Map projections
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The dilemma
Maps are flat, but the Earth is not!
Producing a perfect map is like peeling
an orange and flattening the peel without distorting
a map drawn on its surface.
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For example:
The Public Land Survey System
• As surveyors worked
north along a central
meridian, the sides of
the sections they were
creating converged
• To keep the areas of
each section ~ equal,
they introduced
“correction lines” every
24 miles
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Like this
Township Survey
Kent County, MI
1885
http://en.wikipedia.org/wiki/Image:Kent-1885-twp-co.jpg
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One very practical result
The jog created by these “correction lines”, where the
old north-south line abruptly stopped and a new one
began 50 or 60 yards east or west, became a feature of
the grid, and because back roads tend to follow
surveyors’ lines, they present an interesting driving
hazard today. After miles of straight gravel or
blacktop, the sudden appearance of a correction line
catches most drivers by surprise, and frantic tire marks
show where vehicles have been thrown into hasty 90dgree turns, followed by a second skid after a short
stretch running west or east when the road head north
again onto the new meridian.
Andro Linklater. 2002. Measuring America. Walker & Co., NY. P. 162
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Geographical (spherical) coordinates
Latitude & Longitude
(“GCS” in ArcMap)
 Both measured as
angles from center of
Earth
 Reference planes:
- Equator for latitude
- Prime meridian for
longitude
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Lat/Long. are not Cartesian coordinates

They are angles
measured from the
center of Earth

They can’t be used
(directly) to plot
locations on a plane
Understanding Map Projections. ESRI, 2000 (ArcGIS 8). P. 2
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Parallels and Meridians
Parallels: lines of
latitude.
Meridians: lines of
longitude.
 Everywhere parallel
 Converge toward
the poles
 1o always ~ 111 km
(69 miles)
 Some variation due
to ellipsoid (110.6 at
equator, 111.7 at
pole)
 1o =111.3 km at 1o
= 78.5
“ at 45o
=
“ at 90o
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Overview of the cartographic process
1. Model surface of Earth
mathematically
2. Create a geographical
datum
3. Project curved surface
onto a flat plane
4. Assign a coordinate
reference system
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1. Modeling Earth’s surface

Ellipsoid: theoretical
model of surface
- not perfect sphere
- used for horizontal
measurements

Geoid: incorporates effects of gravity
- departs from ellipsoid because of different
rock densities in mantle
- used for vertical measurements
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Ellipsoids: flattened spheres

Degree of flattening
given by f = (a-b)/a
(but often listed as 1/f)

Ellipsoid can be local
or global
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Local Ellipsoids

Fit the region of
interest closely

Global fit is poor

Used for maps at
national and local
levels
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Examples of ellipsoids
Local Ellipsoids
Inverse flattening (1/f)
Clarke 1866
294.9786982
Clarke 1880
293.465
N. Am. 1983
Global Ellipsoids
International 1924
297
GRS 80 (Geodetic Ref. Sys.)
298.257222101
WGS 84 (World Geodetic Sys.)
298.257223563
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2. Then what’s a datum?

Datum: a specific
ellipsoid + a set of
“control points” to
define the position
of the ellipsoid “on
the ground”

Either local or
global

> 100 world wide
Some of the datums stored
in Garmin 76 GPS receiver
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North American datums
Datums commonly used in the U.S.:
- NAD 27: Based on Clarke 1866 ellipsoid
Origin: Meads Ranch, KS
- NAD 83: Based on GRS 80 ellipsoid
Origin: center of mass of the Earth
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Datum Smatum
NAD 27 or 83 – who
cares?

One of 2 most
common sources of
mis-registration in
GIS

(The other is getting
the UTM zone wrong
– more on that later)
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3. Map Projections
Why use a projection?
1.
A projection permits
spatial data to be
displayed in a
Cartesian system
2.
Projections simplify
the calculation of
distances and areas,
and other spatial
analyses
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Properties of a map projection

Area

Distance

Shape

Direction
Projections that
conserve area are
called equivalent
Projections that
conserve shape are
called conformal
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Two rules:
Rule #1: No projection can preserve all four
properties. Improving one often makes another
worse.
Rule #2: Data sets used in a GIS must be in the
same projection. GIS software contains routines
for changing projections.
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Classes of projections
a.
Cylindrical
b.
Planar
(azimuthal)
c.
Conical
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Cylindrical projections

Meridians & parallels
intersect at 90o

Often conformal

Least distortion
along line of contact
(typically equator)
http://ioc.unesco.org/oceanteacher/resourcekit/Module2/GIS/Module/Module_c/module_c4.html

Ex. Mercator - the ‘standard’ school map
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Transverse Mercator projection

Mercator is hopelessly
poor away from the
equator

Fix: rotate the
projection 90° so that
the line of contact is a
central meridian (NS)

Ex. Universal
Transverse Mercator
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Planar projections

a.k.a Azimuthal

Best for polar regions
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Conical projections

Most accurate along
“standard parallel”

Meridians radiate
out from vertex
(often a pole)

Ex. Albers Equal Area

Poor in polar regions –
just omit those areas
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Compromise projections
Robinson world projection
 Based on a set of
coordinates rather
than a mathematical
formula
 Shape, area, and
distance ok near origin
and along equator
http://ioc.unesco.org/oceanteacher/r
esourcekit/Module2/GIS/Module/Mo
dule_c/module_c4.html
 Neither conformal nor equivalent (equal area).
Useful only for world maps
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More compromise projections
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What if you’re interested in oceans?
http://www.cnr.colostate.edu/class_info/nr502/lg1/map_projections/distortions.html
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“But wait: there’s more …”
http://www.dfanning.com/tips/map_image24.html
All but upper left:
http://www.geography.hunter.cuny
.edu/mp/amuse.html
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Buckminster Fuller’s “Dymaxion”
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