Map Projections

1
Map Projections

• Francisco Olivera, Ph.D., P.E.
• Srikanth Koka
• Department of Civil Engineering
• Texas AM University

2
Overview

• Geodetic Datum
• Map Projections
• Coordinate systems
• Viewing,Defining,Changing Projections

3
Definition

• 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 and spheres to complex ellipsoids.
• Flat earth models can be accurate over 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.

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Shape of the Earth
... when it is actually an ellipsoid, slightly
larger in radius at the equator than at the poles.
We think of the earth as a sphere ...
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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
F2
O
a
F1
X
?
?

• For the earth
• Major axis a 6378 km
• Minor axis b 6357 km
• Flattening ratio f 1/300

P
P
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Earth Surfaces
Sea surface
Ellipsoid
Topographic surface
Geoid
Geoid is a surface of constant gravity.
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Standard Ellipsoids
Ref Snyder, Map Projections, A working manual,
USGS Professional Paper 1395, p.12
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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.

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Elevation
P
z zp
Topographic Surface
z 0
Mean Sea level Geoid
Elevation is measured from the Geoid
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Standard Vertical Geodetic Datum

• 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).

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Overview

• Geodetic Datum
• Map Projections
• Coordinate systems
• Viewing,Defining,Changing Projections

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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 the
  projected onto a flat surface.

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Map Projection
Projection
Scale
Scale Fraction Map distanceGlobe distance
Representative Fraction Globe distanceEarth
distance
(e.g. 124,000)
(e.g. 0.9996)
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Distortion Projected Maps

• 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 that
  preserving other.

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Distortion Projected Maps

• Conformal projections Preserves local shapes.
• Equal area projections Preserves the area of
  displayed
• Equidistant projections Preserves the distances
  between certain points.

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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. Examples
  (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.

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Conic Projections
Albers and Lambert
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Cylindrical Projections
Mercator
Transverse
Oblique
Tangent Secant
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Azimuthal
Lambert
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Albers Equal-Area Conic
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Lambert Conformal Conic
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Universal Transverse Mercator
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Lambert Azimuthal Equal-Area
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Overview

• Geodetic Datum
• Map Projections
• Coordinate systems
• Viewing,Defining,Changing Projections

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Coordinate Systems

• A coordinate system is used to locate a point of
  the surface of the earth.

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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 earths surface.

The z-coordinate in Global Cartesian and
Projected coordinates is defined geometrically
and in Geographic coordinates gravitationally.
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Global Cartesian Coordinates
Z
Greenwich Meridian
O

Y
X
Equator
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Geographic Coordinates
(0ºN, 0ºE) Equator, Prime Meridian
Longitude line (Meridian)
Latitude line (Parallel)
N
N
W
E
W
E
S
S
Range 90ºS - 0º - 90ºN
Range 180ºW - 0º - 180ºE
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If the Earth were a Sphere
Length on a Meridian AB R Df (same for all
latitudes)
r
D
r
Dl
C
B
Df
0 N
R
Length on a Parallel CD r Dl R Cosf
Dl (varies with latitude)
A
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If the 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

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Cartesian Coordinates
A planar cartesian coordinate system is defined
by a pair of orthogonal (x,y) axes drawn through
an origin.
Y
X
Origin
(xo,yo)
(fo, lo)
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Geographic Transformations

• Moving data between coordinate systems may
  include transformation between the geographic
  coordinate systems.
• United states uses a grid-based method for
  conversions NADCON
• NADCON NAD 1927 and other older GCS to NAD 1983

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Coordinate Systems

• Universal Transverse Mercator (UTM) - a global
  system developed by the US Military Services.
• State Plane - civilian system for defining legal
  boundaries.

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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) in the
  Northern Hemisphere.
• Units are meters

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UTM Zone 14
-99
-102
-96
6
Equator
Origin
-90
-120
-60
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State Plane (USA only)

• Defined for each State in the United States.
• East-West States (e.g. Texas) use Lambert
  Conformal Conic, North-South States (e.g.
  California) use Transverse Mercator.
• Texas has five zones (North, North Central,
  Central, South Central, South) to give accurate
  representation.
• Greatest accuracy for local measurements

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Overview

• Geodetic Datum
• Map Projections
• Coordinate systems
• Viewing,Defining,Changing Projections

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Viewing the Layer Projection

• In ArcCatalog, right click on the layer, and then
  click Properties.
• In the window that opens up, click on the Fields
  tab, and then on Geometry under Data Type column.
• Finally, in the Field Properties frame, click on
  the button located on the Spatial Reference row
  to open the Spatial Reference Properties window.
• In the Spatial Reference Properties window, the
  layer projection is displayed in the Coordinate
  System tab, and the coordinate rage in the X/Y
  Domain tab.

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Defining the Layer Projection

• In the Spatial Reference Properties window, one
  can Select, Import, create a New, Modify or Clear
  the layer projection.
• For projecting on-the-fly, the layer projection
  has to be defined.

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Viewing/Modifying the Data Frame Projection

• The Data Frame projection is user-defined. All
  Data Frames have a projection. If not modified by
  the user, the projection is the one of the first
  layer added to it.
• To view/modify the Data Frame projection, in
  ArcMap, right-click on the Data Frame name and
  then click on Properties to open the Data Frame
  Properties window.
• In the Data Frame Properties window, one can
  select, Import or create a New projection.

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Projection On-The-Fly

• All layers (with projection defined) are
  automatically projected on-the-fly to the data
  frame coordinate system when they added to the
  data frame.
• If the data frame projection is modified, all its
  layers (with projection defined) are
  automatically projected on-the-fly.
• A layer, whose projection is not defined, cannot
  be projected on-the-fly and is displayed
  according to its coordinate values.
• If a layer, whose projection is not defined, is
  added to a data frame that has the same
  projection, the layer will be displayed in the
  correct location.
• A layer can be exported either with its original
  projection or with data frames projection.