MAP PROJECTIONS

MAP PROJECTIONS

Present by : Nidal ABURAJAB

Basic Definitions

Geodesy - the shape of the earth and definition of earth datum

Map Projection - the transformation of a curved surface of earth to a flat map

Coordinate systems - (x,y) coordinate systems for map data

BRIEF INTRODUCTION TO MAP PROJECTIONS

Map Projection is defined as an orderly system (a mathematical model) of projecting portions of the earth’s surface to a plane or flat surface – a map.

To produce a map of the world or part of it, we make use of map projections.

Map projection is transformation between curved reference surface of the Earth & the flat plane of map.

Map projection can also be defined as a set of equations which allows us to transform a set of Geographic coordinates (Longitude, Latitude) to Cartesian Coordinates of the two dimensional surface of Earth

Map Projection

‘A map projection is a means of representing the lines of latitude and longitude of the globe on a flat sheet of paper’

J.A. Steers

"a systematic drawing of parallels of latitude and meridians of longitudes on a plane surface for the whole earth or a part of it on a certain scale so that any point on the earth surface may correspond to that on the drawing."

BASIC PROBLEM OF MAP PROJECTION

The transformation from the curved surface of the Earth to the flat plane of the map is never completed successfully.

Distortion after flattening a curved surface.

MAP PROJECTION CONCEPTS The concept of projecting is required to express three

dimensional space as a two dimensional map. One easy way to understand is visualize projecting a light

through the Earth onto a surface.

Only a few map projections are actually produced by direct projection.

SCALE DISTORTION

It is impossible to project the Earth on a piece of paper without location distortions.

•It is important to know the extent to which the scale varies on map.

No scale distortion No scale distortion No scale distortion

Earth surfaceEarth surface

Most distortion Most distortion

Tangent map plane Secant map plane

Most distortion

Most distortion

Projection Properties

On the globe, places and areas contain the properties of shape, area, direction and distance. A projection may retain one or two of these but not all, and specifically NOT both area and shape.

a. Area

A projection that maintains 'area' is called equal area or equivalent. That means that two equal areas on the globe are equal areas on the map. This is achieved by sacrificing shape. Stretching in one direction to counter for earth curvature must be compensated by compaction in the other. b. Shape

A given shape, such as a square, maintains its shape across the map but at the cost of area and is called conformal or orthomorphic. For example a 2x2 square becomes a 1x1or 4x4 square. Stretching in one direction is matched by stretching in the other.

c. Distance

Not all distances can be retained, but distances can be correct in one direction from a line. This line is called a standard line. Distance can also be correct in all directions from a point. In these cases, the projection is termed equidistant.

d. Direction (azimuth)

Azimuth is the compass bearing between two points. Not all azimuth can be maintained (or there would be no distortion). It can be maintained in all directions only from a point, these projections are called azimuthal.

MAP PROJECTION

- Any Transformation between the curved reference surface of the earth & the flat plane of the map.

- Transforms a set of ellipsoidal geographic coordinates on the reference of the earth surface to a set of Cartesian coordinates representing positions on the 2-D surface of the map.

x, y = ƒ (, ) = Geographic to Cartesian

, = ƒ (x, y) Inverse

Projections- used for

Conic (Albers Equal Area, Lambert Conformal Conic) - good for East-West land areas

Cylindrical (Transverse Mercator) - good for North-South land areas

Azimuthal (Lambert Azimuthal Equal Area) - good for global views

Earth to Globe to Map

Representative Fraction

Globe distanceEarth distance

=

Map Scale: Map Projection:

Scale Factor

Map distanceGlobe distance

=

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

Cylindrical Projections

•The normal aspect for cylindrical projections assumes that the cylinder is tangent to the globe along the Equator. In this orientation, the graticule appears as a rectangular grid. In the tangent case, the Equator is true to scale and distortion increases with distance from the Equator.

•In the secant case, the standard parallels which lie equidistant north and south of the Equator are true to scale and distortion increases with distance from the standard lines.

Conic Projections(Albers, Lambert)

Cylindrical Projections(Mercator)

Transverse

Oblique

Azimuthal (Lambert)

Albers Equal Area Conic Projection

Lambert Conformal Conic Projection

POLYCONIC PROJECTION Class : Neither Conformal nor equal area. Projection principle : It consists of many (poly) cones

involved. The cones are tangent to each parallel, so that meridians are curved & parallels are also curved.

Uses :- 1. All topographic maps in India are in Polyconic Projection.

2. Used for large scale mapping of USA till 1950 only. 3. Not a suitable projection for digital environment due to

problem of rolling fit .

Conic Projection Polyconic Projection

CYLINDERICAL PROJECTION

Parallels and Meridians are straight line intersecting at right angles. It is conformal i.e. direction are true. Areas & distances are exaggerated near poles.

Since directions are true it was used for navigational purpose since 16th century. Now often inappropriately used as world map (as pole & equator are of same size).

0-180 +180

0

-90

+90

Cylinder

UNIVERSAL TRANSVESE MERCATOR (UTM)

UTM projection is now widely used projection for topographical mapping purposes.

It is conformal cylindrical projection. It covers almost entire world (80o S to 84o N) in 60 zones of 6o

Longitude. Reference Latitude (o), is the equator. Each zone has its own Central Meridian. Each zone has its own Cartesian coordinate system.

Universal Transverse Mercator Projection

Coordinate systems Universal Transverse Mercator

Every place on earth falls in a particular zone

Examples of different projections

Albers (Conic)

Shape Shape along the standard parallels is accurate and minimally distorted in the region between the standard parallels and those regions just beyond. The 90-degree angles between meridians and parallels are preserved, but because the scale along the lines of longitude does not match the scale along lines of latitude, the final projection is not conformal.

Area All areas are proportional to the same areas on the Earth.Direction Locally true along the standard parallels.Distance Distances are best in the middle latitudes. Along parallels, scale is reduced between the standard

parallels and increased beyond them. Along meridians, scale follows an opposite pattern.


Lambert Azimuthal Equal Area (Planar)

Shape Shape is true along the standard parallels of the normal aspect (Type 1), or the standard lines of the transverse and oblique aspects (Types 2 and 3). Distortion is severe near the poles of the normal aspect or 90° from the central line in the transverse and oblique aspects.

Area There is no area distortion on any of the projections.Direction Local angles are correct along standard parallels or standard lines. Direction is distorted elsewhere.Distance Scale is true along the Equator (Type 1), or the standard lines of the transverse and oblique aspects

(Types 2 and 3). Scale distortion is severe near the poles of the normal aspect or 90° from the central line in the transverse and oblique aspects.


Mercator(Cylindrical)

Shape Conformal. Small shapes are well represented because this projection maintains the local angular relationships.

Area Increasingly distorted toward the polar regions. For example, in the Mercator projection, although Greenland is only one-eighth the size of South America, Greenland appears to be larger.

Direction Any straight line drawn on this projection represents an actual compass bearing. These true direction lines are rhumb lines, and generally do not describe the shortest distance between points.

Distance Scale is true along the Equator, or along the secant latitudes.


Miller(Cylindrical)

Shape Minimally distorted between 45th parallels, increasingly toward the poles. Land masses are stretched more east to west than they are north to south.

Area Distortion increases from the Equator toward the poles. Direction Local angles are correct only along the Equator. Distance Correct distance is measured along the Equator.


Mollweide(Pseudo-cylindrical)

Shape Shape is not distorted at the intersection of the central meridian and latitudes 40° 44' N and S. Distortion increases outward from these points and becomes severe at the edges of the projection.

Area Equal-area.Direction Local angles are true only at the intersection of the central meridian and latitudes 40° 44' N and S.

Direction is distorted elsewhere.Distance Scale is true along latitudes 40°44' N and S. Distortion increases with distance from these lines and

becomes severe at the edges of the projection.


Orthographic

Shape Minimal distortion near the center; maximal distortion near the edge. Area The areal scale decreases with distance from the center. Areal scale is zero at the edge of the

hemisphere.Direction True direction from the central point. Distance The radial scale decreases with distance from the center and becomes zero on the edges. The scale

perpendicular to the radii, along the parallels of the polar aspect, is accurate.


Robinson(Pseudo-cylindrical)

Shape Shape distortion is very low within 45° of the origin and along the Equator.Area Distortion is very low within 45° of the origin and along the Equator.Direction Generally distorted.Distance Generally, scale is made true along latitudes 38° N and S. Scale is constant along any given latitude,

and for the latitude of opposite sign.

Map Projection 4

DIFFERENT MAP PROJECTIONS

Maps in Geographic Coordinates

- Digital map data can also be stored in ‘Geographic coordinates i.e. Latitude and Longitude

– Geographic coordinate system is becoming more popular in digital environment where contiguous maps are displayed on screen and measurements can be made in meters in most of GIS software.

– 1:250,000 (1 x 1) or 1:50,000 scale maps get displayed as perfect squares.

– The geographic system has become so popular that most of modern GIS softwares give an option whether map is to be displayed in geographic system (unprojected) or projected.

BRIEF INTRODUCTION TO DATUM

The datum for horizontal control is called ELLIPSOID.

The datum for vertical control is called GEOID.

Shape of the Earth

We think of the earth as a sphere

It is actually an ellipsoidal, slightly larger in radius at

the equator than at the poles

Ellipse

P

F2

O

F1

a

b

X

Z

An ellipse is defined by:Focal length = Distance (F1, P, F2) isconstant for all pointson ellipseWhen = 0, ellipse = circle

For the earth:Semi Major axis, a = 6377 kmSemi Minor axis, b = 6356 kmFlattening ratio, f = (a-b)/a ~ 1/300

Ellipsoid or SpheroidRotate an ellipse around an axis

O

X

Z

Ya ab

Rotational axis

ELLIPSOID The physical surface of earth which is highly irregular & hence unsuitable for

computation. It is approximated by various mathematical surfaces which is known as ellipsoid.

Earth has been surveyed many times over the years . Since the shape of the earth's surface differs from one geographic area to another, different ellipsoids are used to describe particular areas to get the best fit of the area being mapped.

Ellipsoid is used as a reference surface for the determination of a location (x,y coordinates or Ф, λ).

N

India

N America Ellipsoid 1

Ellipsoid 2

N

GEOID

• Geoid is equipotential surface at Mean Sea Level (MSL). It is smooth & continuous but not geometric. This surface is perpendicular to the direction of the gravity field of the Earth at all points.

Due to irregular mass distributionGeoid is irregular. It is reference surface for height.

Geoid

TYPE OF ELLIPSOIDS

India uses Everest Ellipsoid. With the help of artificial satellites, earth has been observed as a

whole and WGS 84 is the best fit surface for the entire globe.Name a (m) b (m) 1/f Used in

Everest(1830) 6377301 6356100 300.8017 IndiaAiry (1830) 6377563 6356257 299.32 BritainWGS 84 6378137 6356752 298.257 Global

Ref: Snyder, Map Projections, A working manual, US

Representations of the Earth

Earth surface

EllipsoidSea surface

Geoid

Mean Sea Level is a surface of constant gravitational potential called the Geoid

Geoid and Ellipsoid

Ocean

Geoid

Earth surface

Ellipsoid

Gravity Anomaly

Types of Coordinate Systems

(1) Global Cartesian coordinates (x,y,z) for the whole earth

(2) Geographic coordinates (, z) (3) Projected coordinates (x, y, z) on a local

area of the earth’s surface

Global Cartesian Coordinates (x,y,z)

O

X

Z

Y

GreenwichMeridian

Equator

•

Geographic Coordinates (, z)

Latitude () and Longitude () defined using an ellipsoid, an ellipse rotated about an axis

Elevation (z) defined using geoid, a surface of constant gravitational potential

Latitude and Longitude on a Sphere

Meridian of longitude

Parallel of latitude

X

Y

ZN

EW

=0-90

°S

P

OR

=0-180°E

=0-90°N

•

Greenwichmeridian

=0°

•

Equator =0°

•

•=0-180°W

- Geographic longitude - Geographic latitude

R - Mean earth radius

O - Geocenter

MAP PROJECTIONS

Documents

Transcript of MAP PROJECTIONS