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

GreenwichMeridian

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

W E

(0ºN, 0ºE) Equator, Prime Meridian

Geographic Coordinates

90 W120 W 60 W

30 N

0 N

60 N

Geographic Coordinates

Meridian of longitude

Parallel of latitude

X

Y

ZN

EW

=0-90

°S

P

OR

=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

Sm

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

rr

A

BC

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° = 3.1416/180° = 0.0175 radians

• For the meridian: L = R = 6370 Km * 0.0175 = 111 km

• For the parallel: L = R Cos= 6370 * Cos30° * 0.0175 = 96.5 km

• Meridians converge as poles are approached

If Earth were a Sphere ...

Cartesian Coordinates

(o, o)

(xo,yo)

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.

(Xshift, Yshift) = (xo,yo) = (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

r3r2

r4

Numberof Satellites

1234

Object Defined

SphereCircle

Two PointsSingle Point