Map projections CS 128/ES 228 - Lecture 3a

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

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 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: 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 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 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 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 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 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

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 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 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 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 conserve shape are called conformal CS 128/ES 228 - Lecture 3a

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

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 Cylindrical Planar (azimuthal) Conical CS 128/ES 228 - Lecture 3a

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

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 Robinson world projection Based on a set of coordinates 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 CS 128/ES 228 - Lecture 3a

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 …” 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” CS 128/ES 228 - Lecture 3a