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
http://www.texasflyer.com/ms150/img/rider
s05.jpg
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The Paris meridian

Surveyed by Delambre
& Méchain (1792-98)

Used to establish the
length of the meter &
estimate the curvature
of the Earth

Paris meridian used by
French as 0o longitude
until 1914
Alder, K. 2002. The measure of all things: the seven-year
odyssey and hidden error that transformed the world. The
Free Press, NY. Frontispiece.
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The new meridian*

In 1884, at the International Meridian Conference in
Washington, DC, the Greenwich Meridian was
adopted as the prime meridian of the world. France
abstained.

The French clung to the Paris Meridian (now longitude
2°20′14.025″ east) as a rival to Greenwich until 1911 for
timekeeping purposes and 1914 for navigation.

To this day, French cartographers continue to
indicate the Paris Meridian on some maps.
http://en.wikipedia.org/wiki/Paris_Meridian
* for most of the world
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Geographical (spherical) coordinates
Latitude & Longitude
(“GCS” in ArcMap)
 Both measured as
angles from the
center of Earth
 Reference planes:
- Equator for latitude
- Prime meridian
(through Greenwich,
England) 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 0o
= 78.5
“ at 45o
=
“ at 90o
CS 128/ES 228 - Lecture 3a
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The foundation of cartography
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 (leave for next lecture)
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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
http://exchange.manifold.net/manifold/manuals/5_userman/m
fd50The_Earth_as_an_Ellipsoid.htm
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Examples of ellipsoids
Local Ellipsoids
Inverse flattening (1/f)
Clarke 1866
294.9786982
Clarke 1880
293.465
N. Am. 1983
(uses GRS 80, below)
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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An early projection
Leonardo da Vinci [?], c. 1514
http://www.odt.org/hdp/
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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
displayed 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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Beware of Mercator world maps
In 1989, seven North
American professional
geographic organizations
… adopted a resolution
that called for a ban on all
rectangular coordinate
maps due to their
distortion of the planet. .
http://geography.about.com/library/weekly/aa031599.htm
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Transverse Mercator projection

Mercator is hopelessly
distorted away
from the equator

Fix: rotate 90° so that
the line of contact is a
central meridian (N-S)

Ex. Universal Transverse
Mercator (UTM) Works
well for narrow strips (N-S)
of the globe
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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)

Poor in polar regions
– just omit those areas

Ex. Albers Equal Area. Used in
most USGS topographic maps
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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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