

NAD'27 NORTH ZONE

NAD'27 SOUTH ZONE

NAD'83 NORTH ZONE

NAD'83 SOUTH ZONE

2,00

0,00

0 m

NO

RT

HIN

G

EASTING

NAD'83 SINGLE ZONE

4921245 ft

4101038 ft

328083 ft

2460623 ft

1640415 ft

820207 ft

8202

07 f

t

1640

415

ft

2460

623

ft

3280

83 f

t

4101

038

ft

4921

245

ft

5741

453

ft

6561

660

ft

0 ft

0 ft

Projected Coordinate Systems

Equatorial Plane

Topographic Surface(Ground)

Ellipsoid

E1

Axi

s of

Rot

atio

n

S1

G1

G2

E2

S2

G3

E3S3

E1 < G1 < S1

G2 < E2 < S2

E3 < S3 < G3Projection Grid

Nor

th S

tand

ard

Par

alle

l

South Standard P

arallel

E = Distance on ellipsoidG = Distance on gridS = Distance on surface

SF = Grid Scale Factor

= Geodetic latitude

SF=1

SF<1

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 PDF1/1.000059148

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

• 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/Survey-Coordination.

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.