Transcript Projections
Map projections CS 128/ES 228 - Lecture 3a 1 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 2 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 3 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 4 One very practical result http://www.texasflyer.com/ms150/img/rider s05.jpg CS 128/ES 228 - Lecture 3a 5 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 6 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 7 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 8 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 9 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 0 10 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 11 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 12 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 13 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 14 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 15 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 16 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 17 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 18 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 19 Properties of a map projection Area Distance Shape Direction Projections that conserve area are called equivalent Projections that conserve shape are called conformal CS 128/ES 228 - Lecture 3a 20 An early projection Leonardo da Vinci [?], c. 1514 http://www.odt.org/hdp/ CS 128/ES 228 - Lecture 3a 21 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 22 Classes of projections a. Cylindrical b. Planar (azimuthal) c. Conical CS 128/ES 228 - Lecture 3a 23 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 CS 128/ES 228 - Lecture 3a 24 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 25 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 26 Planar projections a.k.a Azimuthal Best for polar regions CS 128/ES 228 - Lecture 3a 27 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 28 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 CS 128/ES 228 - Lecture 3a 29 More compromise projections CS 128/ES 228 - Lecture 3a 30 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 31 “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 32 Buckminster Fuller’s “Dymaxion” CS 128/ES 228 - Lecture 3a 33