Map Projections (2/2) Francisco Olivera, Ph.D., P.E. Center for Research in Water Resources University of Texas at Austin

Overview Geodetic Datum Map Projections Coordinate systems Global Positioning System

Coordinate Systems A coordinate system is used to locate a point of the surface of the earth.

Coordinate Systems Global Cartesian coordinates (x,y,z) for the whole earth. Geographic coordinates ( , , z) for the whole earth. Projected coordinates (x, y, z) on a local area of the earth’s surface. The z-coordinate in Global Cartesian and Projected coordinates is defined geometrically; and in Geographic coordinates gravitationally.

Global Cartesian Coordinates O X Z Y Greenwich Meridian Equator

Geographic Coordinates P Meridian Equator plane Prime Meridian

Geographic Coordinates Longitude line (Meridian) N S W E Range: 180ºW - 0º - 180ºE Latitude line (Parallel) Range: 90ºS - 0º - 90ºN N S WE (0ºN, 0ºE) Equator, Prime Meridian

Geographic Coordinates 90 W 120 W 60 W 30 N 0 N 60 N

Geographic Coordinates Meridian of longitude Parallel of latitude  X Y Z N E W  =0-90°S P O R =0-180°E  =0-90°N Greenwich meridian = 0° Equator  = 0° =0-180°W - Geographic longitude  - Geographic latitude R - Earth radius O - Geocenter

Geographic Coordinates Earth datum defines the standard values of the ellipsoid and geoid. Latitude (  ) and longitude ( ) are defined using an ellipsoid (i.e., an ellipse rotated about an axis). Elevation (z) is defined using a geoid (i.e, a surface of constant gravitational potential).

Latitude  Take a point S on the surface of the ellipsoid and define there the tangent plane mn. Define the line pq through S and normal to the tangent plane. Angle pqr is the latitude , of point S  S m n q p r

Longitude 0°E, W 90°W (-90 °) 180°E, W 90°E (+90 °) -120° -30° -60° -150° 30° -60° 120° 150°  = the angle between a cutting plane on the prime meridian and the cutting plane on the meridian through the point, P P

If Earth were a Sphere... 0 N  R r r A B C  Length on a Meridian: AB = R  (same for all latitudes) Length on a Parallel: CD = r  = R Cos  (varies with latitude) D

Example: What is the length of a 1º increment on a meridian and on a parallel at 30N, 90W? Radius of the earth R = 6370 km. Solution: A 1º angle has first to be converted to radians:  radians = 180°, so 1º =  /180° = /180° = radians For the meridian:  L = R  = 6370 Km * = 111 km For the parallel:  L = R Cos  = 6370 * Cos30° * = 96.5 km Meridians converge as poles are approached If Earth were a Sphere...

Cartesian Coordinates (  o, o ) (x o,y o ) X Y Origin A planar cartesian coordinate system is defined by a pair of orthogonal (x,y) axes drawn through an origin.

Coordinate Systems Universal Transverse Mercator (UTM) - a global system developed by the US Military Services. State Plane - civilian system for defining legal boundaries.

Universal Transverse Mercator Uses the Transverse Mercator projection. 60 six-degree-wide zones cover the earth from East to West starting at 180° West. Each zone has a Central Meridian ( o ). Reference Latitude (  o ) is the equator. (X shift, Y shift ) = (x o,y o ) = (500,000, 0) in the Northern Hemisphere. Units are meters

UTM Zone 14 Equator -120° -90 ° -60 ° -102°-96° -99° Origin 6°

State Plane Defined for each State in the United States. East-West States (e.g. Texas) use Lambert Conformal Conic, North-South States (e.g. California) use Transverse Mercator. Texas has five zones (North, North Central, Central, South Central, South) to give accurate representation. Greatest accuracy for local measurements

Overview Geodetic Datum Map Projections Coordinate systems Global Positioning System

Global Positioning System (GPS) 24 satellites in orbit around the earth. Each satellite is continuously radiating a signal at speed of light. GPS receiver measures time lapse  t since signal left the satellite, and calculates the distance to it  r = c  t. Position obtained by intersection of radial distances  r from each satellite. Differential correction improves accuracy.

Global Positioning System (GPS) r1r1 r3r3 r2r2 r4r4 Number of Satellites Object Defined Sphere Circle Two Points Single Point