Projections

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Map projections
CS 128/ES 228 - Lecture 3a
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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.
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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Like this
Township Survey
Kent County, MI
1885
http://en.wikipedia.org/wiki/Image:Kent-1885-twp-co.jpg
CS 128/ES 228 - Lecture 3a
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One very practical result
http://www.texasflyer.com/ms150/img/rider
s05.jpg
CS 128/ES 228 - Lecture 3a
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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.
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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)
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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.
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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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
CS 128/ES 228 - Lecture 3a
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Buckminster Fuller’s “Dymaxion”
CS 128/ES 228 - Lecture 3a
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