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Map projections

Producing a perfect map is like peeling an orange and flattening the peel without distorting the map. This lecture explores the challenges of representing the Earth's curved surface on a flat map, including the use of meridians, ellipsoids, and datums.

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Map projections

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  1. Map projections CS 128/ES 228 - Lecture 3a

  2. 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

  3. 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

  4. 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

  5. One very practical result http://www.texas-flyer.com/ms150/img/riders05.jpg CS 128/ES 228 - Lecture 3a

  6. The Paris meridian • Surveyed by Delambre & Méchain (1792-98) • Used to establish thelength 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. Examples of ellipsoids CS 128/ES 228 - Lecture 3a

  16. 2. Then what’s a datum? Some of the datums stored in Garmin 76 GPS receiver • 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 CS 128/ES 228 - Lecture 3a

  17. 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

  18. 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

  19. 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

  20. 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

  21. An early projection Leonardo da Vinci [?], c. 1514 http://www.odt.org/hdp/ CS 128/ES 228 - Lecture 3a

  22. 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

  23. Classes of projections • Cylindrical • Planar (azimuthal) • Conical CS 128/ES 228 - Lecture 3a

  24. 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

  25. 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 CS 128/ES 228 - Lecture 3a

  26. 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 CS 128/ES 228 - Lecture 3a

  27. Planar projections • a.k.a Azimuthal • Best for polar regions CS 128/ES 228 - Lecture 3a

  28. 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

  29. 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

  30. More compromise projections CS 128/ES 228 - Lecture 3a

  31. 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

  32. “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

  33. Buckminster Fuller’s “Dymaxion” CS 128/ES 228 - Lecture 3a

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