It deals with the basics of map projection, one of the pillars of scientific cartography.

1. Map Projections ―concepts, classes
and usage

2. Maps are flat, but the Earth is not!
Producing a perfect map is like peeling an orange and
flattening the peel without distorting a map drawn on
its surface.
A map projection is a mathematical model of a set of
rules or for transforming locations from the 3D Earth
onto a 2D display.
This conversion necessarily distorts some aspect of the
earth's surface, such as area, shape, distance, or
direction.

3. Why use a projection?
1. A projection permits spatial
data to be displayed in a
Cartesian system.
2. Projections simplify the
calculation of distances and
areas, and other spatial
analyses.
P = Point on 3D Globe
Geographical Co-ordinates = (λ, Ф)
Spherical Co-ordinates = (θ, r)
Ṕ´= Transformed point on 2D Plane
Rectangular Co-ordinates = (x, y)
Polar Co-ordinates = (ρ, z)
Transformation Functions
{X = f1 (λ, Ф), y = f2 (λ, Ф)}
{ρ = f3 (θ, r), z = f4 (θ, r)}
f1, f2, f3, f4 are real, single valued, continuous and differentiable functions

4. Hence, no flat representation of the earth can be completely
accurate.
Many different projections have been developed, each suited to a
particular purpose.
Map projections differ in the way they handle four properties: area,
angles, distance and direction.
Accordingly, they are called equal-area (authalic, homolographic or
equivalent), orthomorphic (true-shape or conformal),
equidistant, and azimuthal projections.
Rules:
1. No projection can preserve all four simultaneously, although
some combinations can be preserved, such as Area and
Direction.
2. No projection can preserve both Area and Angles, however. The
map-maker must decide which property is most important and
choose a projection based on that.

5. Basics of Map Projection
• Every projection has its own set of advantages and disadvantages.
• There is no such thing as "best" projection.
• Distortions in shape, scale, distance, direction, and area always occur.
• Some projections minimize distortions in some of these properties at the
expense of maximizing errors in others.
• Some projection are attempts to only moderately distort all of these
properties.
The mapmaker must select the one best suited to the needs, reducing
distortion of the most important features. They have devised almost
limitless ways to project the image of the globe onto a flat surface (paper).
“Every map user and maker should have a basic understanding of
map projections” to —
• Create spatial data (collecting GPS data)
• Import into GIS and overlay with other layers
• Acquire spatial data from other sources
• Display the GPS data using maps

6. Classes of Map projections
Physical models:
• Cylindrical Projections (cylinder)
- Tangent case
(Normal, Equatorial, Oblique)
- Secant case
• Conic Projections (cone)
- Tangent case
(Normal, Equatorial, Oblique)
- Secant case
• Azimuthal or planar projections
(plane)
- Tangent case
(Normal, Equatorial, Oblique)
- Secant case
Distortion properties:
• Conformal (preserves
local angles and shape)
• Equal area or equivalent
(preserves area)
• Equidistant (preserves
scale along a center line)
• Azimuthal (preserves
directions)

7. Planar Cylindrical Conical
Normal
Equatorial
Oblique

8. Coordinate Systems ―
3D Earth
Sphere : (Longitude, Latitude, Altitude)
Spheroid / Ellipsoid: (Longitude, Latiitude, Altitude)
2D Plane
Cartesian: (u, v)
Polar: (d, z)
Rectangular: (x, y)

9. Cylindrical Projections
• Meridians and Parallels intersect at 90o,
• Often Conformal,
• Least Distortion along Equator,
• Example: Plate Carree, Mercator, Galls, etc.

10. Plate Carree
Projection
Lambert’s Cylindrical Equal Area Projection

11. Miller’s Cylindrical Projection

12. Gall’s Orthographic
Projection
Gall’s Stereographic
Projection

13. Transverse Mercator Projection
• Mercator is hopelessly distorted
away from the equator towards
high latitudes.
• Fix: rotate 90° so that
the line of contact is a central
meridian (N-S).
• Example: Universal Transverse
Mercator (UTM) Works well for
narrow strips (N-S) of the globe.

14. Planar Projections
• Preserves Azimuth from
the Center
• Best for Polar Regions
• Gnomonic Chart
• Celestial Hemisphere
• Conformality or
Stereograms

15. Azimuthal Equidistant
Equatorial Projection
Azimuthal Equal Area
Equatorial Projection
Polar Zenithal Equal Area Projection

16. Conical Projections
• Most accurate along
“standard parallels”.
• Meridians radiate out from
vertex (often a pole).
• Poor in polar regions – just
omit those areas.
• Examples: Albers Equal
Area. Used in most USGS
topographic maps.

17. Conical Equidistant
Projection
Lambert’s Conical Equal Area
Projection
Alber’s Conical Equal Area
Projection

18. Euler Projection
Lambert’s
Conformal Conic
Projection
Braun’s Stereographic Conic
Projection

19. American Polyconic Projection
Rectangular Polyconic Projection

20. Werner’s Projection
Bonne’s Projection

21. Compromise Projections
1. Robinson’s World Projection based on a set of
co-ordinates rather than a mathematical formula.
2. Shape, area, and distance ok near origin and along equator.
3. Neither conformal nor equivalent (equal area). Useful only for
world maps.

22. What if you’re interested in oceans?

23. “But wait: there’s more …”

24. Buckminster Fuller’s “Dymaxion”

25. Cassini’s Projection

26. Braun’s Stereographic
Cylindrical Projection
Sinusoidal Projection

27. Mollweide’s Projection
Collignon Diamond
Projection

28. Quartic Authalic
Projection
McBride–Thomas IV
Projection

29. Eckert – IV Projection
Robinson Projection

30. Goode’s Homolosine
Projection
Boggs Eumorphic
Projection

31. Winkel II Projection

32. Wiechel’s Projection

33. Aitoff Projection
Hammer’s Projection

34. Wagner IX Projection
Eckert – Greifendroff
Projection

35. Winkel Tripple Projection
Lagrange Projection

36. Eisenlohr Projection
August Projection

37. Peirce’s Quincuncial Projection

38. Guyou Projection
Adam’s Projection

39. Xarax’s World Projection
Van der Grinten’s III
Projection
Van der Grinten’s IV Projection

40. Maurar Globular Projection
Orthoapsidal Projection

41. Arden - Close Projection
Oblique
Hammer Projection
Briesemeister’s Projection

42. Interrupted
Sinusoidal
Projection

43. Jager’s Projection
Petermann’s Projection

44. Berghaus Projection

45. Conoalactic Projection
Maurer’s S231 Projection Maurer’s S233 Projection

46. Tetrahedral Projection

47. Interrupted
Molleide’s Projection
Interrupted
Goode’s Homolosine Projection
Interrupted
Bogg’s Eumorphic Projection

48. Kent – Halstead’s
Projection
Halstead’s Composite
World Projection

49. Tetrahedral Globe
Cubic Globe

50. Octahedral Projection

51. Cahill’s Butterfly Projection

52. Waterman’s Projection

53. Gnomonic Projection
on
a Cuboctahedron
Gnomonic Projection
on
a Icosahedron

54. Snyder’s Polyhedral
Projection
Gnomonic Projection
on
a Dodecahedron

55. Rhombicuboctahedral Projection

56. Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Cuboctahedron
Rhombicuboctahedron
Map Fold Out

57. Trapezoidal Projection
Ortelius’s Projection
Apian Projection

58. Nicolosi Projection
Octant Projection

59. Mercator Projection
Oblique Mercator Projection

60. Universal Transverse Mercator (UTM) Coordinate
System
• UTM system is transverse-secant cylindrical projection, dividing the surface of the
Earth into 6 degree zones with a central meridian in the center of the zone. each
one of zones is a different Transverse Mercator projection that is slightly rotated to
use a different meridian. UTM zone numbers designate 6 degree longitudinal strips
extending from 80 degrees South latitude to 84 degrees North latitude. UTM is a
conformal projection, so small features appear with the correct shape and scale is
the same in all directions. (all distances, directions, shapes, and areas are
reasonably accurate ). Scale factor is 0.9996 at the central meridian and at most
1.0004 at the edges of the zones.
• UTM coordinates are in meters, making it easy to make accurate calculations of
short distances between points (error is less than 0.04%)
• Used in USGS topographic map, and digital elevation models (DEMs)
• Although the distortions of the UTM system are small, they are too great for some
accurate surveying. zone boundaries are also a problem in many applications,
because they follow arbitrary lines of longitude rather than boundaries between
jurisdictions.

61. CM: central meridian
AB: standard meridian
DE: standard meridian
-105
-108
-102
Transverse Mercator Projection

62. Zone 1
International Date
Line - 180
Equator
Zone 18o
Universal Transverse Mercator-
Grid

63. UTM Zone Numbers

64. Universal Polar Stereographic (UPS)
Coordinate System
• The UPS is defined above 84⁰N latitude and south
of 80⁰ S latitude.
• The eastings and northings are computed using a
polar aspect stereographic projection.
• Zones are computed using a different character
set for south and north Polar regions.

66. THANK YOU
prof ashis sarkar
Presidency University, Kolkata
profdrashis@gmail.com