Map Projections (1/2)

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

Definition

A geodetic datum defines the size and shape of the earth, and the origin and orientation of the axis used to define the location of points.

Over time, geodetic data have evolved from simple flat surfaces and spheres to complex ellipsoids.

Flat earth models can be accurate over short distances (i.e., less than 10 Km), spherical earth models for approximate global distance calculations, and ellipsoidal earth models for accurate global distance calculations.

Shape of the Earth

We think of the earth as a sphere ...

... when it is actually an ellipsoid, slightly larger in

radius at the equator than at the poles.

P

O a

b

X

Ellipse

Z

An ellipse is defined by:•Focal length = •Flattening ratio: f = (a-b)/a

•Distance F1-P-F2 is constant for all

points P on ellipse•When = 0 then ellipse = circle

For the earth:•Major axis: a = 6378 km•Minor axis: b = 6357 km•Flattening ratio: f = 1/300

F1F2

P

Ellipsoid or Spheroid

O

X

Z

Ya ab

Rotational axis

Rotate an ellipse around one of its axis.

Standard Ellipsoids

Ellipsoid Majoraxis, a (m)

Minoraxis, b (m)

Flatteningratio, f

Clarke(1866)

6,378,206 6,356,584 1/294.98

GRS80 6,378,137 6,356,752 1/298.57

Ref: Snyder, Map Projections, A working manual, USGS Professional Paper 1395, p.12

Standard Horizontal Geodetic Data

NAD27 (North American Datum of 1927) uses the Clarke (1866) ellipsoid.

NAD83 (North American Datum of 1983) uses the GRS80 ellipsoid.

WGS84 (World Geodetic System of 1984) uses GRS80.

Earth Surfaces

Geoid is a surface of constant gravity.

Topographic surface

EllipsoidSea surface

Geoid

Earth Surfaces

Ocean

Geoid

Topographic surface

EllipsoidGravity Anomaly

Elevation

P z = zp

z = 0

Mean Sea level = Geoid

Topographic Surface

Elevation is measured from the Geoid

Standard Vertical Geodetic Datum

A vertical datum defines elevation z, taking into account a map of gravity anomalies between the ellipsoid and the geoid.

NGVD29 (National Geodetic Vertical Datum of 1929).

NAVD88 (North American Vertical Datum of 1988).

Overview

Geodetic Datum

Map Projections

Coordinate systems

Global Positioning System

Map Projections

A map projection is a mathematical algorithm to transform locations defined on the curved surface of the earth into locations defined on the flat surface of a map.

Map Projection

Representative Fraction

Globe distanceEarth distance

Scale Projection

(e.g. 1:24,000) (e.g. 0.9996)

Scale Fraction

Map distanceGlobe distance

Types of Projections

Conic: Screen is a conic surface. Lamp at the center of the earth. Examples: Albers Equal Area, Lambert Conformal Conic. Good for East-West land areas.

Cylindrical: Screen is a cylindrical surface. Lamp at the center of the earth. Examples: (Transverse Mercator). Good for North-South land areas.

Azimuthal: Screen is a flat surface tangent to the earth. Lamp at the center of the earth (gnomonic), at the other side of the earth (stereographic), or far from the earth (orthographic). Examples: Lambert Azimuthal Equal Area. Good for global views.

Conic Projections

Albers and Lambert

Cylindrical Projections

Transverse

Oblique

Tangent Secant

Mercator

Azimuthal

Lambert

Albers Equal-Area Conic

Lambert Conformal Conic

Universal Transverse Mercator

Lambert Azimuthal Equal-Area

Distortion Projected Maps

In the process of transforming a curved surface into a flat surface, some geometric properties are modified.

The geometric properties that are modified are:

Area (important for mass balances)

Shape

Direction

Length

The difference between map projections has to do with which geometric properties are modified.

Depending on the type of analysis, preserving one geometric property might be more important that preserving other.