Projections and Coordinate Systems

Overview

Projections

Examples of different projections

Coordinate systems

Datums

Projections

Overview

• GIS must accurately represent locations of features

from the earth’s surface on a map

• This requires a reference system, or coordinate

system

Projections and Coordinate

Systems

The earth is a spheroid

The best model of the earth is a globe

Drawbacks:

•not easy to carry

•not good for making

planimetric measurement

(distance, area, angle)

Maps are flat

easy to carry

good for measurement

scaleable

A map projection is a method

for mapping spatial patterns

on a curved surface (the

Earth’s surface) to a flat

surface.

Definitions I

• Coordinate System (CS) – provides a frame of

reference to define locations

– Geographic CS – Used for 3D (sphere or globe),

locations defined by latitude and longitude, usually in

decimal degrees

– Projected CS – Used for 2D (maps), locations

defined by x,y measured from some origin (0,0),

usually in meters, sometimes in feet

Definitions II

• Spheroid – A simplified model of the shape of the earth

– Spheroid is used interchangeably with ellipsoid

• Datum – Defines the spheroid being used and aligns the

spheroid, optimizing fit either for a specific area, or

overall globally. This also defines the origin point of the

coordinate system.

Introductio

Definitions III

• Projection – A mathematical equation to convert from

spherical (3-D) to planar (2-D) coordinates

– Every projection will cause some form of spatial

distortion in shape, area, distance, and/or direction

– Common projections used in the US:

� Albers Equal Area for national scale

� State Plane and UTM for state and local scales

North American Datums

• Common North American datums are:

– NAD27, Clarke 1866 spheroid and Meade’s Ranch, KS as

the origin point

– NAD83, GRS80 (GRS = Geodetic Reference System)

spheroid and the earth’s center of mass as it’s origin

– WGS84, WGS84 spheroid and the earth’s center of mass

as it’s origin. This datum is used globally

• In North America, WGS84 is virtually identical to NAD83

– Other datum changes will change coordinates of spatial

features (e.g. NAD83 to NAD27)

Types of Projections

• Imagine a globe with a light source inside that projects

features on the earth’s surface onto a flat surface

• That flat surface can be configured a number of ways – as a

cylinder, a cone, or a plane

�an imaginary light is “projected” onto a “developable surface”

�a variety of different projection models exist

secant cone

tangent cone

cone as developable surface

tangent cylinders

cylinder as developable surface

plane as developable surface

Map projections always introduce error

and distortion

Map projections always introduce error

and distortion

Map projections always introduce error

and distortion

Map projections always introduce error

and distortion

Distortion may be minimized in one or

more of the following properties:

o Shape … conformal

o Distance … equidistant

o Direction … true direction

o Area … equal area

input output

unprojected projected

angles (lat/long) Cartesian coordinates

Exactly what are map projections?

Sets of mathematical equations that convert coordinates from one system to another

y)(x,y)(x, ff

→

How do projections work on a

programmatic level?

o each set of "coordinates" is transformed using a specific

projection equation from one system to another

o angular measurements can be converted to Cartesian

coordinates

o one set of Cartesian coordinates can be converted to a

different measurement framework

Projection, zone, datum (units) X Y

geographic, NAD27 (decimal degrees) -122.35° 47.62°

UTM, Zone 10, NAD27 (meters) 548843.5049 5274052.0957

State Plane, WA-N, NAD83 (feet) 1266092.5471 229783.3093

Geographic “projection”

Examples of different projections

• Albers

(Conic)

Shape Shape along the standard parallels is accurate and minimally distorted in the region between

the standard parallels and those regions just beyond. The 90-degree angles between

meridians and parallels are preserved, but because the scale along the lines of longitude

does not match the scale along lines of latitude, the final projection is not conformal.

Area All areas are proportional to the same areas on the Earth.

Direction Locally true along the standard parallels.

Distance Distances are best in the middle latitudes. Along parallels, scale is reduced between the

standard parallels and increased beyond them. Along meridians, scale follows an opposite

pattern.

Examples of different projections

• Lambert

Azimuthal

Equal

Area

(Planar)

Shape Shape is true along the standard parallels of the normal aspect (Type 1), or the standard

lines of the transverse and oblique aspects (Types 2 and 3). Distortion is severe near the

poles of the normal aspect or 90° from the central line in the transverse and oblique

aspects.

Area There is no area distortion on any of the projections.

Direction Local angles are correct along standard parallels or standard lines. Direction is distorted

elsewhere.

Distance Scale is true along the Equator (Type 1), or the standard lines of the transverse and oblique

aspects (Types 2 and 3). Scale distortion is severe near the poles of the normal aspect or

90° from the central line in the transverse and oblique aspects.

Examples of different projections

•Mercator(Cylindrical)

Shape Conformal. Small shapes are well represented because this projection maintains the local angular

relationships.

Area Increasingly distorted toward the polar regions. For example, in the Mercator projection, although

Greenland is only one-eighth the size of South America, Greenland appears to be larger.

Direction Any straight line drawn on this projection represents an actual compass bearing. These true direction

lines are rhumb lines, and generally do not describe the shortest distance between points.

Distance Scale is true along the Equator, or along the secant latitudes.

Examples of different projections

•Miller(Cylindrical)

Shape Minimally distorted between 45th parallels, increasingly toward the poles. Land masses are stretched

more east to west than they are north to south.

Area Distortion increases from the Equator toward the poles.

Direction Local angles are correct only along the Equator.

Distance Correct distance is measured along the Equator.

Examples of different projections

• Mollweide

(Pseudo-

cylindrical)

Shape Shape is not distorted at the intersection of the central meridian and latitudes 40° 44' N and S.

Distortion increases outward from these points and becomes severe at the edges of the projection.

Area Equal-area.

Direction Local angles are true only at the intersection of the central meridian and latitudes 40° 44' N and S.

Direction is distorted elsewhere.

Distance Scale is true along latitudes 40°44' N and S. Distortion increases with distance from these lines and

becomes severe at the edges of the projection.

Examples of different projections

• Orthographic

Shape Minimal distortion near the center; maximal distortion near the edge.

Area The areal scale decreases with distance from the center. Areal scale is zero at the edge of the

hemisphere.

Direction True direction from the central point.

Distance The radial scale decreases with distance from the center and becomes zero on the edges. The scale

perpendicular to the radii, along the parallels of the polar aspect, is accurate.

Coordinate Systems

Coordinates

� Features on spherical surfaces are not easy to

measure

� Features on planes are easy to measure and

calculate

� distance

� angle

� area

� Coordinate systems provide a measurement

framework

Coordinates

�Lat/long system measures angles on

spherical surfaces

�60º east of PM

�55º north of equator

Lat/long values are NOT Cartesian (X, Y)

coordinates

constant angular deviations do not have constant

distance deviations

1° of longitude at the equator ≠ 1° of longitude

near the poles

Coordinate systems

Examples of different coordinate/projection

systems

� State Plane

� Universal Transverse Mercator (UTM)

Coordinate systems

State Plane

� Codified in 1930s

� Use of numeric zones for shorthand

� SPCS (State Plane Coordinate System)

� FIPS (Federal Information Processing System)

� Uses one or more of 3 different projections:

� Lambert Conformal Conic (east-west orientation )

� Transverse Mercator (north-south orientation)

� Oblique Mercator (nw-se or ne-sw orientation)

Coordinate systems

projection.ppt 42

Pennsylvania State Plane zone definitions

ZoneSPCS

Zone #

FIPS

Zone #Projection

1st Std.

Parallel

2nd Std.

Parallel

Central

MeridianOrigin

False

Easting (m)

False

Northing

(m)

PA_N 5126 3701 Lambert Conformal Conic 40 10 00 40 53 00 -77 45 00 47 00 00 600000 0

PA_S 5151 3702 Lambert Conformal Conic 39 56 00 40 58 00 -120 30 00 39 20 00 600000 0

Coordinate systems

Universal Transverse Mercator

(UTM)

� Based on the Transverse

Mercator projection

� 60 zones (each 6° wide)

� false eastings

� Y-0 set at south pole or equator

Coordinate systems

Every place on earth falls in a particular zone

Pennsylvania is has 2 UTM Zones (18 and 19)

Universal Transverse Mercator (UTM)

Scale

A .30 mm line on a 1:200,000 scale map is almost 2,000 ft on the ground

Map measurement and true ground measurement

( ) ( ) 50' * * 0.025"12"1'

1"

24,000" =

A 1/40th in line on a 1:24,000 scale map is 50 ft on the ground

( ) ( ) ( ) ( ) 1,969' * * * * mm 0.3012"1'

2.54cm1"

mm 101cm

1

2,000,000 =

Scale

Map Scale

Scale refers to the relationship or ratio between a distance on a map and the

distance on the earth it represents. Maps should display accurate distances

and locations, and should be in a convenient and usable size.

Map scales can be expressed as

representative fraction or ratio: 1:100,000 or 1/100,000

graphical scale:

verbal-style scale:

1 inch in map equal to 2000 feet on the ground or

1 inch = 2000 feet

GIS is Scale less

In GIS, the scale can be easily enlarged and

reduced to any size that is appropriate.

However, if we get farther and farther from the

original scale of the layer, problems appear:

details no appear in an enlarged map

too dense in a reduced map

The scale of the original map determines the largest

map scale at which the data can be used.

Road map 1:50,000 scale can NOT be used

accurately at the 1:24,000 scale.

Water coverage at 1:250,000 scale can NOT be

used accurately at the 1:50,000 scale

Scale in attention

1:100,000 scale data from USGS DLG

Data from different sources and scales can vary widely

Scale in attention

1:1,000,000 scale data from DCW (DMA)

Data from different sources and scales can vary widely

Scale in attention

1:2,000,000 scale data from USGS DLG

Data from different sources and scales can vary widely

Scale in attention

one inch on the map equals 200 inches on the ground?

or

one inch on the map equals 200 feet on the ground?

“one to two-hundred”: does this mean

Beware of scale statements

Projections in ArcGIS

Projections in ArcGIS

• All geographic datasets have a GCS

• Many also have a PCS

• Unprojected (GCS) data is generally inappropriate for GIS

analyses

• The first layer added to ArcMap defines the projection and

datum for the data frame. This can be changed in the data

frame properties.

• All subsequent layers are projected on-the-fly (if necessary)

to match the data frame projection. The GCS of the layer

must match the data frame or a transformation is required.

How does ArcGIS handle map projections in data frames?

o Project data frames to see or measure features under different projection parameters

o Applying a projection on a data frame projects data “on the fly.”

o ArcGIS’s data frame projection equations can handle any input projection.

o However, sometimes on-the-fly projected data do not properly overlap.

Applying a projection to a data frame is like putting on a pair of glasses

You see the map differently, but the data have

not changed

How does ArcGIS handle map projections

for data?

� Projecting data creates a new data set on the file

system

� Data can be projected so that incompatibly projected

data sets can be made to match.

� ArcGIS’s projection engine can go in and out of a large

number of different projections, coordinate systems, and

datums.

Projection Metadata

• Stored in:

– A .prj file for shapefiles

– A table in geodatabases

– A prj.adf in coverages and grids

– A header for other raster formats. This can be either a

separate file (.hdr, .tifw) or embedded in the raster file

itself

• Data have a GCS and may have a PCS even if the

metadata is missing. These data have an unknown

coordinate system.

Projection Metadata

• CS information can be viewed in ArcMap (layer

properties>source tab) and ArcCatalog (Description tab)

Data with Unknown

Coordinate Systems

• Cannot be projected on-the-fly

• Will generate a warning when added to ArcMap

• Layer will be added, but may not align with other data

• Can be fixed with Define Projection tool if you know the CS

• If you don’t, you can try:

– contacting the source

– adding the layer to a map with a known CS. If the new

layer is registered with existing layers, it is the same CS.

Projecting Raster Data

• Because projections cause distortions in size and shape, and raster data must have square cells, cells in projected rasters represent different areas on the surface of the earth

• Resampling is used to assign values to projected cells.

– Nearest neighbor is fastest and must be used for categorical data like land cover

– Bilinear interpolation uses a weighted distance average of surrounding cells. Smooths values moderately, useful for continuous data like elevation

– Cubic convolution fits a curve through surrounding points. Can smooth data significantly and is slower

Map Units vs. Display Units

• Coordinates of the dataset are stored in map units.

Geographic generally uses decimal degrees, while a PCS

will usually use meters or feet. Map units can only be

changed by changing the CS of the data (Project tool).

• Display units are independent of map units and are set in

the data frame properties. ArcMap reports coordinate

values and measurements in display units.

Points to Remember

• Choose a PCS for each project. When new data is

acquired, be sure it is in that PCS or convert it using the

Project tool. It is best not to rely on projection on-the-fly in

ArcMap. It is especially important to have all data in the

same PCS when doing analyses.

• Try and avoid changing the CS of raster data

• Use the Project tool to change the CS of data, but use the

Define Projection tool to add metadata about the CS

• CS should be defined for all spatial data