1. Map Projections RG 620 April 20, 2016
Institute of Space Technology, Karachi

2. Converting the 3D Model to 2D Plane

3. Map Projection
Notice how the continents look stretched or squashed depending on the projection 3

4. Map Projection 4

5. 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 5

6. All flat maps are distorted to some degree
Map Projection
All flat maps are distorted to some degree
Can not be accurately depicted on 2-D plane
In the graphic above data near the poles is stretched .
Different projections have different spatial relationships between regions
There is always a distortion in 1 or 2 of its characteristics when projected to a 2-D map 6

7. Map Projection Classification
Based on Distortion Characteristics
Based on Developable Surface 7

8. Map Projection Classification
Based on Distortion Characteristics: According to the property or properties that are maintained by the transformation.
Some map projections attempt to maintain linear scale at a point or along a line, rather than area, shape or direction.
Some preserve area but distortion in shape
Some maintain shapes and angles and have area distortion
Projections may be classified on the basis of their distortion characteristics
Source figure: 8

9. Distortion Conformity Distance Area Direction
The 4 basic characteristics of a map likely to be preserved / distorted depending upon the map projection are:
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
Conformal: feature outlines look same on map as they look on earth. 10

10. Source: Text book

11. Distortion
Transfer of points from the curved ellipsoidal surface to a flat map surface introduces Distortion
Distortions are unavoidable when making flat maps.
Distortion may take different forms in different portions of the map. In one portion of the map features may be compressed and exhibit reduced areas or distances relative to the Earth’s surface measurements, while in another portion of the map areas or distances may be expanded. At few locations distortion may be zero.
Distortion is usually less near the point or line of intersections, where the map surface intersects the imaginary Globe.
Distortion usually increases with increasing distance from the intersection points or lines.

12. 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?
Placement/location of the projection source (light bulb), shape of the developable surface, placement of projection surface at/near the globe all have an impact on type and extent map distortion.
LCC: used for area that are larger in an east-west than north-south direction

13. 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?
For example, a projection may have unacceptable distortions if used to map the entire country, but may be an excellent choice for a large-scale (detailed) map of a county.
A compromise projection distorts all the properties of shape, area, distance, and direction, within some acceptable limit, example is Robison Projection used for World Maps 14

14. 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 15

15. 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
Conformal map projections preserve angles locally.
Mercator by Gerardus Mercator, in 1569
Mercator projection 16

16. Lambert Conformal Conic
superimposes a cone over the sphere of the Earth, with two reference parallels secant to the globe and intersecting it.
Conformal everywhere except at the poles.

17. 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
Scale is not maintained correctly by any projection throughout an entire map. However, there are in most cases, one or more lines on a map along which scale is maintained correctly
Azimuthal Equidistant map centered at Washington DC: shows the correct distance between Washington, DC, and any other point on the projection. It shows the correct distance between Washington, DC, and San Diego and between Washington, DC, and Seattle, but it does not show the correct distance between San Diego and Seattle.
Map on slide is Sinusoidal projection. 18

18. Polar Azimuthal Equidistant

19. Equirectangular or Rectangular Projection
An equirectangular projection is a cylindrical equidistant projection.

20. Source: http://www. geo. hunter. cuny

21. 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
No map projection can be both equivalent (equal area) and conformal
Examples:
Albers Conic Equal-Area
Lambert Azimuthal Equal-Area
In some instances, especially maps of smaller regions, shapes are not obviously distorted, and distinguishing an Equal area projection from a Conformal projection is difficult unless documented or measured 22

22. Albers Conic Equal-Area
It is similar to the Lambert Conformal Conic projection except that Albers Conic Equal Area portrays area more accurately than shape. 23

23. Lambert Azimuthal Equal-Area
Preserves the area of individual polygons while simultaneously maintaining a true sense of direction from the center 24

24. 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
Azimuthal
Accurate direction and therefore true angular relationship from a given center point
In land navigation, an azimuth is defined as a horizontal angle measured clockwise from a north base line or meridian. 25

25. 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
miscellaneous =which include special cases not falling into the other three categories. 26

26. Source: http://www. geo. hunter. cuny

27. 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 28

28. Cylindrical Projection
Source: Longley et al. 2001 29

29. Other Types of Cylindrical Projections
Cylinder may be either tangent to the Earth along a selected line or secant (intersect the Earth) along 2 lines
Transverse Cylindrical
Oblique Cylindrical
Secant Cylindrical 30

30. Examples of Cylindrical Projection
Mercator
Transverse Mercator
Oblique Mercator
Etc. 31

31. 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 32

32. Other Types of Conical Projection
Secant Conical 33

33. Examples of Conical Projection
Albers Equal Area Conic
Lambert Conformal Conic
Equidistant Conic 34

34. 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 35

35. Secant Planar 36

36. Examples of Planar Projection
Orthographic
Stereographic
Gnomonic
Azimuthal Equidistance
Lambert Azimuthal Equal Area 37

37. Summary of Projection Properties38

38. Where at Map there is Least Distortion?
The lines where the cylinder is tangent or secant are the places with the least distortion.

39. Where at Map there is Least Distortion?
The lines where the cylinder is tangent or secant are the places with the least distortion.

40. Where at Map there is Least Distortion

41. 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?

42. Great Circle Distance
Example from Text Book

43. 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

44. Websites on Map Projection
_Projections_by_preservation_of_a_metric_property/id/
projections 45