Cartography and Map projections Different needs of maps ...

Cartography and
Map projections
Different needs of maps
Different map types !
Reference maps
Thematic maps
Special maps
© Copyright 2002 Lund University GIS Centre. All Rights Reserved
What is special with spatial data?
A coordinate system on the model
135°W
150°W
180°
165°E
North Pole
120°W
150°E
They have a position.
105°W
90°W
75°W
135°E
120°E
105°E
Meridians
Longitude
Parallels
Latitude
105°W
120°W
135°W
150°W
180°
165°E
150°E
135°E
Meridians
&
Parallels
Coordinates
x, y (z) in a
coordinate system.
60°W
45°W
30°W
15°W
60°E
45°E
0° 15°E 30°E
75°E
90°E
90°W
South Pole
75°W
60°W
45°W
30°W
15°W
0°
15°E 30°E
45°E
60°E
75°E
120°E
90°E
105°E
A grid net i.e. a
coordinate
system
on the 3D model
Geocentric three-dimensional
Cartesian coordinates
Three dimensional coordinate system with its origo in the
centre of the earth model. Convenient to use in computations
but not for presenting coordinates (since it has no direct link to
earth surface).
Z
Earth rotation axis
Greenwich meridian
Y
Definition of
spherical
latitude
Prime meridian
Geographical coordinates
Expressed in LATITUDE & LONGITUDE
Angles in
degrees º, minutes ’ & seconds ”
from the
equatorial plane &
prime meridian at Greenwich (London)
Earth rotation axis
Latitude
Prime meridian
Earth rotation axis
Latitude
Definition of
geographic
latitude
X
0
Longitude
0
Longitude
1

Coordinates in lat, long
The shape of the Earth
There are 360 º
around the Earth
90º N and S
180º E and W
1º = 60’ (minutes)
1’= 60” (seconds)
Example:
Lund’s old water tower
long: 13º12’14.00” E
= 13 + 12/60 + 14.00/3600
= 13.203889 degrees
lat: 55º42’42.00” N
= 55 + 42/60 + 42.00/3600
= 55.716667 degrees
Geoid = ”Real” shape of the
Earth.
Irregular in shape.
Caused by non uniform
distribution of mass inside the
Earth
We need a model
Some common ellipsoids
We need a model – Ellipsoid best
Ellipsoid
Semi major axis = a,
(m)
1/f (flattening)
GRS 80
6 378 137.0
298.257 222 101
WGS 84
6 378 137.0
298.257 223 563
Bessel 1841
6 377 397.155
299.1528128
Clarke 1866
6 378 206.4
294.9786982
Definition of an Ellipsoid:
Semi major axis = a
Everest 1830
6 377 276.345
300.8017
Semi minor axis = b
Hayford 1910
6 378 388.0
297.0
Flattening = f
( a − b)
f =
a
International
Krassovsky 1940
6 378 388.0
6 378 245.0
297.0
298.3
ESRI, Understanding Map Projections, 2004
The shape of the Earth II
Geoid = Equipotential surface in
the earth gravity field.
Coincides with the mean sea
surface at the oceans.
Map of the geoid separation
(GRS1980 ellipsoid earth centred)
H = Height above sea level
(from levelling instrument).
+10 +100 m m
Mean Sea level
Topographic Surface
H
h
Geoid
N
Ellipsoid
h = Height above ellipsoid (from
GPS).
N = Geoid height = geoid
separation (computed
from satellite orbit
observations and terrestrial
gravity measurements.
00 m
-10 -100 m m
2

Earth centered
ellipsoid (datum)
Geodetic reference systems
or datums
We move the ellipsoid to fit a certain part of the Earth
Geoid
Earth center
of mass
1. Model of the
Earth Geoid
Ellipsoid
2. Coordinate
system on the
Earth model
How do we create a map?
Prime meridian
0
Earth rotation axis
Longitude
Latitude
3. Adjust position
ellipsoid Geoid
4. Map projection Flat Map
5. Plane 2D Cartesian
coordinate system
6. Simplify
Cartographic and
visualisation
techniques.
Lokal ellipsoid
(datum)
Map projections
Properties of map projections
A method to make the Earth model flat.
All map projections implies distortions.
Choose a map projection that suits an
application as good as possible.
= Unavoidable distortions should be
manageable for the applications.
Equal area property
Conformal – (Local) angle preserving
Equidistant along certain lines
Equal area projection
Preserves the relative sizes of geographic features.
Distort the shape of features.
Conformal
Preserves the local shapes (angles). The relative size of
geographic features changes.
Reality
Map
projection
Reality
Map projection
3

Equidistance in map projections
Caution with the term
equidistant map projections.
A map projection can be equidistant
for a couple of lines, but
no map projection is equidistant along all lines.
The light bulb metaphor
= perspective approach to map
projections
Geometric shape of
the map ”paper”.
Placement of the
paper ”on” the Earth,
aspect.
Placement of the light
bulb.
Different maps.
ESRI, Understanding Map Projections, 2004
Distortions at a
minimum where the
paper is close to the
Earth.
Basic geometrical shapes
of map projections
Azimuthal
Aspect of map projections
There are three main geometrical
forms of map projections:
Polar Equatorial Oblique
Planar projections (azimuthal projections)
Cylindric
Cylindric projections
Conic projections
Normal
Transverse
Oblique
Conic
Normal
Normal
Oblique
Secant (cutting) map projections
Light bulb location (azimuthal)
Azimuthal Secant
Normal cylindric secant
Transverse cylindric secant
ESRI, Understanding Map Projections, 2004
Conic Secant
Gnomonic Stereographic Orthographic
4

Planar projections
Cylindrical projections
Conic projections
Map projection shapes and properties
Light beam
A
a
To get specific properties of
the resulting map we need to
modify the light beam.
The direction is changed
when it cuts the Earth
surface on its way from the
light source to the map
(paper).
Central meridian
This is done mathematically.
Projection parameters I
Projection parameters II
Each map projection has it own set of
projection parameters.
These parameters describe the
details of the relationship between the
geographic coordinates and the map
projection coordinates.
That is they are necessary to fully
define the function f x and f y in:
x =
y =
f
f
x
y
( ϕ , λ )
( ϕ , λ )
Example:
Gauss-Krűger
projection
(Transverse
cylinder)
y
F.E.
False easting – A linear value applied to the
origin of the x-coordinates
False northing – A linear value applied to the
origin of the y-coordinates
x
Area to be mapped
Central meridian – Defines
where the cylinder touches the
earth model. Also defines the
origin of the x-coordinates.
Scale factor – A unitless value applied to the
center point or line of a map projection
5

Choice of map projections I
Choice of map projections III
Large scale mapping
Mapping of small areas
Atlases
Some geographic analysis
World maps for illustration
Robinson projection
Conformal projections should
always be used.
Equal area projections is
often the best choice.
Special projection might be
used (compromise between
equal area and conformal
properties)
Which conformal projection is best for a
certain area?
Large area in North-South
direction:
Transverse cylindrical projection
(e.g. Gauss-Krűger or UTM)
Large area in East-West
direction:
Conical projection
(e.g. Lambert Conformal Conic
projection)
Definition of the Swedish plane coordinate system
Spherical
coordinate system
on the Earth
105°W
120°W
135°W
150°W
180°
165°E
150°E
135°E
A transverse
cylindrical
projection
90°W
120°E
Coordinates in
latitute
and
longitude
75°W
60°W
45°W
75°E
90°E
105°E
Parallels and
meridians are no
longer straight
lines in a Cartesian
coordinate grid
30°W
60°E
15°W
0°
15°E 30°E
45°E
We need a
plane
Cartesian
coordinate
system
15°W 0° 15°E 45°E 60°E
60°N 60°N
45°N 45°N
We define
an origo
15°W 0° 15°E 45°E 60°E
60°N 60°N
45°N 45°N
30°N
30°N
Example:
The Swedish
national
system:
RT90, 2.5
gon W
15°N
15°N
15°N
15°N
0° 0°
0° 0°
15°W
0°
15°E
30°E
45°E
15°W
0°
15°E
30°E
45°E
6

0
0
0
0
0
0
We define
an origo
7500000
-4500000
-3000000 -1500000
0
1500000 3000000
15°W 0° 15°E 45°E 60°E
60°N 60°N
7500000
We define
an origo
7500000
-3000000
-1500000
0
1500000 3000000 4500000
15°W 0° 15°E 45°E 60°E
60°N 60°N
7500000
We create
the grid
6000000
4500000
45°N 45°N
30°N
6000000
4500000
We create
the grid
6000000
4500000
45°N 45°N
30°N
6000000
4500000
We get
negative
coordinates
in western
Sweden !
3000000
1500000
15°N
0° 0°
15°N
3000000
1500000
We move the
north/south
axis far west
of Sweden
=
False
Easting
3000000
1500000
15°N
0° 0°
15°N
3000000
1500000
15°W
0°
15°E
30°E
45°E
15°W
0°
15°E
30°E
45°E
-4500000
-3000000
-1500000
0
1500000
3000000
-3000000
-1500000
0
1500000
3000000
4500000
-3000000
-1500000
0
1500000
3000000
4500000
7500000
15°W 0° 15°E 45°E 60°E
60°N 60°N
7500000
RT90, 2.5
gon W
6000000
45°N 45°N
6000000
4500000
30°N
4500000
Plane, cartesian
coordinate system
3000000
15°N
15°N
3000000
National e.g.
RT 90 in Sweden
1500000
0° 0°
1500000
Coordinates in
x and y, meters
15°W
0°
15°E
30°E
45°E
-3000000
-1500000
0
1500000
3000000
4500000
Plane,
cartesian
local
coordinate
systems
e.g.
Lund’s
coordinate
system
LUND
Universal Transverse Mercator
UTM
A system of map projections that is defined over the whole globe.
All the projections are conformal.
* In polar region (φ>84 o N or φ>80 o S) a planar polar stereographic
map projection is used.
* In other regions a transverse mercator map projection is used
(cylindrical)
7

UTM II
UTM III
Projection parameters:
Central meridian:
Centred in the current zone.
Scale factor: 0.9996
False Easting: 500 000 m
False Northing:
0 m - Norhern hemisphere)
10 000 000 m – Southern hemisphere)
The End
8