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Map projections. 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. For example:. The Public Land Survey System.

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Map projections

Map projections

CS 128/ES 228 - Lecture 3a


The dilemma

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


For example

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


Like this

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


One very practical result

One very practical result

http://www.texas-flyer.com/ms150/img/riders05.jpg

CS 128/ES 228 - Lecture 3a


Geographical spherical coordinates

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


Lat long are not cartesian coordinates

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


Parallels and meridians

Parallels and Meridians

Parallels: lines of latitude.

  • Everywhere parallel

  • 1o always ~ 111 km (69 miles)

  • Some variation due to ellipsoid (110.6 at equator, 111.7 at pole)

Meridians: lines of longitude.

  • Converge toward the poles

  • 1o =111.3 km at 0o = 78.5 “ at 45o

    = 0 “ at 90o

CS 128/ES 228 - Lecture 3a


The foundation of cartography

The foundation of cartography

  • Model surface of Earth mathematically

  • Create a geographical datum

  • Project curved surface onto a flat plane

  • Assign a coordinate reference system

(leave for next lecture)

CS 128/ES 228 - Lecture 3a


1 modeling earth s surface

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


Ellipsoids flattened spheres

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


Local ellipsoids

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/mfd50The_Earth_as_an_Ellipsoid.htm

CS 128/ES 228 - Lecture 3a


Examples of ellipsoids

Examples of ellipsoids

CS 128/ES 228 - Lecture 3a


2 then what s a datum

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

CS 128/ES 228 - Lecture 3a


North american datums

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

CS 128/ES 228 - Lecture 3a


Datum smatum

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)

CS 128/ES 228 - Lecture 3a


3 map projections

3. Map Projections

Why use a projection?

  • A projection permits spatial data to be displayed in a Cartesian system

  • Projections simplify the calculation of distances and areas, and other spatial analyses

CS 128/ES 228 - Lecture 3a


Properties of a map projection

Area

Shape

Projections that conserve area are called equivalent

Distance

Direction

Projections that conserveshape are called conformal

Properties of a map projection

CS 128/ES 228 - Lecture 3a


An early projection

An early projection

Leonardo da Vinci [?], c. 1514

http://www.odt.org/hdp/

CS 128/ES 228 - Lecture 3a


Two rules

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


Classes of projections

Classes of projections

  • Cylindrical

  • Planar (azimuthal)

  • Conical

CS 128/ES 228 - Lecture 3a


Cylindrical projections

Cylindrical projections

  • Meridians & parallels intersect at 90o

  • Often conformal

  • Least distortion along line of contact (typically equator)

  • Ex. Mercator- the ‘standard’ school map

http://ioc.unesco.org/oceanteacher/resourcekit/Module2/GIS/Module/Module_c/module_c4.html

CS 128/ES 228 - Lecture 3a


Transverse mercator projection

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)

CS 128/ES 228 - Lecture 3a


Planar projections

Planar projections

  • a.k.a Azimuthal

  • Best for polar regions

CS 128/ES 228 - Lecture 3a


Conical projections

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


Compromise projections

Compromise projections

  • Robinson world projection

  • Based on a set ofcoordinates rather than a mathematical formula

  • Shape, area, and distance ok near origin and along equator

  • Neither conformal nor equivalent (equal area). Useful only for world maps

http://ioc.unesco.org/oceanteacher/resourcekit/Module2/GIS/Module/Module_c/module_c4.html

CS 128/ES 228 - Lecture 3a


More compromise projections

More compromise projections

CS 128/ES 228 - Lecture 3a


What if you re interested in oceans

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


But wait there s more

“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


Buckminster fuller s dymaxion

Buckminster Fuller’s “Dymaxion”

CS 128/ES 228 - Lecture 3a


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