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Chapter 2 - Map Projection

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Chapter 2 - Map Projection

9-1-2004

Week 1

- Same coordinate system is used on a same “View” of ArcView or same “Data Frame” in ArcMap.
- Projection - converting digital map from longitude/latitude to two-dimension coordinate system.
- Re-projection - converting from one coordinate system to another

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

b=semiminor 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

- The location reference system for spatial features on the Earth’s 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).

- 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 (-85o30’36”)

- New York City’s La Guardia Airport is located at (73o54’,40o46’). Convert this DMS to DD.

- New York City’s 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.

- 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)

- 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, b=40.77, 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

- 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)

- Map Scale = map distance / earth distance
- RF (representative fraction) – such as 1:25,000, 1:50,000…
- 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?

- 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

- 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.

- Overlay shapes on maps (fig 2-14)
- Tissot’s indicatrix (fig 2-15)

- S=max. areal distortion, = 1.0, no area distortion
- a=b conformal proj. S varies
- ab not conformal

Simple Case

Secant Case

Conic

Cylindrical

Azimuthal

- 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.

- darker areas represent greater distortion

source of data: Dent, 1999

- 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 projection’s 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.

- 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.

- Plane coordinate systems are used in large-scale mapping such as at a scale of 1:24,000.
- accuracy in a feature’s 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

- 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)

- 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)

- divide state into 6x6 mile squares or townships. Each township was further partitioned into 36 square-mile parcels of 640 acres, called sections
- Link for downloading PLSS from Wyoming
- http://www.sdvc.uwyo.edu/clearinghouse/howto.html
- Exercise: download a county’s PLSS from Wyoming and load it to the ArcMap or ArcView.