CS 128/ES Lecture 3a1 Map projections

CS 128/ES Lecture 3a2 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 Lecture 3a3 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 Lecture 3a4 Like this Township Survey Kent County, MI

CS 128/ES Lecture 3a5 One very practical result flyer.com/ms150/img/rider s05.jpg

CS 128/ES Lecture 3a6 The Paris meridian Surveyed by Delambre & Méchain ( ) Used to establish the length of the meter & estimate the curvature of the Earth Paris meridian used by French as 0 o longitude until 1914 Alder, K The measure of all things: the seven-year odyssey and hidden error that transformed the world. The Free Press, NY. Frontispiece.

CS 128/ES Lecture 3a7 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.prime meridian 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. * for most of the world

CS 128/ES Lecture 3a8 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 Lecture 3a9 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 Lecture 3a10 Parallels and Meridians Parallels: lines of latitude.  Everywhere parallel  1 o always ~111 km (69 miles)  Some variation due to ellipsoid (110.6 at equator, at pole) Meridians: lines of longitude.  Converge toward the poles  1 o =111.3 km at 0 o = 78.5 “ at 45 o = 0 “ at 90 o

CS 128/ES Lecture 3a11 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 Lecture 3a12 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 Lecture 3a13 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 Lecture 3a14 Local Ellipsoids Fit the region of interest closely Global fit is poor Used for maps at national and local levels fd50The_Earth_as_an_Ellipsoid.htm

CS 128/ES Lecture 3a15 Examples of ellipsoids Local EllipsoidsInverse flattening (1/f) Clarke Clarke N. Am. 1983(uses GRS 80, below) Global Ellipsoids International GRS 80 (Geodetic Ref. Sys.) WGS 84 (World Geodetic Sys.)

CS 128/ES Lecture 3a16 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 Lecture 3a17 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 Lecture 3a18 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 Lecture 3a19 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 Lecture 3a20 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 Lecture 3a21 An early projection Leonardo da Vinci [?], c. 1514

CS 128/ES Lecture 3a22 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 Lecture 3a23 Classes of projections a. Cylindrical b. Planar (azimuthal) c. Conical

CS 128/ES Lecture 3a24 Cylindrical projections Meridians & parallels intersect at 90 o Often conformal Least distortion along line of contact (typically equator) Ex. Mercator - the ‘standard’ school map

CS 128/ES Lecture 3a25 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..ban

CS 128/ES Lecture 3a26 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 Lecture 3a27 Planar projections a.k.a Azimuthal Best for polar regions

CS 128/ES Lecture 3a28 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 Lecture 3a29 Compromise projections esourcekit/Module2/GIS/Module/Mo dule_c/module_c4.html 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

CS 128/ES Lecture 3a30 More compromise projections

CS 128/ES Lecture 3a31 What if you’re interested in oceans?

CS 128/ES Lecture 3a32 “But wait: there’s more …” All but upper left:

CS 128/ES Lecture 3a33 Buckminster Fuller’s “Dymaxion”