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
4. 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
5. 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
6. Deviations (undulations) between the
Geoid and the WGS84 ellipsoid
Source: http://www.kartografie.nl/geometrics/Introduction/introduction.html
7. Shape of the Earth
We think of the
earth as a sphere ...
... when it is actually an
ellipsoid/Spheroid, slightly larger
in radius at the equator than at
the poles.
8. Earth Models
Flat Earth
Survey over a short distance ~ 10km
Spherical Earth
Approximate global distance
Ellipsoid Earth
Accurate global distance
Geoid
Surface of equal gravitational potential
10. 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.
11. Ellipse
Z
An ellipse is defined by:
• Focal length = ε
• Flattening ratio: f = (a-b)/a
• Distance F1-P-F2 is constant for all
points P on ellipse
• When ε = 0 then ellipse = circle
b
F1
ε
For the earth:
•Major axis: a = 6378 km
•Minor axis: b = 6357 km
•Flattening ratio: f = 1/300
P
P
O
ε
a
F2
X
13. Why use different spheroids?
The earth's surface is not perfectly symmetrical, so the
semi-major and semi-minor axes that fit one
geographical region do not necessarily fit another.
Satellite technology has revealed several elliptical
deviations. For one thing, the most southerly point on
the minor axis (the South Pole) is closer to the major
axis (the equator) than is the most northerly point on
the minor axis (the North Pole).
14. The earth's spheroid deviates slightly for different
regions of the earth.
Ignoring deviations and using the same spheroid for
all locations on the earth could lead to errors of
several meters, or in extreme cases hundreds of
meters, in measurements on a regional scale.
•EVEREST (India, Nepal, Pakistan, Bangladesh)
•WGS84 (GPS World-wide)
•Clark 1866 (North America)
•GRS80 (North America)
•International 1924 (Europe)
•Bessel 1841 (Europe)
15. Ellipsoids used in India
Everest – 1830
Colonel Sir George Everest is the name of
the surveyor general of India
• The Great Trigonometrical Survey India
Sometimes referred as Indian Datum
Has been modified several times since
WGS – 1984
Used by GPS
Need for a common global system
16. Great Trigonometrical Survey
Idea
• 1745 – General Watson
1802 – real commencement
Baseline
• 40006.4 feet, on a plane near Saint Thomas' Mount,
Madras
• From this a series of triangles were formed
– 30 to 40 miles in length
The objective is to find the curvature of the earth
accurately at different latitude and longitude
Everest 1830 - Spheroid
17.
18. 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.
Earth-centered
Local
19. 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
20. 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.
23. Datum
A mathematical model must be related to real-world
features.
a datum is a reference point or surface against which
position measurements are made, and an associated
model of the shape of the earth for computing
positions
A smooth mathematical surface that fits closely to the
mean sea level surface throughout the area of
interest. The surface to which the ground control
measurements are referred.
Provides a frame of reference for measuring
locations on the surface of the earth.
24. How do I get a Datum?
To determine latitude and longitude, surveyors level
their measurements down to a surface called a geoid.
The geoid is the shape that the earth would have if all its
topography were removed.
Or more accurately, the shape the earth would have if
every point on the earth's surface had the value of mean
sea level.
Indian Datum:
Kalianpur hill in
Madhya Pradesh
Spheroid: Everest
25. Horizontal vs Vertical Datums
Horizontal datums are the reference values for a
system of location measurements (E.g. Lat,
Long).
The horizontal datum is the model used to
measure positions on the earth
Vertical datums are the reference values for a
system of elevation measurements.
A vertical datum is used for measuring the
elevations of points on the earth's surface
26. Vertical Datum
Vertical data are
• either tidal level
– based on sea levels,
– Tidal datum
• gravimetric,
– based on a geoid, or geodetic,
• based on the same ellipsoid models of the
earth used for computing horizontal datums.
29. Geographic Coordinate System
Spherical Earth’s surface
-radius 6371 km
Meridians (lines of longitude)
- passing through Greenwich,
England as prime meridian or
0º longitude.
Parallels (lines of latitude)
- using equator as 0º latitude.
degrees-minutes-seconds
(DMS),
decimal degrees (DD)
True direction, shape,
distance, and area
30. Latitude and Longitude on a Sphere
Meridian of longitude
Z
Greenwich
meridian
λ=0°
N
Parallel of latitude
°
-90
ϕ =0
P
•
N
W
λ =0
-180
X
ϕ
O
•
°W
•
λ
Equator
•
R
ϕ =0°
λ=0-180°E
°S
90
=0
ϕ
E
λ - Geographic longitude
ϕ - Geographic latitude
Y
R - Mean earth radius
O - Geocenter
31. Length on Meridians and Parallels
(Lat, Long) = (φ, λ)
Length on a Meridian:
AB = Re ∆φ
(same for all latitudes)
Length on a Parallel:
CD = R ∆λ = Re ∆λ Cos φ
(varies with latitude)
R ∆λ
30 N
0N
Re
R
C
∆φ B
Re
A
D
32. Example: What is the length of a 1º increment along
on a meridian and on a parallel at 30N, 90W?
Radius of the earth = 6370 km.
Solution:
• A 1º angle has first to be converted to radians
π radians = 180 º, so 1º = π/180 = 3.1416/180 = 0.0175
radians
• For the meridian, ∆L = Re ∆φ = 6370 ∗ 0.0175 = 111 km
• For the parallel, ∆L = Re ∆λ Cos φ
= 6370 ∗ 0.0175 ∗ Cos 30
= 96.5 km
• Parallels converge as poles are approached
33. Curved Earth Distance
(from A to B)
Shortest distance is along a
“Great Circle”
Z
A “Great Circle” is the
intersection of a sphere with a
plane going through its
center.
1. Spherical coordinates
converted to Cartesian
coordinates.
2. Vector dot product used to
calculate angle α from
latitude and longitude
B
A
α
•
Y
X
3. Great circle distance is Rα,
where R=6370 km2
R cos−1 (sin φ
1
sin φ2 + cos φ1 cos φ2 cos(λ1 − λ2 )
Longley et al. (2001)
35. Projection
Real-world features must be projected with minimum
distortion from a round earth to a flat map; and given
a grid system of coordinates.
A map projection transforms latitude and longitude
locations to x,y coordinates.
36. 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.
37. Cartesian Coordinate System
Planar coordinate systems are based on
Cartesian coordinates.
Projection:
Spherical to Cartesian
Coordinates with
Minimal distortion
38. Why use a Projection?
Can only see half the earth’s surface at a time.
Unless a globe is very large it will lack detail and
accuracy.
Harder to represent features on a flat computer
screen.
Doesn’t fold, roll or transport easily.
39. Types of Projections
Conic (Albers Equal Area, Lambert
Conformal Conic) - good for East-West
land areas
Cylindrical (Transverse Mercator) good for North-South land areas
Azimuthal (Lambert Azimuthal Equal
Area) - good for global views
44. Map Projection & Distortion
Converting a sphere to a flat surface results in distortion.
Shape (conformal) - If a map preserves shape, then
feature outlines (like country boundaries) look the
same on the map as they do on the earth.
Area (equal-area) - If a map preserves area, then
the size of a feature on a map is the same relative to
its size on the earth. On an equal-area map each
country would take up the same percentage of map
space that actual country takes up on the earth.
Distance (equidistant) - An equidistant map is one
that preserves true scale for all straight lines passing
through a single, specified point. If a line from a to b
on a map is the same distance that it is on the earth,
then the map line has true scale. No map has true
scale everywhere.
45. Direction/Azimuth (azimuthal) – An azimuthal
projection is one that preserves direction for all straight
lines passing through a single, specified point. Direction
is measured in degrees of angle from the north. This
means that the direction from ‘a’ to ‘b’ is the angle
between the meridian on which ‘a’ lies and the great
circle arc connecting ‘a’ to ‘b’. If the azimuth value from
‘a’ to ‘b’ is the same on a map as on the earth, then the
map preserves direction from ‘a’ to ‘b’. No map has true
direction everywhere.
46. Trade-off
On an equidistant map, distances are
true only along particular lines such as
those radiating from a single point
selected as the center of the projection.
Shapes are more or less distorted on
every equal-area map.
Sizes of areas are distorted on
conformal maps even though shapes of
small areas are shown correctly.
47. Polyconic
The projection is based on an
infinite number of cones tangent
to an infinite number of parallels.
The central meridian is straight.
Other meridians are complex curves.
The parallels are non-concentric
circles.
Scale is true along each parallel and
along the central meridian
50. 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
58. Lambert Azimuthal Equal Area
Albers Equal Area Conic
Equidistant Conic
Lambert Conformal Conic
59. Universal Transverse Mercator
Uses the Transverse Mercator projection.
60 six-degree-wide zones cover the earth from East to West
starting at 180° West.
extending from 80 degrees South latitude to 84 degrees North latitude
Each zone has a Central Meridian (λo).
Reference Latitude (φo) is the equator.
(Xshift, Yshift) = (xo,yo) = (500,000, 0).
Units are meters.
GRS 80: Global Reference System ellipsoid of 1980
Potsdam, Germany
North American Datum of 1927 (NAD27) is a datum based on the Clarke ellipsoid of 1866. The reference or base station is located at Meades Ranch in Kansas. There are over 50,000 surveying monuments throughout the US and these have served as starting points for more local surveying and mapping efforts. Use of this datum is gradually being replaced by the North American Datum of 1983.
degrees-minutes-seconds (DMS), MSEC GPS result: 34º04'04.270" N, 106º54'20.870 “ W
decimal degrees (DD): 34.06785 N, 106.90580 W
Parallels are parallel and spaced equally on meridians. Meridians and other arcs of great circles are straight lines (if looked at perpendicularly to the Earth's surface). Meridians converge toward the poles and diverge toward the Equator.
Meridians are equally spaced on the parallels, but their distances apart decreases from the Equator to the poles. At the Equator, meridians are spaced the same as parallels. Meridians at 60° are half as far apart as parallels. Parallels and meridians cross at right angles. The area of the surface bounded by any two parallels and any two meridians (a given distance apart) is the same anywhere between the same two parallels.
The scale factor at each point is the same in any direction.
You cannot flatten out features on an ellipsoid without distorting them. (Imagine viewing a tennis ball in its natural round state, now imagine putting a slit into it and trying to spread it out flat. It cannot be done without stretching, tearing, and altering its appearance substantially.
1º in equator is about 111 km,
each UTM zone is 6º or 666 km.
the distance between AB and CM, or CM and DE are 180 km
scale factor at central meridian 0.9996, at the standard meridian is 1, and most 1.0004 at the edges of the zones
