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Chapter 2 - Map Projection. 9-1-2004 Week 1. Introduction. 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.

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Introduction l.jpg
Introduction

  • 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


Size and shape of the earth l.jpg
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.

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


Geographic grid l.jpg
Geographic Grid

  • 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).


Dms and dd sexagesimal scale l.jpg
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 (-85o30’36”)


Exercise convert new york city s dms to dd l.jpg
Exercise - convert New York City’s DMS to DD

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


Exercise convert new york city s dms to dd7 l.jpg
Exercise - convert New York City’s 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 angular great circle l.jpg
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)


Measure distance on a spherical surface l.jpg
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, 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


Projection to represent the earth as a reduced model of reality l.jpg
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)


Scale l.jpg
Scale reality

  • 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?


Map projections l.jpg
Map projections reality

  • 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


Projections l.jpg
Projections reality

  • 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.


Measuring distortion l.jpg
Measuring Distortion reality

  • 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

  • ab not conformal


Slide15 l.jpg

Simple Case reality

Secant Case

Conic

Cylindrical

Azimuthal


Standard line the line of tangency between the projection surface and the reference globe l.jpg
Standard line - realitythe 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.


Planes of deformation l.jpg
Planes of deformation reality

  • darker areas represent greater distortion

source of data: Dent, 1999


Commonly used map projections l.jpg
Commonly used map projections reality

  • 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.


Datum l.jpg
Datum reality

  • 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.


Coordinate systems l.jpg
Coordinate Systems reality

  • 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


Slide21 l.jpg
UTM reality

  • 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)


The spc system l.jpg
The SPC System reality

  • 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)


Slide23 l.jpg
PLSS reality

  • 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.