Projections

Map Projections
• Geodesy - the shape of the earth and definition
of earth datums
• Map Projection - the transformation of a curved
earth to a flat map
• Coordinate systems - (x,y) coordinate systems
for map data
Geodesy

Geodesy
• Determination the Earth’s size (geometry), shape
(gravity) and figure (surface).
• Determination of the Earth’s motions (in space: polar
motion, variations in rotation rate), its deformations (e.g.,
plate tectonic motion, plate boundary deformation,
volcanoes, land subsidence), and gravity variations.
• Definition and maintenance of terrestrial reference
frames (datums) for precise 3D positioning, thus
providing the backbone for mapping, surveying, and GIS.
Geodesy

Triangulation from Dunkirk to Barcelona
Jean Baptiste Delambre measured
the stations between Dunkirk and
Rodez, France. The southern segment,
from Rodez to Barcelona, was measured
by Pierre Méchain. They began the
project in 1792.
Geodesy

Spherical Earth
• Authalic Sphere
• Basic figure for mapping
• Radius = 6371 km
• Meridians from pole to
pole
• Equator and small circles
perpendicular to the
meridians
• Geographic grid of
meridians and small
circles
Geodesy

Prime meridian – Greenwich from 1884
Geodesy

Latitude and Longitude on a Sphere
Greenwich
meridian
λ=0°
Z
N
Meridian of longitude
Parallel of latitude
X
W
λ=0-180°W
O ϕ
•
λ
•
Equator =0°
ϕ
R
P
•
λ=0-180°E
ϕ=0-90°S
ϕ=0-90°N
E
•
λ - Geographic longitude
ϕ - Geographic latitude
Y
R - Mean earth radius
O - Geocenter
Geodesy

Spherical Earth Approximation
Z = Pole of Rotation
Y: Right-handed
X = Greenwich Meridian
Geodesy

What kind of ellipsoid
• Cassini's report to the Academy, that the length of a degree seemed to
get shorter towards the pole, generated an intense controversy
between French and English scientists and resulting in arc
measurement expeditions to Lapland (1736/37, average latitude 66°
20’) and Peru/Ecuador (1739-1743, (average latitude 1° 31' S).
Oblate
Prolate
Geodesy

French triangulation
in Lapland
1736
Geodesy

Ellipsoid or Spheroid
Rotate an ellipse around an axis
The earth is flattened slightly
at the poles and bulges
somewhat at the equator
a
Z
b
O
a
Y
X
Rotational axis
Geodesy

Definition of Latitude, φ
m
S
p
O
q
φ
r
n
(1) Take a point S on the surface of the ellipsoid and define
there the tangent plane, mn
(2) Define the line pq through S and normal to the
tangent plane
(3) Angle pqr which this line makes with the equatorial
plane is the latitude φ, of point S
Geodesy

Geoid – Figure of the Earth
CM
CE
By the early 19th century,
scientists like Laplace (1802),
Gauss (1828), Bessel (1837)
recognized that the assumption of
an ellipsoidal earth model was
untenable under sufficiently high
observational accuracy. One
could no longer ignore the
deviation of the physical plumb
line, to which measurements refer,
from the ellipsoidal normal.
Geodesy

Different ellipsoids for different areas
on the globe
Geodesy

Standard Ellipsoids
Ellipsoid
Airy
(1830)
Clarke
(1866)
WGS 84
(1984)
Major Minor Flattening
axis, a (m) axis, b (m) ratio, f
6377,563 6356,257 1/299,32
6,378,206 6,356,584 1/294.98
6,378,137 6,356,752 1/298.57
Ref: Snyder, Map Projections, A working manual, USGS
Professional Paper 1395, p.12
Geodesy

Position shifts from datum differences
Geodesy

What is a Projection
• If you could project light from a source through the
earth's surface onto a two-dimensional surface, you
could then trace the shapes of the surface features
onto the two-dimensional surface.
• This two-dimensional surface would be the basis for
your map.
Geodesy

Projections always distort the map
Sides of different lenght
Rectangular
form
Geodesy

Map projections
• The characteristics normally considered in choosing a
map projection are as follows:
1. Area - equal-area
area 2. Shape – conformal
3. Scale – one or more lines on the map along which the
scale remains true
4. Direction – conformal, azimuthal
5. Special – gnomonic 6. Method of construction
Geodesy

Earth to Globe to Map
Map Scale:
Representative Fraction
=
Globe distance
Earth distance
(e.g. 1:50,000)
Geodesy
=
Map Projection:
Scale Factor
Map distance
Globe distance
(e.g. 0.9996)

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
Geodesy

Conic Projections
(Albers, Lambert)
Geodesy

Cylindrical Projections
(Mercator)
Normal
Transverse
Oblique
Geodesy

Azimuthal
(Lambert)
Geodesy

Projection onto a Flat Surface
Geodesy

Lambert equal-area projection
Geodesy

Conic equal-area projection
Geodesy

Mercator (Normal cylindric)
Conformal
Geodesy

Great circle navigation
Geodesy

Mollweide
Equal-area
Geodesy

Geodesy

South America
Geodesy

Types of Coordinate Systems
• (1) Global Cartesian coordinates (X,Y,Z)
for the whole earth
• (2) Geographic coordinates (φ,(
λ, z)
• (3) Projected coordinates (x, y, z) on a
local area of the earth’s s surface
• The z-coordinate z
in (1) and (3) is
defined geometrically; ; in (2) the z-z
coordinate is defined gravitationally
Geodesy

Global Cartesian Coordinates
Greenwich
Meridian
(X,Y,Z)
Z
O
•
Y
X
Equator
Geodesy

Geographic Coordinates (φ, λ, z)
• Latitude (φ)(
) and Longitude (λ)(
) defined
using an ellipsoid, , an ellipse rotated about
an axis
• Elevation (z) defined using geoid, , a
surface of constant gravitational potential
• Earth datums define standard values of
the ellipsoid and geoid
Geodesy

Coordinate System
A planar coordinate system is defined by a pair
of orthogonal (x,y) axes drawn through an origin
Y
Origin
X
(φ o ,λ o )
(x o ,y o )
Geodesy

Cartesian Coordinate System
Planar coordinate systems are based on
Cartesian coordinates.
Geodesy

Any projected data that you add to ArcMap, , or that
you project within ArcMap, , is associated with a
projected coordinate system (PCS) in addition to its
underlying Geographic Coordinate System (GCS).
Geodesy

Universal Transverse Mercator
(UTM)
A comprehensive system for identifying locations and making
measurements over most of the earth's surface.
• Divide the world into sixty vertical strips, each spanning
six degrees of longitude. Apply a custom Transverse
Mercator projection to each strip and use false eastings
and northings to make all projected coordinates positive.
• Data that crosses zones is subject to distortion.
Geodesy

UTM-zones
Geodesy

UTM-zones
Sweden lies in 6
zones
Geodesy

Geodesy

Geodesy

UPS – (Universal Polar Stereographic grid)
Geodesy

ArcGIS Reference Frames
• Defined for a feature
dataset in
ArcCatalog
• Coordinate System
• Projected
• Geographic
• X/Y Domain
• Z Domain
Geodesy

Coordinate Systems
• Geographic
coordinates (decimal
degrees)
• Projected coordinates
(length units, ft or
meters)
Geodesy

Geodesy

Datum Transformations
7-parameter transformation
NAD27 to NAD 83
Geodetic tools available at:
http://www.ngs.noaa.gov/TOOLS/
Geodesy

Summary Concepts
• Two basic locational systems: geometric
or Cartesian (x, y, z) and geographic or
gravitational (φ,(
λ, z)
• Mean sea level surface or geoid is
approximated by an ellipsoid to define an
earth datum which gives (φ,(
λ) and
distance above geoid gives (z)
Geodesy