Lecture # 7
Coordinate systems and
georeferencing
Coordinate Systems
ƒ Coordinate reference on a plane
ƒ Coordinate reference on a sphere
Coordinate reference on a plane
ƒ Coordinates are a convenient way of locating
points on a two-dimensional plane
ƒ Two usually employed systems are
z
Rectangular cartesian coordinate
• system that makes use of linear measurements in two
directions from a pair of fixed axes, usually
orthogonal (or rectangular)
z
Polar coordinate
• position defined with reference to the origin or pole,
the polar axis, radius vector (distance), and the
vectorial angle
Cartesian (x,y)
Polar coordinates
O- origin or pole
A
OA- radius vector
r
QOA- vectorial angle
φ
O
Q
Coordinate reference on a sphere
ƒ For unique references for locations on the
surface of earth, a system of three
dimensional polar coordinates may be used
ƒ Two orthogonal planes are chosen, which
intersect the origin (centre of earth)
z
z
one plane goes through equator
second plane is perpendicular to the equatorial
plane (Greenwich, UK, 1884)
Referencing location on the
earth’s surface
ƒ latitude and longitude reference system
Prime
Meridian
Equator
Parallels of
Latitude
Meridians of
Longitude
Graticular
Network
Grids vs. Graticules
ƒ Graticule - 3 dimensional network of lines
that wrap around the earth (lat./long)
ƒ Grid - flat 2 dimensional network of lines
(x, y plane)
A grid is on a flat piece of paper, while a
graticule is on the round earth
Referencing location on the
earth’s surface
ƒ latitude φ: angle from the equator to the
parallel
ƒ longitude λ: angle from Greenwich meridian
Referencing location on the
earth’s surface
Geographic Coordinate System
NORTH POLE
INTERNATIONAL
DATE LINE
+90° Latitude
-180° Longitude
CENTRAL
MERIDIAN
LATITUDE LINES
(PARALLELS)
EQUATOR
LONGITUDE LINES
(MERIDIANS)
0° Lat.
Earth Surfaces
ƒ Ellipsoid or spheroid - smooth mathematical
models of the shape of the earth
ƒ Geoid - Geoid models attempt to represent the
surface of the entire earth over both land and
ocean as though the surface resulted from gravity
alone
ƒ Topographic surface - the actual surface of the
land and sea at some moment in time
The earth as a spheroid
Geoid
Ellipsoid
Reference Ellipsoid
Ellipsoidal Parameters
b
a
a - semi-major axis
b - semi-minor axis
f = (a-b)/a - flattening
used to establish a
datum: reference point
for large scale mapping
Heights
ƒ Elevation - Mean Sea Level - the average
surface of the oceans. Tidal forces and
gravity differences from different locations
cause variations of hundreds of meters.
ƒ Ellipsoid height - the height from the
ellipsoid
ƒ Orthometric height - the height from the
geoid.
Datums
ƒ The datum is “where zero is”, it is where
you start measuring from
ƒ A datum is a mathematical model of the
earth we use to calculate the coordinates
on any map, chart, or survey system
Geodesy
ƒ Modern geodetic datums range from flat-earth models used
for plane surveying to complex models used for
international applications which completely describe the
size, shape, orientation, gravity field, and angular velocity
of the earth
ƒ Cartography, surveying, navigation, and astronomy all
make use of geodetic datums, and the science of geodesy is
the central discipline for the topic
ƒ Referencing geodetic coordinates to the wrong datum can
result in position errors of hundreds of meters
Geodetic Datums
ƒ Geodetic datums define the size and shape of the
earth and the origin and orientation of the
coordinate systems used to map the earth
ƒ Hundreds of different datums have been used to
frame position descriptions since the first
estimates of the earth's size were made by
Aristotle
ƒ Datums have evolved from those describing a
spherical earth to ellipsoidal models derived from
years of satellite measurements
Datum Selection
ƒ Different nations, states, and agencies use
different datums as the basis for coordinate
systems used to identify positions in geographic
information systems, precise positioning systems,
and navigation systems. The diversity of datums in
use today and the technological advancements that
have made possible global positioning
measurements with sub-meter accuracies requires
careful datum selection and careful conversion
between coordinates in different datums
Why do the themes on a map
need to have the same datum?
ƒ Datum shift- the coordinates for a point on
the Earth’s surface in one datum will not
match the coordinates from another datum
for the same point
ƒ A shift exists between datums because each
datum has a different origin. Sometimes the
shift is obvious, but sometimes it is very
subtle
History of Datums
ƒ Flat Earth models are still in use for
plane surveying over distances less
than 10 km; because earth’s curvature
is insignificant
ƒ Spherical Earth models are still used
for short range navigation and global
distance approximation
ƒ Ellipsoidal Earth models
Two Types of Datum
ƒ Horizontal Datum
ƒ forms the basis for computations of
horizontal control surveys in which
the curvature of the Earth is
considered
ƒ Vertical Datum
ƒ almost always use mean sea level for
elevation
History of the North American
Datum
ƒ New England Datum - 1879 first official US
geodetic datum referenced to Clarke 1866
ellipsoid
ƒ Renamed United States Standard Datum in
1901 after network extensions to the south
and west. Official origin was the
triangulation station in Meades Ranch,
Kansas.
History of the North American
Datum, Cont.
ƒ 1913 Canada and Mexico agreed to base
there triangulations on the US system and it
was renamed the North American Datum
ƒ After many adjustments to the new
networks the system was called the North
American 1927 Datum (NAD27). Its origin
is still Meades Ranch, Kansas and it is
computed on the Clarke 1866 ellipsoid
History of the North American
Datum, Cont.
ƒ The NAD27 system was adjusted in 1982-83
to become a geocentric datum. Geocentric
means the center of the ellipsoid is
referenced to Earth’s center of mass
ƒ This new system is North American Datum
1983 (NAD83). Its origin is Earth’s center of
mass and it is referrenced to the GRS80
ellipsoid
ƒ Because it is geocentric, it is the datum
commonly used by GPS
Projections
ƒDefinition: a mathematical transformation
that projects the Earth’s surface from 3D to
some 2D representation that can be drawn on
paper.
ƒSince this cannot be done without distortion,
a characteristic must be chosen to be portrayed
accurately at the expense of others, or a
compromise of several characteristics. This
can be done in infinite number of ways.
Map Projections
ƒ Curved surface of the earth needs to be
“flattened” to be presented on a map
ƒ projection is the method by which the
curved surface is converted into a flat
representation
Map Projections
ƒ defined as a mathematical function to
convert between the surface location on the
earth and the projected location on the map
ƒ conversion from a geographic (spherical)
reference system to a planar (Cartesian)
system;
e.g., lat/long -> x/y
Map Projections
• we can literally think of it as a light source
located inside the globe which projects the
features on the earth’s surface onto the flat
map
p’
map
q’
p
earth’s surface
q
Three Levels of Recognition
ƒ Class
ƒ Aspect
ƒ Property
Basic Classes of map projections
ƒ Cylindrical
ƒ Conical
ƒ Azimuthal
The Aspect of Map Projection
ƒ The Normal Aspect
ƒ The Transverse Aspect
ƒ The Oblique Aspect
Properties of Map Projection
ƒ Conformality
ƒ Equivalence (equal area)
ƒ Equidistance
Cylindrical Projections
Conic Projections
Azimuthal Projections
Aspect
Azimuthal Projections
Distortion in Map Projections
ƒ some distortion is inevitable
ƒ less distortion if maps show only small
areas, but large if the entire earth is shown
ƒ projections are classified according to
which properties they preserve:
area, shape, angles, distance
Conformal
When the scale of a map at any point on the
map is the same in any direction, the
projection is conformal. Meridians (lines of
longitude) and parallels (lines of latitude)
intersect at right angles. Shape is preserved
locally on conformal maps.
Equal area projections
ƒ area on the map is proportional to the true
area on the earth’s surface
ƒ required when area measures are made
ƒ popular in GIS
Equal area projections
Equidistant projections
ƒ represent the distances to other locations
from either one or two points correctly
UTM
ƒ coordinates are usually measured in meters
from the central meridian (x) and the
equator (y)
ƒ minimal distortions of area, angles distance
and shape at large and medium scales
ƒ very popular for medium scale mapping
GPS Coordinates
ƒ Hundreds of geodetic datums are in use
around the world.
ƒ The Global Positioning system is based on
the World Geodetic System 1984 (WGS84).
ƒ Parameters for simple XYZ conversion
between many datums and WGS-84 are
published by the Defense Mapping Agency.
Lat/Long can also be represented in
planar form
(but is not technically a projection)