Map projections CS 128ES 228 Lecture 3 a

Map projections CS 128/ES 228 - Lecture 3 a 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 3 a 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 3 a 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 3 a 4

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. 2002. Measuring America. Walker & Co. , NY. P. 162 CS 128/ES 228 - Lecture 3 a 5

Geographical (spherical) coordinates Latitude & Longitude (“GCS” in Arc. Map) § Both measured as angles from center of Earth § Reference planes: - Equator for latitude - Prime meridian for longitude CS 128/ES 228 - Lecture 3 a 6

Lat/Long. are not Cartesian coordinates n They are angles measured from the center of Earth n They can’t be used (directly) to plot locations on a plane Understanding Map Projections. ESRI, 2000 (Arc. GIS 8). P. 2 CS 128/ES 228 - Lecture 3 a 7

Parallels and Meridians Parallels: lines of latitude. Meridians: lines of longitude. § Everywhere parallel § Converge toward the poles § 1 o always ~ 111 km (69 miles) § Some variation due to ellipsoid (110. 6 at equator, 111. 7 at pole) § 1 o =111. 3 km at 1 o = 78. 5 “ at 45 o = “ at 90 o CS 128/ES 228 - Lecture 3 a 0 8

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 228 - Lecture 3 a 9

1. Modeling Earth’s surface n Ellipsoid: theoretical model of surface - not perfect sphere - used for horizontal measurements n Geoid: incorporates effects of gravity - departs from ellipsoid because of different rock densities in mantle - used for vertical measurements CS 128/ES 228 - Lecture 3 a 10

Ellipsoids: flattened spheres n Degree of flattening given by f = (a-b)/a (but often listed as 1/f) n Ellipsoid can be local or global CS 128/ES 228 - Lecture 3 a 11

Local Ellipsoids n Fit the region of interest closely n Global fit is poor n Used for maps at national and local levels CS 128/ES 228 - Lecture 3 a 12

Examples of ellipsoids Local Ellipsoids Inverse flattening (1/f) Clarke 1866 294. 9786982 Clarke 1880 293. 465 N. Am. 1983 Global Ellipsoids International 1924 297 GRS 80 (Geodetic Ref. Sys. ) 298. 257222101 WGS 84 (World Geodetic Sys. ) 298. 257223563 CS 128/ES 228 - Lecture 3 a 13

2. Then what’s a datum? n Datum: a specific ellipsoid + a set of “control points” to define the position of the ellipsoid “on the ground” n Either local or global n > 100 world wide Some of the datums stored in Garmin 76 GPS receiver CS 128/ES 228 - Lecture 3 a 14

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 3 a 15

Datum Smatum NAD 27 or 83 – who cares? n One of 2 most common sources of mis-registration in GIS n (The other is getting the UTM zone wrong – more on that later) CS 128/ES 228 - Lecture 3 a 16

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 3 a 17

Properties of a map projection n Area n Distance n Shape n Direction Projections that conserve area are called equivalent Projections that conserve shape are called conformal CS 128/ES 228 - Lecture 3 a 18

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 228 - Lecture 3 a 19

Classes of projections a. Cylindrical b. Planar (azimuthal) c. Conical CS 128/ES 228 - Lecture 3 a 20

Cylindrical projections n Meridians & parallels intersect at 90 o n Often conformal n Least distortion along line of contact (typically equator) http: //ioc. unesco. org/oceanteacher/resourcekit/Module 2/GIS/Module_c/module_c 4. html n Ex. Mercator - the ‘standard’ school map CS 128/ES 228 - Lecture 3 a 21

Transverse Mercator projection n Mercator is hopelessly poor away from the equator n Fix: rotate the projection 90° so that the line of contact is a central meridian (N-S) n Ex. Universal Transverse Mercator CS 128/ES 228 - Lecture 3 a 22

Planar projections n a. k. a Azimuthal n Best for polar regions CS 128/ES 228 - Lecture 3 a 23

Conical projections n Most accurate along “standard parallel” n Meridians radiate out from vertex (often a pole) n Ex. Albers Equal Area n Poor in polar regions – just omit those areas CS 128/ES 228 - Lecture 3 a 24

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/Module 2/GIS/Module/Mo dule_c/module_c 4. html § Neither conformal nor equivalent (equal area). Useful only for world maps CS 128/ES 228 - Lecture 3 a 25

More compromise projections CS 128/ES 228 - Lecture 3 a 26

What if you’re interested in oceans? http: //www. cnr. colostate. edu/class_info/nr 502/lg 1/map_projections/distortions. html CS 128/ES 228 - Lecture 3 a 27

“But wait: there’s more …” http: //www. dfanning. com/tips/map_image 24. html All but upper left: http: //www. geography. hunter. cuny. edu/mp/amuse. html CS 128/ES 228 - Lecture 3 a 28

Buckminster Fuller’s “Dymaxion” CS 128/ES 228 - Lecture 3 a 29