Chapter 2 Map Projection

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Chapter 2 - Map Projection

• Week 2
• Spring 2004

2
Introduction

• Same coordinate system is used on a same view of
  ArcView or same layer in ArcMap
• Projection - converting digital map from
  longitude/latitude to two-dimension coordinate
  system.
• Re-projection - converting from one coordinate
  system to another

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Size and Shape of the Earth

• Shape of the Earth is called geoid
• The sciences of earth measurement is called
  Geodesy
• ellipsoid - reference to the Earth shape.

bsemiminor axis (polar radius)
f (a-b)/a - flattening 1/298.26 for GRS1980,
and 1/294.98 for Clarke 1866
a semimajor axis (equatorial radius)
The geoid bulges at the North Pole and is
depressed at the South Pole
4
Geographic Grid

• The location reference system for spatial
  features on the Earths surface, consisting of
  Meridians and Parallels.
• Meridians - lines of longitude for E-W direction
  from Greenwich (Prime Meridian)
• Parallels - line of latitude for N-S direction
• North and East are positive for lat. and long.
  such as Cookeville is in (-85.51, 36.17).

5
DMS and DD (sexagesimal scale)

• Longitude/Latitude can be measured in DMS or DD,
• For example in downtown Cookeville, a point with
  (-85.51, 36.17) which is in DD. To convert DD to
  DMS, we will have to do several steps for
  example, to convert -85.51 to DMS,
• 0.51 60 30.6, this add 30 to minute and leave
  0.6.
• 0.6 60 36, this add 36 to seconds. Thus, the
  longitude is (-85o3036)

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Exercise - convert New York Citys DMS to DD

• New York Citys La Guardia Airport is located at
  (73o54,40o46). Convert this DMS to DD.

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Exercise - convert New York Citys DMS to DD

• New York Citys La Guardia Airport is located at
  (73o54,40o46). Convert this DMS to DD.
• 54/60 0.9 and 46/60 0.77
• (73.90, 40.77) is the answer.

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GPS (Global Positional System)
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Applications of GPS

• Location Identification find out where you are
  in a big city or highway
• Travel find out distance to the next exit
• Survey streets, building, school, etc.
• Forest where are you now?
• Boating/Fishing locations
• Aviation landing, navigation
• Defense Industry where is the target?

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At least 3 satellites are required
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SVs (Space Vehicles)

• GPS Constellation consists of 24 satellites that
  orbit the earth in 12 hours. There are more than
  24 operational satellites as new ones are
  launched to replace older.
• Six orbital planes (4 sat. each), 20,200 km
  altitude, equally spaced (60o apart), and
  inclined at about 55o with respect to the
  equatorial plane, which provides 5-8 SVs visible
  from any point on the earth

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Length of Parallels/Angular/Great Circle

• Length of parallels cos(?) length of equator
• Meridians and parallels intersect at right
  angles.
• Loxodrome meridians, parallels and equator all
  have constant compass bearing.
• Great circle arc shortest distance between 2
  points on earth, formed by passing a plane
  through the center of the sphere.
• All meridians and equator are great circle.
• Small Circle circles on the grid are not great
  circle. Parallels of latitude of small circle
  (except equator).
• Travel along N-S is the shortest, but not E-W
  (except along equator)
• Azimuth angel between great circle and meridian
  (fig 2.10)

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Measure Distance on a Spherical Surface

• cos D sin a sin b cos a cos b cos c
• where D is the distance between A and B in
  degrees
• a is the latitude of A, b is the latitude of B
  and c is the difference in longitude between A
  and B.
• Multiply D by by the length of one degree at the
  equator,which is 69.17 miles. For example
• Between Cookeville and New York City, we have a
  36.17, b40.77, and c -85.51 - (-73.90) -
  11.61
• cos D sin36.17 sin 40.77 cos 36.17 cos
  40.77 cos (-11.61) 0.988, cos-1 0.988
  8.885
• Distance 8.885 69.17 615 miles

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Projection to represent the earth as a reduced
model of reality

• Transformation of the spherical surface to a
  plane surface. Graticule meridians and
  parallels on a plane surface.
• Projection Process (fig 2.12)
• Best fit (earth geoid)
• Reference ellipsoid
• Generating globe
• Map projection (2D surface)

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Scale

• Map Scale map distance / earth distance
• RF (representative fraction) such as 125,000,
  150,000
• Compute the scale with 10-in radius globe
• Scale Bar, Verbal Scale (1 in 2 miles)
• Determine scale of 1 inch to 4 miles
• Scale problem distance between two points is 5
  mile, what is the scale of a map on which the
  points is 3.168 inches apart?

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Map projections

• Distortion caused by tearing, shearing and
  compression from 3D to 2D.
• For a large scale map, distortion is not a major
  problem. However, the mapped is larger, then
  distortion will occur.
• Conformal - preserves local shapes
• Equivalent - preserves size
• Equidistant - maintain consistency of scale for
  certain distance
• Azimuthal - retains accurate direction
• Conformal and Equivalent - mutually exclusive,
  otherwise a map projection can have more than one
  preserved property

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Projections

• Equal-Area Mapping - distort Shape, but
  important in thematic mapping, such as in
  population density map.
• Conformal Mapping shapes of small areas are
  preserved, meridian intersect parallels at right
  angles. Shapes for large areas are distorted.
• Equidistance Mapping preserve great circle
  distances. True from one point to all other
  points, but not from all points to all points.
• Azimuthal Mapping true directions are shown
  from a central point to other points, not from
  other points to other points. This projection is
  not exclusive, it can occur with equivalency,
  conformality and equidistance.

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Measuring Distortion

• Overlay shapes on maps (fig 2-14)
• Tissots indicatrix (fig 2-15)

• Smax. areal distortion, 1.0, no area
  distortion
• ab conformal proj. S varies
• a?b not conformal

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Simple Case
Secant Case
Conic
Cylindrical
Azimuthal
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Standard line - the line of tangency between the
projection surface and the reference globe

• Simple case has one standard line where secant
  case has two standard lines.
• Scale Factor(SF) - the ratio of the local scale
  to the scale of the reference globe
• SF 1 in standard line.
• Central line - the center (origin) of a map
  projection
• To avoid having negative coordinates , false
  easting and false northing are used in GIS. Move
  origin of map to SW corner of the map.

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Planes of deformation

• darker areas represent greater distortion

source of data Dent, 1999
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Commonly used map projections

• Transverse Mercator - use standard meridians,
  required parameters central meridian, latitude
  of origin (central parallel) false easting, and
  false northing.
• Lambert Conformal Conic - good choice for
  mid-latitude area of greater east-west than
  north-south extent (U.S. Tn,,,,). Parameters
  required first/second standard parallels,
  central meridian, latitude of projections
  origin, false easting/northing.
• Albers Equal-Area Conic - requires same
  parameters as Lambert Conformal
• Equidistant Conic - preserves distance property
  along all meridians and one or two standard
  parallels.

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Datum

• Spheroid or ellipsoid- a model that approximate
  the Earth - datum is used to define the
  relationship between the Earth and the ellipsoid.
• Clarke 1866 - was the standard for mapping the
  U.S. NAD 27 is based on this spheroid, centered
  at Meades Ranch, Kansas.
• WGS84 (GRS80) - from satellite orbital data. More
  accurate and it is tied into a global network and
  GPS. NAD 83 is based on this datum.
• Horizontal shift between NAD 27 and NAD can be
  large (fig 2.10)
• USGS 7.5 minute quad map is based on NAD 27.

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Coordinate Systems

• Plane coordinate systems are used in large-scale
  mapping such as at a scale of 124,000.
• accuracy in a features absolute position and its
  relative position to other features is more
  important than the preserved property of a map
  projection.
• Most commonly used coordinate systems UTM, UPS,
  SPC and PLSS

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UTM

• See the back of front cover for UTM zones.
• Divide the world into 60 zones with 6o of
  longitude each,covering surface between 84oN and
  80oS.
• Use Transverse Mercator projection with scale
  factor of 0.9996 at the central meridian. The
  standard meridian are 180 km east and west of the
  central meridian.
• false origin at the equator and 500,000 meters
  west of the central meridian in N Hemisphere, and
  10,000,000 m south of the equator and 500,000 m
  west of the central meridian.
• Maintain the accuracy of at least one part in
  2500 (within one meter accuracy in a 2500 m
  line)

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The SPC System

• Developed in 1930.
• To maintain required accuracy of one in 10,000,
  state may have two ore more SPC zones. (see the
  front side of the back cover)
• Transvers Mercator is used for N-S shapes,
  Lambert conformal conic for E-W direction.
• Points in zone are measured in feet origianlly.
• State Plane 27 and 83 are two systems. State
  Plane 83 use GRS80 and meters (instead of feet)

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PLSS

• divide state into 6x6 mile squares or townships.
  Each township was further partitioned into 36
  square-mile parcels of 640 acres, called sections

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Homework 2, due Next Friday midnight (2/6/04)

• Take GPS and get 10 distinct readings from your
  choices (such as house, farm, church, school,
  grocery stores..etc)
• Write down these ten readings in DD and UTM Zone
  16 (20 points)
• Page 36 39, Changs book (20 points)
• Do the following modifications
• Copy folder u\4210\chang\chap02 to your personal
  folder(u\4210\students\hw\hw1) and start working
  on this homework using files from this folder. If
  you have trouble doing this, please either ask me
  or your classmates.
• In Step 3 of Task 1, save the output of the
  projected file to your own folder
  (u\4210\students\yourname\hw\hw2)
• Refer all the files mentioned in the task to your
  own chap02 folder.