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 http: //www. texasflyer. com/ms 150/img/rider s 05. jpg CS 128/ES 228 - Lecture 3 a 5

The Paris meridian n Surveyed by Delambre & Méchain (1792 -98) n Used to establish the length of the meter & estimate the curvature of the Earth n Paris meridian used by French as 0 o 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 3 a 6

The new meridian* n In 1884, at the International Meridian Conference in Washington, DC, the Greenwich Meridian was adopted as the prime meridian of the world. France abstained. n 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. n 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 3 a 7

Geographical (spherical) coordinates Latitude & Longitude (“GCS” in Arc. Map) § 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 3 a 8

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 9

Parallels and Meridians Parallels: lines of latitude. Meridians: lines of longitude. § Everywhere parallel § 1 o always ~111 km (69 miles) § Some variation due to ellipsoid (110. 6 at equator, 111. 7 at pole) § Converge toward the poles § 1 o =111. 3 km at 0 o = 78. 5 “ at 45 o = “ at 90 o CS 128/ES 228 - Lecture 3 a 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 3 a 11

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 12

Ellipsoids: flattened spheres n n 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 3 a 13

Local Ellipsoids n Fit the region of interest closely n Global fit is poor n Used for maps at national and local levels http: //exchange. manifold. net/manifold/manuals/5_userman/m fd 50 The_Earth_as_an_Ellipsoid. htm CS 128/ES 228 - Lecture 3 a 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 3 a 15

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 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 3 a 17

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 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 3 a 19

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 20

An early projection Leonardo da Vinci [? ], c. 1514 http: //www. odt. org/hdp/ CS 128/ES 228 - Lecture 3 a 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 3 a 22

Classes of projections a. Cylindrical b. Planar (azimuthal) c. Conical CS 128/ES 228 - Lecture 3 a 23

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 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/aa 031599. htm CS 128/ES 228 - Lecture 3 a 25

Transverse Mercator projection n Mercator is hopelessly distorted away from the equator n Fix: rotate 90° so that the line of contact is a central meridian (N-S) n Ex. Universal Transverse Mercator (UTM) Works well for narrow strips (N-S) of the globe CS 128/ES 228 - Lecture 3 a 26

Planar projections n a. k. a Azimuthal n Best for polar regions CS 128/ES 228 - Lecture 3 a 27

Conical projections n Most accurate along “standard parallel” n Meridians radiate out from vertex (often a pole) n Poor in polar regions – just omit those areas n Ex. Albers Equal Area. Used in most USGS topographic maps CS 128/ES 228 - Lecture 3 a 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/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 29

More compromise projections CS 128/ES 228 - Lecture 3 a 30

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 31

“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 32

Buckminster Fuller’s “Dymaxion” CS 128/ES 228 - Lecture 3 a 33