1. 2.10 Under what conditions are mixed cells a problem in raster data models? In what ways may the problem of mixed cells be addressed?
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2. 2.18
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3. 2. 21 Why do we need to compress data
2.21 Why do we need to compress data? Which is most commonly compressed, raster data or vector data? Why?
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4. 2.24 b a c d e f s
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5. Geodesy & Map Projections
Chapter 3
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6. Introduction To effectively use GIS, it is important to understand:
How coordinate systems are established for the surface of the Earth.
How coordinates are measures on the Earth’s curved surface.
How these coordinates are converted for use on flat maps
To understand these things we need some knowledge of geodesy and map projections.
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7. Geodesy
Geodesy is the science of measuring the size and shape of the earth.
All measurements are relative to some reference, and the best estimates of this reference have changed over time.
Maps use a two dimensional reference system, but this doesn’t work well for long distances or over the whole Earth.
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8. Defining a Spatial Referencing System
Every spatial feature needs to be referenced to a location for GIS use
Spatial reference systems provide a framework to define positions on the Earth‘s surface.
Steps
Define the size and shape of the Earth.
Establish a datum – reference surface from which other points can be measured.
Develop a spatial reference system:
Origin
Orientation of the axes
Units of measure
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9. Difficulty in Defining Coordinates for the Earth
Three complicating factors:
A flat map must distort geometry in some way.
The irregular shape of the Earth.
The imperfections of our measurements.
Because of these three factors we may have several sets of coordinates for the same location.
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10. Lecture 3

11. Early Measurements The Earth as a Sphere
4% error
Other early scholars were no where near as close.
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12. Defining the Ellipsoid
Newton reasoned that the Earth was not a sphere.
Efforts were then focused on measuring the size of the ellipsoid.
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13. Local or Regional Ellipsoid
Origin, r1, and r2 of ellipsoid specified such that separation between ellipsoid and Geoid is small
These ellipsoids have names, e.g., Clarke 1880, or Bessel
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14. Lecture 3

15. The Earth is NOT an Ellipsoid
(only very close in shape)
The Earth has irregularities in it - deviations from a perfectly ellipsoidal shape
These deviations are due to differences in the gravitational pull of the Earth
Deviations are NOT the surface topography
Not mountains and valleys but deviations in the basic shape
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16. The Earth’s True Shape is Best Described as a GEOID
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17. The Geoid is a measured surface (not mathematically defined)
Found via surface instruments (gravimeters)
towed behind boats, planes, or carried in vehicles.
Or, from measurements of satellite paths
(ephemerides)
May be thought of as an approximation
of mean sea level
An ephemeris (plural: ephemerides; from the Greek word ἐφήμερος ephemeros "daily") is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds are used for astronomy and astrology. Even though this was also one of the first applications of mechanical computers, an ephemeris will still often be a simple printed table.
The position is given to astronomers in a spherical polar coordinate system of right ascension and declination or to astrologer in longitude along the zodiacal ecliptic, and sometimes declination. Astrological positions may be given for either noon or midnight.
An astronomical ephemeris may also provide data on astronomical phenomena of interest to astrologers and astronomers such as eclipses, apparent retrogradation/planetary stations, planetary ingresses, sidereal time, positions for the Mean and True nodes of the moon, the phases of the Moon, and sometimes even the position(s) of Chiron, and other minor celestial bodies.
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18. Measuring Elevations (Dynamic Height) Lecture 3
Dynamic heights are important in studying the flow of water,
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19. Datum
A set of points on Earth for which the horizontal and vertical positions have been accurately measured.
These form a mathematical surface from which all other points can be measured.
Many countries have government bodies charged with making these measurements; e.g. National Geodetic Survey
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20.
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21. Lecture 3

22. Data Sheet
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23. Defining a Datum Horizontal Datum Vertical Datum Specify the ellipsoid
Specify the coordinate locations of features on this ellipsoidal surface
Vertical Datum
Specify the ellipsoid
Specify the Geoid – which set of measurements will you use, or which model
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24. Two Main Eras of Datums
There are two horizontal control networks commonly referred to
North American Datum of 1927 (also NAD27)
North American Datum of 1983 (also NAD83), to replace NAD27
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25. NAD27 vs NAD83
Kansas
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26. Datum “Adjustment”
A datum adjustment is a calculation of the coordinates of each benchmark – this is how we specify the “reference surface”
Not straightforward, because of contradictions
Errors in distance, angle measurements
Improvements in our measurements of Geoid, best spheroid
Improvements in computing capabilities
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27. Datum Versions
NAD83 (19860
NAD83(HARN) – High Accuracy Reference Networks
NAD83(1996)
NAD83(2007)
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28. Vertical Datums Two major vertical datums,
 Like horizontal, but referenced to standard elevation and established using vertical leveling
 Two major vertical datums,
 North American Vertical Datum of 1927 (NAVD29), and an update,
 North American Vertical Datum of 1988 (NAVD88)
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29. Global Ellipsoid
Selected so that these have the best fit “globally”, to sets of measurements taken across the globe.
Generally have less appealing names, e.g. WGS84, or ITRF 2000
World Geodetic Survey
International Terrestrial Reference Frame (ITRF)
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30. Datum Transformations
Converting coordinates from one datum to another requires a transformation.
Simple formulas do not exist for NAD27 to NAD83 conversions.
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31. The Evolution of N. Am. Datums
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32. Review Geographic coordinates in decimal degrees:
Latitude is positive above the equator and negative below.
Longitude is positive east of the Greenwich meridian and negative west of the Greenwich meridian.
Two eras of North American datums:
NAD 27 and NAD 83.
Must transform from one to the other.
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33. There are many different versions of the NAD83 datum.
These should not be combined unless you are certain that the errors will be small.
There are also vertical datums.
Elevations may be ellipsoidal or orthometric heights.
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34. GIS Use and Datums
GIS projects should not mix datums unless you are sure that the datum shifts are small relative to the analysis.
Datum transformations are estimated relationships that are developed with a specific data sets.
All datum transformations will introduce error into the data set.
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35. Map Projections
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36. Map projections are used to transfer or
“project” geographical coordinates onto a
flat surface. .
There are many projections/coordinate systems:
Maine example:
NAD 27 Universal Transverse Mercator – Zone 19N
NAD 27 Maine State Plane (based on the TM projection)
East Zone
West Zone
NAD 83 Universal Transverse Mercator– Zone 19N
NAD 83 Maine State Plane (based on the TM projection)
Central Zone
State plane is based on two projections TM –long NS & Lambert CC for zones long EW
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37. Lecture 3 Geodetic/geographic
A particular place on earth will have different values depending upon the choice of datum, coordinate system and units of measure.
One common source of incompatibility is between the North American Datums of 1927 and To start with, they used different reference ellipsoids, then the readjusted local surveys.
For these reasons, it is very important to know the full spatial reference system (reference ellipsoid and datum) when putting together data from different sources.
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38. Lecture 3

39. Projections may be categorized by:
The location of projection source
The projection surface
Surface orientation
Distortion properties
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40. Categorized by the Location of Projection Source
Gnomonic - center of globe
Stereographic - at the antipode
Orthographic - at infinity
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Source:

41. The projection surface: Cone – Conic Cylinder - Cylindrical
Plane - Azimuthul
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42. The projection surface: Cone – Conic Cylinder - Cylindrical
Plane - Azimuthul
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43. Projection Surfaces – “developable”
it is "surface" that can be flattened onto a plane without distortion (i.e. "stretching" or "compressing").
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44. The Tangent Case vs. The Secant Case
In the tangent case the cone, cylinder or plane just touches the Earth along a single line or at a point.
In the secant case, the cone, or cylinder intersects or cuts through the Earth as two circles.
Whether tangent or secant, the location of this contact is important because it defines the line or point of least distortion on the map projection.
This line of true scale is called the standard parallel or standard line.
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45. Standard Parallel
The line of latitude in a conic or cylindrical projection where the cone or cylinder touches the globe.
A tangent conic or cylindrical projection has one standard parallel.
A secant conic or cylindrical projection has two standard parallels.
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46. The Orientation of the Surface
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47. Projections Categorized by Orientation:
Equatorial - intersecting equator
Transverse - at right angle to equator
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48. Specifying Projections
The type of developable surface (e.g., cone)
The size/shape of the Earth (ellipsoid, datum), and size of the surface
Where the surface intersects the ellipsoid
The location of the map projection origin on the surface, and the coordinate system units
Lamberts Conformal Conic
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49. Defining a Projection – LCC (Lambert Conformal Conic)
The LCC requires we specify an upper and lower parallel – 20o & 60o
An ellipsoid – GRS 1980
A central meridian – 96o
A projection origin – Lat. 40o
origin
central
meridian
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50. Conformal Projections
Locally preserves angles/shape.
Any two lines on the map follow the same angles as the corresponding original lines on the Earth.
Projected graticule lines always cross at right angles.
Area, distance and azimuths change.
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51. Equidistant Projections
A map is equidistant when the distances between points differs from the distances on Earth by the same scale factor.
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52. Equivalent/Equal Area Projection
Equivalent/equal area projections maintain map areas proportional to the same areas of the Earth.
Shape and scale distortions increase near points 90o from the central line.
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53. “Standard” Projections
Governments (and other organizations) define “standard” projections to use
Projections preserve specific geometric properties, over a limited area
Imposes uniformity, facilitates data exchange, provides quality control, establishes limits on geometric distortion.
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54. National Projections
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55. Lecture 3

56. Map Projections vs. Datum Transformations
A map projections is a systematic rendering from 3-D to 2-D
Datum transformations are from one datum to another, 3-D to 3-D or 2-D to 2-D
Changing from one projection to another may require both.
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57. From one Projection to Another
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58. Lecture 3

59. Common GIS Projections
Mercator- A conformal, cylindrical projection tangent to the equator.  Originally created to display accurate compass bearings for sea travel. An additional feature of this projection is that all local shapes are accurate and clearly defined.
Transverse Mercator - Similar to the Mercator except that the cylinder is tangent along a meridian instead of the equator. The result is a conformal projection that minimizes distortion along a north-south line, but does not maintain true directions.
Lambert Equal Area - An equidistant, conic projection similar to the Lambert Conformal Conic that preserves areas.
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60. Lambert Conformal Conic – A conic, confromal projection typically intersecting parallels of latitude, standard parallels, in the northern hemisphere.  This projection is one of the best for middle latitudes because distortion is lowest in the band between the standard parallels. It is similar to the Albers Conic Equal Area projection except that the Lambert Conformal Conic projection portrays shape more accurately than area.
Albers Equal Area Conic - This conic projection uses two standard parallels to reduce some of the distortion of a projection with one standard parallel. Shape and linear scale distortion are minimized between standard parallels.
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61. Map Projections Summary
Projections specify a two-dimensional coordinate system from a 3-D globe
All projections cause some distortion
Errors are controlled by choosing the proper projection type, limiting the area applied
There are standard projections
Projections differ by datum – know your parameters
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62. Coordinate Systems
Once map data are projected onto a planar surface, features must be referenced by a planar coordinate system.
Coordinates in the GIS are measured from the origin point. However, false eastings and false northings are frequently used, which effectively offset the origin to a different place on the coordinate plane.
The three most common systems you will encounter in the USA are:
State Plane
Universal Transverse Mercator (UTM)
Public Land Survey System (PLSS) – non-coordinate systems
"Geographic" coordinates, latitude and longitude, are in fact not "projected."  Most coordinate systems used on the earth consist of two coordinates: these are generally called an "easting" and a "northing."
False Eastings and Northing - This is done in order to achieve several purposes:
Minimize the possibility of using negative coordinate values (to make calculations of distance and area easier).
Lower the absolute value of the coordinates (to make the values easier to read, transcribe, calculate, etc.).
Coordinate systems
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63. State Plane Coordinate Systems
Uses Lambert conformal conic (LCC) and Transverse Mercator (TM, cylindrical)
LCC when long dimension East-West
TM when long dimension N-S
May be mixed, as many zones used as needed
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64. State Plane Coordinate System
Each state partitioned into zones
Each zone has a different projection specified
Distortion in surface measurement less than 1 part in 10,000 within a zone
California
State Plane
Zones
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65. State Plane Coordinate System Zones
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66. Maine State Plane
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67. State Plane Coordinate System
e.g., Maine East State Plane Zone
Projection: Transverse_Mercator
False_Easting:
False_Northing:
Central_Meridian:
Scale_Factor:
Latitude_Of_Origin:
Linear Unit: Meter ( )
Geographic Coordinate System: GCS_North_American_1983
Angular Unit: Degree ( )
Prime Meridian: Greenwich ( )
Datum: D_North_American_1983
Spheroid: GRS_1980
Semimajor Axis:
Semiminor Axis:
Inverse Flattening:
FIPS – Federal Information Processing Standards
Scale Factor – scale of 1:10,0000 actual 10,000/
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68. UTM – Universal Transverse Mercator
UTM define horizontal positions world-wide by dividing the surface of the Earth into 6o zones.
Zone numbers designate the 6o longitudinal strips extending from 80o south to 84o north.
Each zone has a central meridian in the center of the zone.
The UTM system is not a single map projection. The system instead employs a series of sixty zones, each of which is based on a specifically defined secant transverse Mercator projection.
Since UTM projections distort the regions above 84 degrees north latitude and below 80 degrees south latitude far too much, they are not used on maps referencing the UTM grid.
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69. Universal Transverse Mercator – UTM System
The UTM system divides the surface of Earth between 80°S and 84°N latitude into 60 zones, each 6° of longitude in width and centered over a meridian of longitude. Zone 1 is bounded by longitude 180° to 174° W and is centered on the 177th West meridian. Zone numbering increases in an easterly direction.
Each of the 60 longitude zones in the UTM system is based on a transverse Mercator projection, which is capable of mapping a region of large north-south extent with a low amount of distortion. By using narrow zones of 6° (up to 800 km) in width, and reducing the scale factor along the central meridian by only (to , a reduction of 1:2500) the amount of distortion is held below 1 part in 1,000 inside each zone. Distortion of scale increases to at the outer zone boundaries along the equator.
In each zone, the scale factor of the central meridian reduces the diameter of the transverse cylinder to produce a secant projection with two standard lines, or lines of true scale, located approximately 180 km on either side of, and approximately parallel to, the central meridian (ArcCos = 1.62° at the Equator). The scale factor is less than 1 inside these lines and greater than 1 outside of these lines, but the overall distortion of scale inside the entire zone is minimized.
Overlapping grids
Distortion of scale increases in each UTM zone as the boundaries between the UTM zones are approached. However, it is often convenient or necessary to measure a series of locations on a single grid when some are located in two adjacent zones. Around the boundaries of large scale maps (1:100,000 or larger) coordinates for both adjoining UTM zones are usually printed within a minimum distance of 40 km on either side of a zone boundary. Ideally, the coordinates of each position should be measured on the grid for the zone in which they are located, but because the scale factor is still relatively small near zone boundaries, it is possible to overlap measurements into an adjoining zone for some distance when necessary.
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70. UTM Zone Details Each Zone is 6 degrees wide
Zone location defined by a
central meridian
Origin at the Equator, 500,000m
west of the zone central Meridian
Coordinates are always positive
(offset for south Zones)
Coordinates discontinuous across
zone boundaries
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71. Universal Transverse Mercator Projection – UTM Zones for the U.S.
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72. UTM Zone 19N Projection: Transverse_Mercator
False_Easting:
False_Northing:
Central_Meridian:
Scale_Factor:
Latitude_Of_Origin:
Linear Unit: Meter ( )
Geographic Coordinate System: GCS_North_American_1983
Angular Unit: Degree ( )
Prime Meridian: Greenwich ( )
Datum: D_North_American_1983
Spheroid: GRS_1980
Semimajor Axis:
Semiminor Axis:
Inverse Flattening:
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73. False Easting/Northing
False easting – the value added to the x coordinates of a map projection so that none of the values being mapped are negative.
False northing are values added to the y coordinates.
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74. Central Meridian Every projection has a central meridian.
The line of longitude that defines the center and often the x origin of the projected coordinate system.
In most projections, it runs down the middle of the map and the map is symmetrical on either side of it.
It may or may not be a line of true scale. (True scale means no distance distortion.)
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75. Central Meridian
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76. Coordinate Systems Notation Latitude/Longitude
Degrees Minutes Seconds 45° 3' 38" N
Degrees Minutes (decimal) 45° ' N
Degrees (decimal) °
State Plane (feet) + 2,951,384.24
UTM (meters) + 4,996,473.72
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77. Public Land Survey System
A township is 36 sq. mi.
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78. Coordinate Systems in ArcGIS
Define
Project
Select a standard projection.
Create a custom projection.
Import a projection.
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79. Principal Scale
Scale the earth down to a globe whose size is compatible with the size of the map plane. This is called the generating globe.
The radius between the generating globe and the real earth is called the principal scale.
The representative fraction refers to this scale.
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80. Concepts of Scale
Representative fraction – when the scale value is given as a fraction with a numerator as 1.
When the representative fraction is relatively large we refer to the map as a large scale map.
More detail
Smaller area
Which is the larger fraction?
A small scale map covers a larger area and shows less detail
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81. Large Scale Map 1:24,000 scale (one inch = about 0.4 miles) Lecture 3
A large scale map refers to one which shows greater detail because the representative fraction (i.e. 1/25,000) is a larger fraction than a small scale map which would have an RF of 1/250,000 to 1/7,500,000. Large scale maps will have a RF of 1:50,000 or greater (i.e. 1:10,000). Those between 1:50,000 to 1:250,000 are maps with an intermediate scale.
Relative terms depending on the application
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82. Small Scale Map 1:500,000 scale (one inch = about 8 miles) Lecture 3
The direct relationship between map dimension and actual dimension implied by the representative fraction is misleading because of the distortion involved in transferring dimensions from the curved surface of the earth to a planar map which prevents the possibility of a constant scale through out the map
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83. Scale Factor 0 > scale factor < =1
The ratio of the actual scale at a particular place on the map to the stated scale on the map.
Usually the tangent line or secant lines.
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84. Assignment ReadChapter 3. Problems: 5, 6, 7, 9, 3,10, 22, 28, 29
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