Map projections
Rüdiger Gens
Coordinate systems
Geographic coordinates
f
a: semi-major axis
Map projections
b: semi-minor axis
f: flattening = (a-b)/a
Expresses as a fraction
1/f = about 300
b
a
Geographic
latitude
Geodetic
latitude
– geographical coordinates imply
spherical Earth model
– geodetic coordinates imply
ellipsoidal Earth model
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Sphere versus spheroid
Source: ArcGIS help file
y assumption that the earth is a sphere is
possible for small-scale maps (smaller than
1:5000000)
y to maintain accuracy for larger-scale maps
(scales of 1: 1000000 or larger) a spheroid is
necessary
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Common Spheroids
y
y
y
y
y
y
Bessel 1841
Clarke 1866, Clarke 1880
GEM 6, GEM 10C
GRS 1967, GRS 1980
International 1924, International 1967
WGS 72, WGS 84
GEOS 639 – InSAR and its applications (Fall 2006)
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Reference surfaces
Map projections
Source: R. Gens, UAF
y three reference surfaces
x topography
x geoid
x ellipsoid
GEOS 639 – InSAR and its applications (Fall 2006)
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Reference surfaces
Map projections
Source: R. Gens, UAF
y topography represents the physical surface
of the Earth
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Reference surfaces
Map projections
Source: R. Gens, UAF
y geoid defined as level surface of gravity
field with best fit to mean sea level
x maximum difference between geoid and mean
sea level about 1 m
GEOS 639 – InSAR and its applications (Fall 2006)
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Reference surfaces
Map projections
Source: R. Gens, UAF
y ellipsoid defines mathematical surface
approximating the physical reality while
simplifying the geometry
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Datum
y describes the relationship between a particular
local ellipsoid and a global geodetic reference
system (e.g. WGS84)
y local datum defines the best fit to the Earth's
surface for particular area (e.g. NAD27)
Source: ESRI
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Common Datums
y
y
y
y
y
y
World Geodetic System 1972 (WGS 72)
World Geodetic System 1984 (WGS 84)
North American Datum 1927 (NAD 27)
North American Datum 1983 (NAD 83)
European Datum 1950 (ED 50)
South American Datum 1969 (SAD 69)
GEOS 639 – InSAR and its applications (Fall 2006)
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Datum
y datum
Map projections
x describes the relationship between a particular local
ellipsoid and a global geodetic reference system
y coordinate system
x shape and size given by the ellipsoid
x position given by the fixing of the origin
x fixing of the origin defines a datum
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Geographic coordinate system
y A point is referenced by its
longitude and latitude values
y Longitude and latitude are
angles measured from the
earth’s center to a point on
the earth’s surface
Source: ESRI
Source: ESRI
GEOS 639 – InSAR and its applications (Fall 2006)
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Coordinate systems
Cartesian coordinates
Map projections
z
y
P
y geodetic coordinates
inappropriate for satellite
imagery
→ cartesian
coordinates
x
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Map projections
y problem of mapping three-dimensional
coordinates related to a particular datum on a
flat surface
x maps are two-dimensional
x impossible to convert spheroid into flat plane without
distortions of shape, area, distance, or direction
→ map projections
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Cylindrical projections
y cylinder that has its
entire circumference
tangent to the Earth’s
surface along a great
circle (e.g. equator)
GEOS 639 – InSAR and its applications (Fall 2006)
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Cylindrical projections
y The map projection has distorted the graticule
(data near the poles is stretched)
Map projections
Source: ESRI
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So..
Map projections
y Where would you use a cylindrical projection?
y And why?
At the equatorial regions
As the projection is true at the equator
and distortion increases towards the
poles
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Cylindrical: Examples
y Mercator projection
y Transverse Mercator projection
y Oblique Mercator projection
Source: ESRI
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Map projections
Conic projections
y Simplest conic projection is
tangent to the surface along a
small circle (called standard
parallel)
y The meridians are projected
onto the conical surface,
meeting at the apex
y Parallel lines of latitude are
projected onto the cone as
rings.
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Conic projections
Map projections
y further you get from the standard parallel,
the more distortion increases.
Source: ESRI
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Conic projections
y secant projection: a more complex conic
projection
y contact the global surface at two locations
y defined by two standard parallels
y less overall distortion than a tangent
projection
Source: ESRI
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So..
Map projections
y Where would you use a conic projection?
y And why?
In mid-latitudes
Projection is true along some parallel
between the poles and equator
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Conic: Examples
y Conic projection with two standard parallels
y Lambert Conformal Conic projection
(preserves angles)
y Albers Conic Equal-Area projection (preserves
areas)
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Azimuthal (Planar) projections
x projecting positions directly to a plane tangent to the
Earth’s surface
GEOS 639 – InSAR and its applications (Fall 2006)
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Map projections
Azimuthal (Planar) projections
y Point of contact specifies the aspect and is the
focus of the projection.
y The focus is identified by a central longitude
and a central latitude.
y Possible aspects are polar, equatorial, and
oblique
Source: ESRI
GEOS 639 – InSAR and its applications (Fall 2006)
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So..
Map projections
y Where would you use a azimuthal projection?
y And why?
In polar regions
Projection is best at their center point and
distortion is generally worst at the edges
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Azimuthal (Planar) projections
Map projections
y Lambert Azimuthal Equal-Area projection
y Stereographic (conformal) projection
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Equidistant projections
kp
Map projections
1
1
km = 1
Sphere
Projection
x scale factor along a meridian (line of equal longitude)
is equal to 1
x shape and area of square are distorted
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Equal-area projections
Map projections
1
kp
1
km
Sphere
Projection
x equal areas are represented by the same map area
regardless of where they occur
GEOS 639 – InSAR and its applications (Fall 2006)
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Conformal projections
kp
Map projections
1
km
1
Sphere
Projection
x angles on a conformal map are the same as
measured on the Earth’s surface
x meridians intersect parallels at right angles
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Summary
Map projections
y map purpose
x for distribution maps: equal area
x for navigation: projections that show azimuths or
angles properly
y size of area
x some projections are better suited for East-West
extent, others for North-South
x for small areas the projection is relatively
unimportant
x for large areas the projection is very important
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