GIS

1. Mathematical Principles
behind Projections

2. To locate Geographic Position
A model (3D) of the earth surface
Spheroid
How the 3D model is related to the shape of
the earth?
Datum
A model to translate the 3D points on a 2D
surface with minimal distortion
Projection
A coordinate system to measure the points
3D or 2D

3. What is a Projection?
Mathematical transformation of 3D objects in a 2D
space with minimal distortion
This two-dimensional surface would be the basis for
your map.

4. Engineering Drawing
Types of Projections
Orthographic
Isometric
Perspective
Oblique

6. For any map, the most important
parameters of accuracy can be
expressed as:
can distances be accurately measured?
are directions preserved?
are shapes preserved?
are area ratios preserved?
which regions suffer the most, and which
kind of, distortion?

7. Although all properties cannot be preserved,
a reasonably small spherical patch can be
approximated by a flat sheet with acceptable
distortion
In most projections, at least one specific
region (usually the center of the map) suffers
little or no distortion.
If the represented region is small enough , the
projection choice may be of little importance.

8. Geometric Projection
geometric interpretation as light rays
projected from a source intercept the
Earth and, according to laws of
perspective, "draw" its features on a
surface (Plane, cylinder, or cone)
Algorithmic Projection
are described purely by mathematical
formulae
• E.g. equal-distant, equal area etc.,

10. Graticule

11. Graticules as a guide for distortion
along any meridian, the distance on the map
between parallels should be constant
along any single parallel, the distance on the
map between meridians should be constant;
for different parallels,
Distance between meridians should decrease
to zero towards the poles
therefore, any two grid "cells" bounded by the
same two parallels should enclose the same
area

12. Geometric Projection

13. Distance Property preserved
The Azimuthal Equidistant Projection
North-polar aspect (Arctic at the centre)
• ρ = (π / 2 − φ)R and θ = λ
South-polar aspect (Antarctic at the centre)
• ρ = (π / 2 + φ)R and θ = -λ
Only the radial distance from the centre
of the map to any object is preserved

14. Azimuthal Equidistant projection (North polar aspect)

15. Azimuthal Equidistant projection (South polar aspect)

16. Inverse Mapping

17. Azimuthal Equal-area Projection
N
E
N

18. Azimuthal Equal area projection (North polar aspect)

20. Self-Study
http://www.progonos.com/furuti/index.html
http://mathworld.wolfram.com/MapProjection.html
http://mathworld.wolfram.com/Ellipse.html
http://mathworld.wolfram.com/Ellipsoid.html
Assignment
Map Projection
• Derive equations for azimuthal equal-distance and equal area
from the equatorial aspect
Reference:
• http://www.progonos.com/furuti/index.html
• Books:
– Engineering Surveying, Higher Surveying

21. One projection to another

22. Source: http://www.kartografie.nl/geometrics/Introduction/introduction.html

23. Source: http://www.kartografie.nl/geometrics/Introduction/introduction.html

24. Datum Transformation

25. Earth is not a sphere
Earth mass is not distributed uniformly, so
the gravitational pull is not uniform
Due to rotation, equator is slightly bulged,
and poles are slightly flattened (1/300)
• Oblate Spheroid
Terrain is not uniform

26. Highest spot on earth?
What is the tallest peak on earth?
Mount Everest, at 8,850 meters above
MSL
What is the highest spot on earth where
you are the closest to the outer space?
Mount Chimborazo, in the Andes,
• 6,100 meters above MSL
• But is sitting on a bulge which makes it 2,400
meters taller than Everest
• Everest is sitting down on the lower side of the
same bulge
Source: http://www.npr.org/templates/story/story.php?storyId=9428163

27. Deviations (undulations) between the
Geoid and the WGS84 ellipsoid
Source: http://www.kartografie.nl/geometrics/Introduction/introduction.html

28. Earth Surfaces
Sea surface
Geoid
Ellipsoid
Topographic
surface
Geoid is a surface of constant gravity.

29. Taking into account all these
irregularities is difficult
Some irregularities can be ignored
For e.g. terrain although important locally,
terrain levels are minuscule in planetary
scale
• the tallest land peak stands less than 9km
above sea level, or nearly 1/1440 of Earth
diameter
• the depth of the most profound sea abyss is
roughly 1/1150 diameter.

30. For maps covering very large areas,
especially worldwide,
the Earth may be assumed perfectly
spherical, since any shape imprecision is
dwarfed by unavoidable errors in data and
media resolution.
Conversely, for very small areas terrain
features dominate and measurements
can be based on a flat Earth

31. Datum
Horizontal Datum or Geodetic Datum
Reference frame for locating points on Earth’s
surface.
Defines origin & orientation of latitude/longitude
lines.
Defined by spheroid and spheroid’s position
relative to Earth’s center.

32. 2 types of datums:
Earth-centered
Local

33. Local datum
coordinate system
Earth-centered datum
coordinate system
Earth’s S urface
Earth-centered datum (WGS84)
Local datum (NAD27)

34. Datum Transformation

35. Cartesian Coordinate System
WGS -84 is
Earth Centered
Earth Fixed;
Origin to with
+/- 10 cm
Z
Ellipsoid
sized a & f
Y
X

36. Datums and Defining
Parameters
To translate one datum to another we must know the
relationship between the chosen ellipsoids in terms of
position and orientation. The relationship is defined by 7
constants.
A. 3 - Distance of the ellipsoid center from the center of the
earth (∆X, ∆Y, ∆Z)
B. 3 - Rotations around the X, Y, and Z of the Cartesian
Coordinate System Axes (ε, ψ, ω)
C. 1 - Scale change (S) of the survey control network
2 - The size and shape of the ellipsoid (semi major axis
a and flattening f approximately 1/298

37. Translations (3 Parameters)
Movement of points along an Axis
∆X
∆Y
∆Z

38. Rotations (3 Parameters)
Movement of points around an Axis
ε
ψ
ω

39. Scale (1 Parameter)
Changing the distance between points
S

40. Datum Transformation
Simple Three-parameter transformation
Geocentric translation
Seven-parameter transformation
Helmert 7-parameter transformation
Standard Molodensky formulas

42. Geodetic Lat/Lon/Height
Earth Centered X/Y/Z

44. 3 Parameters
X’
Y’
Z’
=
X
Y
Z
∆X
+ ∆Y
∆Z

45. 3 Parameter Determination
Most Transformation Parameters can be found in the NIMA
technical report “Department of Defense World Geodetic
System 1984” (TR 8350.2) NSN: 7643-01-402-0347

48. 7 Parameters

50. Projection: C
Datum: D
(Local or global datum e.g.
Indian or WGS84
Projection: A
Datum: B
(Local Datum e.g. Indian)
Inverse mapping
Geographic
Coordinates
Datum: B
Datum
Transformation
Reproject to C
Geographic
Coordinates
Datum: D
Geographic
Coordinates
Datum: WGS 84
Datum
Transformation

51. Self-Study
http://www.progonos.com/furuti/MapProj/CartIndex/ca
rtIndex.html
http://mathworld.wolfram.com/Ellipse.html
http://mathworld.wolfram.com/Ellipsoid.html

52. References:
http://www.colorado.edu/geography/gcraft/notes/datum/datum.html
https://www.navigator.navy.mil/navigator/gis.htm
http://earth-info.nga.mil/GandG/coordsys/datums/index.html
http://www.kartografie.nl/geometrics/Introduction/introduction.html