Map Projections (2/2)

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

2. Overview • Geodetic Datum • Map Projections • Coordinate systems • Global Positioning System

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

4. Coordinate Systems • Global Cartesian coordinates (x,y,z) for the whole earth. • Geographic coordinates (f, l, 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.

5. Global Cartesian Coordinates Z Greenwich Meridian O • Y X Equator

6. Geographic Coordinates Prime Meridian P Equator Meridian plane

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

8. 60 N 30 N 60 W 120 W 90 W 0 N Geographic Coordinates

9. =0-180°W =0-90°S Geographic Coordinates Z Meridian of longitude Greenwich meridian l = 0° N Parallel of latitude P  - Geographic longitude • =0-90°N  - Geographic latitude  E W O • Y R - Earth radius R  • • O - Geocenter =0-180°E X Equatorf = 0°

10. Geographic Coordinates • Earth datum defines the standard values of the ellipsoid and geoid. • Latitude (f) and longitude (l) 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).

11. m p S n f r q Latitude f • 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 f, of point S

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

13. If Earth were a Sphere ... Length on a Meridian: AB = R Df (same for all latitudes) r D r Dl C B Df 0 N R Length on a Parallel: CD = r Dl = R Cosf Dl (varies with latitude) A

14. If Earth were a Sphere ... • 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: • p radians = 180°, so 1º = p/180° = 3.1416/180° = 0.0175 radians • For the meridian: DL = R Df = 6370 Km * 0.0175 = 111 km • For the parallel: DL = R Cosf Dl = 6370 * Cos30° * 0.0175 = 96.5 km • Meridians converge as poles are approached

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

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

17. 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 (lo). • Reference Latitude (fo) is the equator. • (Xshift, Yshift) = (xo,yo) = (500,000, 0) in the Northern Hemisphere. • Units are meters

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

19. State Plane • Defined for each State in the United States. • East-West States (e.g. Texas) use Lambert ConformalConic, 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

20. Overview • Geodetic Datum • Map Projections • Coordinate systems • Global Positioning System

21. 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 Dt since signal left the satellite, and calculates the distance to it Dr = c Dt. • Position obtained by intersection of radial distances Dr from each satellite. • Differential correction improves accuracy.

22. Global Positioning System (GPS) Dr2 Dr3 Number of Satellites 1 2 3 4 Object Defined Sphere Circle Two Points Single Point Dr4 Dr1