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 of a map likely to be
preserved / distorted depending upon the map
projection are:
1.
2.
3.
4.
Conformity
Distance
Area
Direction
• In any projection at least 1 of the 4 characteristics
can be preserved (but not all)
• Only on globe all the above properties are
preserved
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 90o
• 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/cgibin/query/h?ammem/gmd:@field(NUMBER+@band(g5761b+ct001576))