Map projections and datums

Maps are flat

flat

Earth is curved

Map Distortion

• No map is as good as a globe.

• A map can show some of these features – True direction or azimuth– True angle– True distance– True area– True shape

But not all of them!

Coordinate System

common frame of reference

for all data on a map

GIS needs Coordinate Systems to:

• perform calculations

• relate one feature to another

• specify position in terms of distances and directions from fixed points, lines, and surfaces

Coordinate Systems

Cartesian coordinate systems: perpendicular distances and directions from fixed axes define positions

Polar coordinate systems: distance from a point of origin and an angle define positions

Each coordinate system uses a different model to map the Earth’s surface to a plane

GCSGeographic Coordinate Systems

• Degrees of latitude and longitude

• Spherical polar coordinate system

• “Real” distance varies

Spherical Coordinates

• Any point uniquely defined by angles passing through the center of the sphere

Equator

Meridian

The Graticule

• Map grid (lines of latitude and longitude)

• A transformation of Earth’s surface to a plane, cylinder or cone that is unfolded to a flat surface

Decimal Degrees

30º 30' 0" = 30.5º

42º 49' 50" = 42.83º

35º 45' 15" = ?35.7541

Standard Geographic Features

Parallels of Latitude

Equator

Slicing the Earth into pieces

Measuring Parallels

Give the slices values

Lines of Longitude

Establish a way of slicing the Earth from pole to pole

Antimeridian AMeridian A

Prime Meridian

Establishes an orthogonal way of slicing the earth

Longitude

Values of pole-to-pole slices

North America

Earth Grid

Comparing the parallels and the lines

Latitude and Longitude

Combining the parallels and the lines

Grid for US

Parallels and Lines for USWhat is wrong with this map?

Sphere vs. EllipsoidGlobes versus Earth

Shape of the Earth

• Approximated by an ellipsoid

• Rotate an ellipse about its minor axis = earth’s axis of rotation

• Semi-major axis a = 6378 km

• Semi-minor axis b = 6356 km

b

a

NP

SP

Ellipsoids and Geoids

• The rotation of the earth generates a centrifugal force that causes the surface of the oceans to protrude more at the equator than at the poles.

• This causes the shape of the earth to be an ellipsoid or a spheroid, and not a sphere.

• The nonuniformity of the earth’s shape is described by the term geoid. The geoid is essentially an ellipsoid with a highly irregular surface; a geoid resembles a potato or pear.

The Ellipsoid

The ellipsoid is an approximation of the Earth’s shape that does not account for variations caused by non-uniform density of the Earth.

ExamplesClarke 1866 Clarke 1880

GRS80 WGS60

WGS66 WGS72

WGS84 Danish

Satellite measurements have led to the use of geodetic datums WGS-84 (World Geodetic System) and GRS-1980 (Geodetic Reference System) as the best ellipsoids for the entire geoid.

The Geoid

• The maximum discrepancy between the geoid and the WGS-84 ellipsoid is 60 meters above and 100 meters below.

• Because the Earth’s radius is about 6,000,000 meters (~6350 km), the maximum error is one part in 100,000.

Geodetic Datums

Geodetic Datum

• Defined by the reference ellipsoid to which the geographic coordinate system is linked

• The degree of flattening f (or ellipticity, ablateness, or compression, or squashedness)

• f = (a - b)/a

• f = 1/294 to 1/300

Geodetic Datums

• A datum is a mathematical model

• Provide a smooth approximation of the Earth’s surface.

• Some Geodetic DatumsWGS60 WGS66 Puerto Rico Indian 1975 Potsdam

South American 1956

Tokyo Old Hawaiian

European 1979

Bermuda 1957

Common U S Datums

• NAD27 North American Datum 1927

• NAD83 North American Datum 1983

• WGS84 World Geodetic System 1984

(based on NAD83)

Map Projections

Concept of the Light Source

Making a Map

Projection Families

Types of Projection Families

Standard Point/Line for Projection

Regular Azimuthal

Azimuthal Projections

• Shapes are distorted everywhere except at the center • Distortion increases from center• True directions can be plotted from the center

outward• Distances are accurate from the center point

Azimuthal Projections

• A series of conic projections stacked together• Have curved rather than straight meridians• Not good choice for tiles across large areas

Polyconic Projections

• Good choice for mid-latitude regions of greater east-west than north-south extent

• Scale factor along two standard parallels is 1.0000

• Scale is reduced between the two standard parallels and increased north or south of the two standard parallels

Albers Conic Equal Area Projections

Equal Area Projections

• Projections that preserve area are called equivalent or equal area.

• Equal area projections are good for small scale maps (large areas)

• Examples: Mollweide and Goode• Equal-area projections distort the shape of

objects

Conformal Map Projections

• Projections that maintain local angles are called conformal.

• Conformal maps preserve angles • Conformal maps show small features

accurately but distort the shapes and areas of large regions

• Examples: Mercator, Lambert Conformal Conic

Conformal Map Projections

• The area of Greenland is approximately 1/8 that of South America. However on a Mercator map, Greenland and South America appear to have the same area.

• Greenland’s shape is distorted.

Map Projections

• For a tall area, extended in north-south direction, such as Idaho, you want longitude lines to show the least distortion.

• You may want to use a coordinate system based on the Transverse Mercator projection.

Map Projections

• For wide areas, extending in the east-west direction, such as Nebraska, you want latitude lines to show the least distortion.

• Use a coordinate system based on the Lambert Conformal Conic projection.

Map Projections

• For a large area that includes both hemispheres, such as North and South America, choose a projection like Mercator.

• For an area that is circular, use a normal planar (azimuthal) projection

The UTM System

Universal Transverse Mercator

• 1940s, US Army

• 120 zones (coordinate systems) to cover the whole world

• Based on the Transverse Mercator Projection

• Sixty zones, each six degrees wide

UTM Zones

• Zone 1

Longitude Start and End 180 W to 174 WLinear Units MeterFalse Easting 500,000False Northing 0Central Meridian 177 WLatitude of Origin EquatorScale of Central Meridian 0.9996

UTM Zones

• Zone 2

Longitude Start and End 174 W to 168 WLinear Unit MeterFalse Easting 500,000False Northing 0Central Meridian 171 WLatitude of Origin EquatorScale of Central Meridian 0.9996

UTM Zones

• Zone 13, Colorado, Nebraska Panhandle, etc.

Longitude Start and End 108 W to 102 WLinear Unit MeterFalse Easting 500,000False Northing 0Central Meridian 105 WLatitude of Origin Equator