Transcript Map Projections - Texas A&M University

Map Projections
Francisco Olivera, Ph.D., P.E.
Srikanth Koka
Department of Civil Engineering
Texas A&M University
Overview
Geodetic Datum
Map Projections
Coordinate Systems
Viewing, Defining and Changing Projections
Shape of the Earth
We think of the
earth as a sphere ...
... when it is actually an ellipsoid,
slightly larger in radius at the
equator than at the poles.
Definitions
A geodetic datum defines the size and shape of the earth and
the origin and orientation of the axis used to define the
location of points.
Over time, geodetic data have evolved from simple flat
surfaces to spheres to complex ellipsoids.
Flat earth models can be accurate for short distances (i.e.,
less than 10 km), spherical earth models for approximate
global distance calculations, and ellipsoidal earth models for
accurate global distance calculations.
Ellipse
Z
An ellipse is defined by:
Focal length = 
Flattening ratio: f = (a-b)/a
Distance F1-P-F2 is constant
for all points P on ellipse
When  = 0 then ellipse =
circle
b
F1

For the earth:
Major axis: a = 6378 km
Minor axis: b = 6357 km
Flattening ratio: f = 1/300
P
O

a
F2
X
Ellipsoid or Spheroid
Z
Rotate an ellipse
around one of its
axis.
b
a O a
X
Rotational axis
Y
Standard Ellipsoids
Ellipsoid
Major axis, Minor axis, Flattening
a (m)
b (m)
ratio, f
Clarke (1866)
6,378,206
6,356,584
1/294.98
GRS80
6,378,137
6,356,752
1/298.57
Ref: Snyder, Map Projections, A working manual, USGS Professional
Paper 1395, p.12
Standard Horizontal Geodetic Data
NAD27 (North American Datum of 1927) uses the
Clarke (1866) ellipsoid.
NAD83 (North American Datum of 1983) uses the
GRS80 ellipsoid.
WGS84 (World Geodetic System of 1984) uses
GRS80.
Earth Surfaces
Sea surface
Geoid
Ellipsoid
Topographic
surface
Geoid is a surface of constant gravity.
Earth Surfaces
Topographic surface
Ocean
Geoid
Ellipsoid
Gravity Anomaly
Elevation
P
z = zp
Topographic Surface
z=0
Mean Sea level = Geoid
Elevation is measured from the Geoid
Standard Vertical Geodetic Data
A vertical datum defines elevation z, taking into account a map
of gravity anomalies between the ellipsoid and the geoid.
NGVD29 (National Geodetic Vertical Datum of 1929)
NAVD88 (North American Vertical Datum of 1988)
Overview
Geodetic Datum
Map Projections
Coordinate Systems
Viewing, Defining and Changing Projections
Map Projections
A map projection is a mathematical algorithm to transform
locations defined on the curved surface of the earth into
locations defined on the flat surface of a map.
The earth is first reduced to a globe and then projected onto
a flat surface.
Map Projection
Scale
Projection
Representative Fraction
Scale Fraction
Globe distance
Earth distance
Map distance
Globe distance
(e.g. 1:24,000)
(e.g. 0.9996)
Map Distortion
In the process of transforming a curved surface into a flat surface,
some geometric properties are modified.
The geometric properties that are modified are:
Area (important for mass balances)
Shape
Direction
Length
The difference between map projections has to do with which
geometric properties are modified.
Depending on the type of analysis, preserving one geometric property
might be more important than preserving another.
Map Distortion
Conformal projections: Preserves shapes.
Equal area projections: Preserves area.
Equidistant projections: Preserves the distances between
certain points.
Types of Projections
Conic: Screen is a conic surface. Lamp at the center of the earth.
Examples: Albers Equal Area, Lambert Conformal Conic. Good for
East-West land areas.
Cylindrical: Screen is a cylindrical surface. Lamp at the center of
the earth. Example: Transverse Mercator. Good for North-South land
areas.
Azimuthal: Screen is a flat surface tangent to the earth. Lamp at
the center of the earth (gnomonic), at the other side of the earth
(stereographic), or far from the earth (orthographic). Examples:
Lambert Azimuthal Equal Area. Good for global views.
Conic Projections
Albers and Lambert
Cylindrical Projections
Mercator
Transverse
Oblique
Tangent
Secant
Azimuthal
Lambert
Albers Equal-Area Conic
Lambert Conformal Conic
Universal Transverse Mercator
Lambert Azimuthal Equal-Area
Overview
Geodetic Datum
Map Projections
Coordinate Systems
Viewing, Defining and Changing Projections
Coordinate Systems
A coordinate system is used to locate a point of the surface of
the earth.
Coordinate Systems
Global Cartesian coordinates (x, y, z) for the whole
earth
Geographic coordinates (f, l, z) for the whole earth
Projected coordinates (x, y, z) on a local area of the
earth’s surface
Global Cartesian Coordinates
Z
Greenwich
Meridian
•
X
O
Y
Equator
Geographic Coordinates
90 W
Geographic Coordinates
(0ºN, 0ºE)
Equator, Prime Meridian
Longitude line (Meridian)
Latitude line (Parallel)
N
N
W
E
S
Range: 180ºW - 0º - 180ºE
W
E
S
Range: 90ºS - 0º - 90ºN
Geographic Coordinates
Earth datum defines the standard values of the ellipsoid and
geoid.
Latitude (f) and longitude (l) are defined using an ellipsoid.
Elevation (z) is defined using a geoid.
Latitude f
Take a point S on the
surface of the ellipsoid and
define there the tangent
plane mn.
Define the line pq through
S and normal to the
tangent plane.
Angle pqr is the latitude f,
of point S.
m
S p
n
f
q
r
If Earth were a Sphere ...
Length on a Meridian:
AB = R Df
(same for all latitudes)
Length on a Parallel:
CD = r Dl = R Cosf Dl
(varies with latitude)
r
R
r
Dl
C
B
Df
A
D
If Earth were a Sphere ...
Example:
What is the length of a 1º increment on a meridian and on a parallel at
30N, 90W? Radius of the earth R = 6370 km.
Solution:
• A 1º angle has first to be converted to radians:
p radians = 180°, so 1º = p/180° = 3.1416/180° = 0.0175 radians
• For the meridian: DL = R Df = 6370 Km * 0.0175 = 111 km
• For the parallel: DL = R Cosf Dl = 6370 * Cos30° * 0.0175 = 96.5 km
• Meridians converge as poles are approached
Cartesian Coordinates
A planar cartesian coordinate system is defined by a pair
of orthogonal (x, y) axes drawn through an origin.
Y
X
Origin
(fo, lo)
(xo, yo)
Coordinate Systems
Universal Transverse Mercator (UTM) - a global system
developed by the US Military Services.
State Plane - civilian system for defining legal boundaries.
Universal Transverse Mercator
Uses the Transverse Mercator projection.
60 six-degree-wide zones cover the earth from East to West
starting at 180° West.
Each zone has a Central Meridian (lo).
Reference Latitude (fo) is the equator.
(Xshift, Yshift) = (xo,yo) = (500,000, 0).
Units are meters.
UTM Zone 14
-99°
-102°
-96°
6°
Equator
Origin
-120°
-90 °
-60 °
State Plane
Defined for each state or part of a state in the United States
Texas has five zones (North, North Central, Central, South
Central and South) to give accurate representation.
East-West States (e.g. Texas) use Lambert Conformal Conic;
North-South States (e.g. California) use Transverse Mercator
Greatest accuracy for local measurements
Overview
Geodetic Datum
Map Projections
Coordinate Systems
Viewing, Defining and Changing Projections
Viewing Projection
There are two ways to view
projection details in ArcCatalog.
One is by viewing the projection
details in the map display. To do
this, make sure that dataset is
selected in table of contents and
click on the Metadata tab in the
map display.
The first paragraph defines the
projected coordinate system and
the geographic coordinate
system.
The second paragraph defines
the extent of the dataset.
Viewing Projection
Another way is viewing the
projection details in Spatial
Reference Properties wizard.
To view the projection details
of a dataset, right-click on the
layer and then click
Properties. In the wizard that
opens up, click on the Fields
tab; then on geometry, under
data, type “column.”
Finally, in the Field properties
frame, click on the button
located on the Spatial
Reference row.
Viewing Projection
Defines the extent of
the shapefile and
feature class.
Defining and Changing the Projection
ArcToolBox contains
tools for data conversion
and data management.
Projection definition and
projection change can be
accomplished using
ArcToolBox
ArcCatalog can also be
used for defining the
data projection.
Projecting On-The-Fly
Each data frame in an ArcMap
document has its own projection.
The projection of the data frame can
be defined using Data Frame
Properties/Coordinate System.
Data (with map projection defined)
added to a data frame is re-projected
on-the-fly to the data frame’s
projection.
A layer in a data frame can be
exported either with its original
projection or with data frame’s
projection.
Defining Projection
Before moving feature classes into a
feature dataset, the spatial
reference of the feature dataset
should be defined.
In the Spatial Reference Properties
wizard:
Select can be used to select a
coordinate system available.
Import can be used to define the
projection using an existing
projection from a feature dataset,
feature class or shape file.
New can be used to assign custom
projection.
Projecting Data
ArcToolBox can be used for
changing projections.
To change the projection of a
dataset, its original coordinate
system should be known.