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 The jog created by these “correction lines”, where the old north-south line abruptly stopped and a new one began 50 or 60 yards east or west, became a feature of the grid, and because back roads tend to follow surveyors’ lines, they present an interesting driving hazard today. After miles of straight gravel or blacktop, the sudden appearance of a correction line catches most drivers by surprise, and frantic tire marks show where vehicles have been thrown into hasty 90- dgree turns, followed by a second skid after a short stretch running west or east when the road head north again onto the new meridian. Andro Linklater Measuring America. Walker & Co., NY. P. 162

CS 128/ES Lecture 3a6 Geographical (spherical) coordinates Latitude & Longitude (“GCS” in ArcMap)  Both measured as angles from center of Earth  Reference planes: - Equator for latitude - Prime meridian for longitude

CS 128/ES Lecture 3a7 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 3a8 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 1 o = 78.5 “ at 45 o = 0 “ at 90 o

CS 128/ES Lecture 3a9 Overview of the cartographic process 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

CS 128/ES Lecture 3a10 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 3a11 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 3a12 Local Ellipsoids Fit the region of interest closely Global fit is poor Used for maps at national and local levels

CS 128/ES Lecture 3a13 Examples of ellipsoids Local EllipsoidsInverse flattening (1/f) Clarke Clarke N. Am Global Ellipsoids International GRS 80 (Geodetic Ref. Sys.) WGS 84 (World Geodetic Sys.)

CS 128/ES Lecture 3a14 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 3a15 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 3a16 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 3a17 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 3a18 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 3a19 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 in the same projection. GIS software contains routines for changing projections.

CS 128/ES Lecture 3a20 Classes of projections a. Cylindrical b. Planar (azimuthal) c. Conical

CS 128/ES Lecture 3a21 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 3a22 Transverse Mercator projection Mercator is hopelessly poor away from the equator Fix: rotate the projection 90° so that the line of contact is a central meridian (N- S) Ex. Universal Transverse Mercator

CS 128/ES Lecture 3a23 Planar projections a.k.a Azimuthal Best for polar regions

CS 128/ES Lecture 3a24 Conical projections Most accurate along “standard parallel” Meridians radiate out from vertex (often a pole) Ex. Albers Equal Area Poor in polar regions – just omit those areas

CS 128/ES Lecture 3a25 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 3a26 More compromise projections

CS 128/ES Lecture 3a27 What if you’re interested in oceans?

CS 128/ES Lecture 3a28 “But wait: there’s more …” All but upper left:

CS 128/ES Lecture 3a29 Buckminster Fuller’s “Dymaxion”