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
But not all of them!
True shape
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
GCS
Geographic 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
Meridian

Equator

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
Antimeridian A
Lines of Longitude
Meridian A
Establish a way of slicing the Earth from pole to pole
Prime Meridian
Establishes an orthogonal way of slicing the earth
Longitude
North America
Values of pole-to-pole slices
Earth Grid
Comparing the parallels and the lines
Latitude and Longitude
Combining the parallels and the lines
Grid for US
What is wrong with this map?
Parallels and Lines for US
Sphere vs. Ellipsoid
Globes
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
NP
b
a
SP
Ellipsoids and Geoids
• The rotation of the earth generates a centrifugal
force that causes the surface of the oceans to
protrude (swell) 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.
Examples
Clarke 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 Datums
WGS60
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
Making a Map
Concept of the Light Source
Projection Families
Types of Projection Families
Standard Point/Line for
Projection
Regular Azimuthal
Azimuthal Projections
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
Polyconic Projections
• A series of conic projections stacked together
• Have curved rather than straight meridians
• Not good choice for tiles across large areas
Albert’s Equal Area Conic
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
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
Linear Units
False Easting
False Northing
Central Meridian
Latitude of Origin
Scale of Central Meridian
180 W to 174 W
Meter
500,000
0
177 W
Equator
0.9996
UTM Zones
• Zone 2
Longitude Start and End
Linear Unit
False Easting
False Northing
Central Meridian
Latitude of Origin
Scale of Central Meridian
174 W to 168 W
Meter
500,000
0
171 W
Equator
0.9996
UTM Zones
• Zone 13, Colorado, Nebraska Panhandle,
etc.
Longitude Start and End
Linear Unit
False Easting
False Northing
Central Meridian
Latitude of Origin
108 W to 102 W
Meter
500,000
0
105 W
Equator