Projections

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Projections Scale
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What is a projection?

• The systematic arrangement of the earths
  parallels and meridians onto a plane surface.
• Parallels and meridians become the graticule.
• Graphic illustration http//wwwstage.valpo.edu/ge
  omet/geo/courses/geo215/gis3.htm

3
Distortion

• All projections have some type of distortion
• Area
• Shape
• Size
• Distance
• Direction
• Scale
• Each projection contains only some distortion
  from these factors projections should be chosen
  to minimize distortion in relation to the maps
  purpose

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Projection Families

• Cylindrical
• Conic
• Azimuthal
• Pseudocylindrical (variation of Cylindrical)

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Projection Families

• Based on the configuration of the plane onto
  which the globe is projected
• Each is good for representing select areas of the
  globe
• Each produces a different graticule
• Each allows for different tangency/secant case
  with globe
• Each is suitable for a different purpose

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Cylindrical

• Formed by wrapping a large plane (such as a piece
  of paper) around the globe to form a cylinder,
  which is then unfolded (http//www.colorado.edu/ge
  ography/gcraft/notes/mapproj/gif/cylinder.gif)
• Equator is the normal aspect (or viewpoint) for
  these projections.
• Typically used to represent the entire world.
• Typical grid appearance shows parallels and
  meridians forming straight, perpendicular lines
• Used for large-scale topo mapping since they
  enable measurements of angle and distance
  (conformal)

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Psuedocylindrical

• Cylinders curve inward at the poles
• Grid shows straight parallels and central
  meridian, but all other meridians are concave
  from perspective of the central meridian
• Often used for world maps
• Examples
• Robinson
• Mollweide
• Eckert
• Sinusoidal

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Conic

• Similar wrapping as cylindrical, but plane is a
  cone (http//www.colorado.edu/geography/gcraft/not
  es/mapproj/gif/cone.gif)
• Normal aspect is north or south pole where axis
  of cone (point) sits
• Can only represent one hemisphere
• Used on areas with greater east-west extent than
  north-south (eg, the US)
• Parallels typically forms arcs of circles facing
  up in N. Hemisphere, down in Southern meridians
  either straight or curved and radiate outwards
  from point of cone

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Azimuthal

• Spherical grid projected onto flat plane (also
  called plane projection http//www.colorado.edu/ge
  ography/gcraft/notes/mapproj/gif/cone.gif)
• Poles are normal aspect
• Normally one hemisphere represented
• Grid appears as parallels forming concentric
  circles, with meridians radiating outward from
  center

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Tangency or Case

• Refers to location(s) where projection surface
  touches or cuts through the globe
• Two types
• Tangent case
• Secant case
• Scale deformation is nearly eliminated at point
  or line(s) of tangency, with distortion
  increasing away from tangency
• Therefore, locate tangency on or near area of
  central focus

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Tangent

• Tangent is simplest case (azimuthal, cylindrical,
  or conic surface).
• Touches the globe at one point or line
• Example http//www.colorado.edu/geography/gcraft/
  notes/mapproj/gif/plane.gif

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Secant

• Projection surface cuts through globe to touch at
  two lines
• Useful for reducing distortion of larger land
  areas
• Example http//www.colorado.edu/geography/gcraft/
  notes/mapproj/gif/scone.gif

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Aspect

• Also called perspective or viewpoint
• Polar
• Equatorial
• Oblique
• Transverse
• Sometimes indicated in name of projection
• Should be selected so that area of greatest
  interest is central on projected maps

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Aspect

• Polar
• North or South pole
• http//www.colorado.edu/geography/gcraft/notes/map
  proj/gif/nstereo.gif
• Equatorial
• Over the Equator (often used for world maps)
• http//www.colorado.edu/geography/gcraft/notes/map
  proj/gif/millerc.gif

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Aspect

• Transverse
• Places projection surface 90 degrees from normal
  position, eg, for an equatorial cylindrical
  projection the poles would be the transverse
  aspect
• http//www.colorado.edu/geography/gcraft/notes/map
  proj/gif/transcyl.gif
• Oblique
• Above or on any position between, but not
  including, the equator and poles. May be centered
  on parallel or meridian.
• Useful for centering smaller regions on a map
  projection (eg, India)
• http//www.colorado.edu/geography/gcraft/notes/map
  proj/gif/txlccus.gif

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Central Meridian

• Meridian that passes through center of a
  projection
• Distortion is minimized along this line (choose
  wisely)
• Example http//www.colorado.edu/geography/gcraft/
  notes/mapproj/gif/naaeana.gif

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Perspective (azimuthal)

• Azimuthal projections are considered from one of
  three perspectives
• Imagine a light source shining on the globe and
  the arcs of the parallels and meridians being
  projected onto the flat, tangent surface

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Perspective (Azimuthal)

• Three types
• Gnomonic light source is from center of the
  earth through to spherical surface
• Orthographic From infinity
• Stereographic A point at the opposite end of
  the globe

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Mathematical properties

• Shape (conformal)
• Area (equivalence)
• Distance (equidistance)
• Direction
• Scale (can vary throughout on one map)

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Shape (conformality)

• Deformation of scale increases regularly in all
  directions
• Parallels and meridians intersect at right
  angles, shapes of small areas and angles with
  short sides are preserved
• No angular deformation, true angles are
  maintained, therefore angular measurements can be
  made
• Useful for large-scale mapping, especially for
  navigation eg, topos, navigational charts.
• Commonly used for world reference maps, eg
  Mercator, Lambert Conformal Conic, etc.

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Equivalence (equal area)

• Maintain true relationships of areas
• At a given scale, map is proportional to
  corresponding area on the earth
• Deformation occurs in elliptical fashion away
  from tangency, therefore shapes are distorted
• Maintain true area, useful for comparing regional
  distributions of geographic phenomenon (eg,
  population density, other human-oriented
  statistics)

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Equidistance

• Scale is preserved in the direction perpendicular
  to the line of zero distortion or radially
  outward from a point
• Used for measuring bearings and distances (eg,
  airline networks) and for very small areas
  (portion of a city) without scale distortion
• Small amounts of angular deformation
• Good compromise between conformality and
  equivalence, often used in atlases as base for
  reference maps of countries and continents

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Compromise

• Some projections offer a compromise between
  conformality, equivalence and equidistance.
• These have some distortion of shape, area,
  distance, direction and scale, but all are
  moderate
• Robinson is a good example (http//www.colorado.ed
  u/geography/gcraft/notes/mapproj/gif/robinson.gif
  -- derived graphically instead of mathematically)

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Scale

• Large scale small area fine scale
• Small scale large scale gross scale
• Think of the fraction
• 1/2400 is a larger number than 1/24,000

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Scale

• Represents relationship between map units and
  ground units
• Can be expressed graphically, verbally or as a
  representative fraction (RF). Area is usually
  represented as a circle or square

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Scale Examples

• Verbally
• One in is equal to three miles (1 3 miles)
• Graphic
• Bar scale is the simplest. When map is enlarged
  or reduced, bar scale changes proportionately
• Representative Fraction
• Expressed as a ratio. Units MUST be the same for
  numerator and denominator (you can then use
  whatever measurement youd like inches, feet,
  etc.).
• Numerator (always 1) is map distance, denominator
  is ground distance.
• 12400, 163,360

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Scale conversions

• See USGS handout for examples, http//mac.usgs.gov
  /mac/isb/pubs/factsheets/fs03800.html

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Coordinate Systems

• Ways of describing locations on earth in
  reference to an established grid
• Lat/long is only one of these

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Latitude Longitude

• Also called the geographical grid (or unprojected
  or geographic projection)
• Divides globe into two circles of 0-180 degrees
  each for longitude 0-90 degrees for latitude
• Parallels (latitude) and meridians (longitude)

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Latitude Longitude

• Its easy, but also cumbersome
• Meridians converge at the poles
• Degree of lat decreases from about 111km at the
  equator to 0 at the poles. This makes it poor for
  use as a rectangular grid with x,y coordinates
• Lat/long is not a decimal system (based on 360,
  deg/min/sec system). Conversions can be a pain.

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State Plane Coordinates

• Developed by National Geodetic Survey in 1933
• The US is broken into smaller zones (120), which
  each have its own projection and coordinate
  center and system.
• Youre never far from the standard line.
• Coordinates are very accurate within each zone
  (less than 1ft per 10,000ft of measurement)
• Problem coordinates between zones dont line up,
  so its now useful for areas that cover more than
  one zone

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State Plane Coordinates

• Nearly all states have multiple zones, but zones
  never cross county lines
• Each state uses either Lambert Conformal Conic or
  Transverse Mercator projection
• Locations are identified by x,y coordinates in
  feet.
• To keep all SPC coords positive, the origin for
  each zone is placed off to the southwest of each
  zone. This is not the actual center of the
  projection.
• Actual center is assigned an arbitrarily large
  coordinate (eg, 2,000,000ft East, 400,000ft West)
  this is called the false origin.
• SPC are shown on USGS topos

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Universal Transverse Mercator

• Similar to SPC, but it covers the globe, is
  measured in meters, and has much larger zones.
• Zones extend N-S, almost from pole to pole
• Projection is very accurate alone n-s zone near
  standard line, but severely distorts at large
  distances away from meridian

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UTM

• 60 zones, each 6 degrees wide (60 x 6 360).
• Each zone is accurate in matching true earth
  distance and direction.
• Going across zones is difficult, its meant
  primarily for local and regional measurement
• Mainly first used by Army in 1940s (also included
  on USGS topos)
• X,y coords are given in meters.
• Also has a false origin off to the southwest,
  with n-s center of the zone placed at 500,000m
  East false easting, northings.
• Reading on topos tick says 3445, equals
  3,445,000m N

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Survey systems

• SPC and UTM are good for locating points, but not
  describing areas
• Surveying systems
• Metes and bounds
• Spanish Land Grants
• Other surveys
• US Public Land Survey

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Metes and Bounds

• Used natural landmarks to delineate property
  boundaries
• Commencing from a point one-half mile upstream
  from Smith Bridge on Jones Creek, proceed
  northeast 500 feet to Spring Hill, then northwest
  to the large oak tree, then
• Problems
• Overlapping claims
• Boundary markers disappear
• Not quick and easy

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Spanish Land Grants

• Seen in California and much of the Southwest
• Similar to metes and bounds, but focused on water
  resources (and rights)

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Other surveys

• Main example
• The French used a system of long lots (in
  Louisiana and elsewhere), also focused on water.
  Broke land up into narrow strips off water
  resource.

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US Public Land Survey

• US PLSS System
• Used by US to divvy up land in the West
  (extending to Mississippi) after independence
• Thomas Jefferson and others worked out this
  rectangular system

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PLSS

• An area is given an x,y coordinate system. N-S
  line is the principal meridian, E-W line is the
  baseline.
• Baselines have unique names
• From this, townships are marked of E/W and N/S.
• Townships are 6 miles on a side (36sqmi)
• Designated as x number of Ranges each of west of
  principal meridian x number of Townships north
  or south of the baseline
• Each township is broken into 36 Sections,
  consecutively from 1 to 36 (snakelike pattern)

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PLSS

• Example in textbook, p236-237
• Subdivions http//www.utexas.edu/depts/grg/huebne
  r/grg312/graphics/section.jpg
• For example, a ten acre parcel could be described
  as
• SE 1/4 SE 1/4 SE 1/4 sec. 5, T2N, R3W Boise
  Meridian, Idaho
• Translated  the southeast quarter of the
  southeast quarter of the southeast quarter of
  section 5, Township 2 North, Range 3 West of the
  Boise Meridian.  Descriptions are read left to
  right but locating them is easier if one reads
  right to left or from larger division to smaller.
  

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PLSS

• Most land purchases were for less than one
  section Sections are broken into halves,
  quarters, etc.
• Typical Midwestern farm used to be one quarter
  section, or 160 acres.
• Included on USGS topos (red lines and numbers)
• Still used for property description
• Very noticeable when flying (http//www.utexas.ed
  u/depts/grg/huebner/grg312/graphics/wysections.jpg
  )

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PLSS

• Problems
• Section and township lines are not always exactly
  n/s and e/w
• Some monuments have disappeared
• Sections are not always a full square mile
• Meridians converge to the north, so townships
  dont always line up. E/W correction lines were
  sometimes set up.
• Lines tended to go astray
• Surveyors were paid by number of sections
  surveyed
• Some surveyors just werent careful

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PLSS

• Tutorial
• http//www.dnr.state.wi.us/org/land/forestry/Priva
  te/PLSSTut/plsstut1.htm