Map Projections

RG 620 Week 5 May 08, 2013 Institute of Space Technology, Karachi

Converting the 3D Model to 2D Plane

Map Projection

Map Projection

Map Projection

Projecting Earth's Surface into a Plane

• Earth is 3-D object • The

transformation

of

3-D Earth’s surface

coordinates into

2-D map coordinates

is called

Map Projection

• A map projection uses

mathematical formulas

to relate spherical coordinates on the globe to flat, planar coordinates

Map Projection

All flat maps are

distorted

to some degree

Can not be accurately

depicted on 2-D plane There is always a distortion in 1 or 2 of its characteristics when projected to a 2-D map

Map Projection Classification

1. Based on Distortion Characteristics 2. Based on Developable Surface

Map Projection Classification

1. Based on Distortion Characteristics: According to the property or properties that are maintained by the transformation. i.

Some map projections attempt to maintain linear scale at a point or along a line, rather than area, shape or direction. ii. Some preserve area but distortion in shape iii. Some maintain shapes and angles and have area distortion

Map Projection Classification

2. Based on Developable Surface: Considering the Earth as a transparent sphere with a point source of illumination at the centre.

Distortion

• The

4 basic characteristics preserved / distorted

of a map likely to be depending upon the map projection are: 1. Conformity 2. Distance 3. Area 4. Direction • In any projection

at least 1 of the 4

can be

preserved

(but not all) characteristics • Only on

globe preserved

all the

above properties

are

Distortion

• Transfer of points from the curved ellipsoidal surface to a flat map surface introduces

Distortion

Distortion

• In projected maps distortions are unavoidable • Different map projections distort the globe in different ways • In map projections features are either compressed or expanded • At few locations at map distortions may be zero • Where on map there is no distortion or least distortion?

Map Projection

• Each type of projection has its advantages and disadvantages • Choice of a projection depends on – Application – for what purposes it will be used – Scale of the map • Compromise projection?

Map Projections

1- Properties Based

• Conformal projection preserves

shape

• Equidistance projection preserves

distance

• Equal-area map maintains accurate

relative

sizes • Azimuthal or True direction maps maintains

directions

Map Projection - Conformal

• Maintains shapes and angles in small areas of map • Maintains angles. Latitude and Longitude intersects at 90 o • Area enclosed may be greatly distorted (increases towards polar regions) • No map projection can preserve shapes of larger regions Examples: – Mercator – Lambert conformal conic

Mercator projection

Lambert Conformal Conic

Conformal everywhere except at the poles.

Map Projection - Equidistance

• Preserve distance from some standard point or line (or between certain points) • 1 or more lines where length is same (at map scale) as on the globe • No projection is equidistant to and from all points on a map (1 0r 2 points only) • Distances and directions to all places are true only from the center point of projection • Distortion of areas and shapes increases as distance from center increases Examples: – Equirectangular – distances along meridians are preserved – Azimuthal Equidistant - radial scale with respect to the central point is constant – Sinusoidal projection - the equator and all parallels are of their true lengths

Polar Azimuthal Equidistant

Equirectangular or Rectangular Projection

Map Projection – Equal Area

• Equal area projections preserve area of displayed feature • All areas on a map have the same proportional relationship to their equivalent ground areas • Distortion in shape, angle, and scale • Meridians and parallels may not intersect at right angles Examples: – Albers Conic Equal-Area – Lambert Azimuthal Equal-Area

Albers Conic Equal-Area

Lambert Azimuthal Equal-Area

Preserves the area of individual polygons while simultaneously maintaining a true sense of direction from the center

Map Projection – True Direction

• Gives directions or azimuths of all points on the map correctly with respect to the center by maintaining some of the great circle arcs • Some True-direction projections are also conformal, equal area, or equidistant – Example: Lambert Azimuthal Equal-Area projection

Map Projection

2- based on developable surface

• A developable surface is a simple geometric form capable of being flattened without stretching • Map projections use different models for converting the ellipsoid to a rectangular coordinate system – Example: conic, cylindrical, plane and miscellaneous • Each causes

distortion

in

scale

and

shape

Cylindrical Projection

• Projecting spherical Earth surface onto a cylinder • Cylinder is assumed to surround the transparent reference globe • Cylinder touches the reference globe at equator

Cylindrical Projection

Source: Longley et al. 2001

Other Types of Cylindrical Projections

Transverse Cylindrical Oblique Cylindrical Secant Cylindrical

Examples of Cylindrical Projection

• Mercator • Transverse Mercator • Oblique Mercator • Etc.

Conical Projection

• A conic is placed over the reference globe in such a way that the apex of the cone is exactly over the polar axis • The cone touches the globe at standard parallel • Along this standard parallel the scale is correct with least distortion

Other Types of Conical Projection

Secant Conical

Examples of Conical Projection

• Albers Equal Area Conic • Lambert Conformal Conic • Equidistant Conic

Planar or Azimuthal Projection

• Projecting a spherical surface onto a plane that is tangent to a reference point on the globe • If the plane touches north or south pole then the projection is called polar azimuthal • Called normal if reference point is on the equator • Oblique for all other reference points

Secant Planar

Examples of Planar Projection

• Orthographic • Stereographic • Gnomonic • Azimuthal Equidistance • Lambert Azimuthal Equal Area

Summary of Projection Properties

Where at Map there is Least Distortion?

Where at Map there is Least Distortion

Great Circle Distance

• Great Circle Distance is the shortest path between two points on the Globe • It’s the distance measured on the ellipsoid and in a plane through the Earth’s center. • This planar surface intersects the two points on the Earth’s surface and also splits the spheroid into two equal halves • How to calculate Great Circle Distance?

Great Circle Distance Example from Text Book

Summary – Map Projection

• Portraying 3-D Earth surface on a 2-D surface (flat paper or computer screen) • Map projection can not be done without distortion • Some properties are distorted in order to preserve one property • In a map one or more properties but NEVER ALL FOUR may be preserved • Distortion is usually less at point/line of intersections of map surface and the ellipsoid • Distortion usually increases with increase in distance from points/line of intersections

Websites on Map Projection

• • • • • • • • • • • http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj.html

http://erg.usgs.gov/isb/pubs/MapProjections/projections.html

http://www.soe.ucsc.edu/research/slvg/map.html

http://www.eoearth.org/article/Maps http://geography.about.com/library/weekly/aa031599.htm

http://www.btinternet.com/~se16/js/mapproj.htm

http://www.experiencefestival.com/a/Map_projection_ _Projections_by_preservation_of_a_metric_property/id/4822091 http://webhelp.esri.com/arcgisdesktop/9.2/index.cfm?TopicName=About_ map_projections http://www.nationalatlas.gov/articles/mapping/a_projections.html

http://en.wikipedia.org/wiki/ http://memory.loc.gov/cgi bin/query/h?ammem/gmd:@field(NUMBER+@band(g5761b+ct001576))