Projections Scale by mikeholy

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									Projections & Scale
What is a projection?

   The systematic arrangement of the
    earth’s parallels and meridians onto
    a plane surface.
   Parallels and meridians become the
    “graticule.”
   Graphic illustration:
    http://wwwstage.valpo.edu/geomet
    /geo/courses/geo215/gis3.htm
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 map’s purpose
Projection Families

   Cylindrical
   Conic
   Azimuthal
   Pseudocylindrical (variation of
    Cylindrical)
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
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/geography/gcraft/note
    s/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)
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
Conic
   Similar wrapping as cylindrical, but plane is a
    cone
    (http://www.colorado.edu/geography/gcraft/note
    s/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
Azimuthal
   Spherical grid projected onto flat plane
    (also called plane projection
    http://www.colorado.edu/geography/gcra
    ft/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
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
Tangent

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

   Projection surface cuts through
    globe to touch at two lines
   Useful for reducing distortion of
    larger land areas
   Example:
    http://www.colorado.edu/geograph
    y/gcraft/notes/mapproj/gif/scone.gi
    f
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
Aspect

   Polar
       North or South pole
       http://www.colorado.edu/geography/gc
        raft/notes/mapproj/gif/nstereo.gif
   Equatorial
       Over the Equator (often used for world
        maps)
       http://www.colorado.edu/geography/gc
        raft/notes/mapproj/gif/millerc.gif
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/
        mapproj/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/
        mapproj/gif/txlccus.gif
Central Meridian

   Meridian that passes through center
    of a projection
   Distortion is minimized along this
    line (choose wisely)
   Example:
    http://www.colorado.edu/geograph
    y/gcraft/notes/mapproj/gif/naaeana
    .gif
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
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
Mathematical properties

   Shape (conformal)
   Area (equivalence)
   Distance (equidistance)
   Direction
   Scale (can vary throughout on one
    map)
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.
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)
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
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.edu/geography/gcr
    aft/notes/mapproj/gif/robinson.gif --
    derived graphically instead of
    mathematically)
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
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
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 you’d like: inches, feet,
        etc.).
       Numerator (always 1) is map distance, denominator
        is ground distance.
       1:2400, 1:63,360
Scale conversions

   See USGS handout for examples,
    http://mac.usgs.gov/mac/isb/pubs/
    factsheets/fs03800.html
Coordinate Systems

   Ways of describing locations on
    earth in reference to an established
    grid
   Lat/long is only one of these
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)
Latitude & Longitude

   It’s 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.
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.
   You’re 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
    don’t line up, so it’s now useful for areas
    that cover more than one zone
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
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
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, it’s 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
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
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
Spanish Land Grants

   Seen in California and much of the
    Southwest
   Similar to metes and bounds, but
    focused on water resources (and
    rights)
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.
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
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)
PLSS
   Example in textbook, p236-237
   Subdivions:
    http://www.utexas.edu/depts/grg/huebner/grg31
    2/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.
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.edu/depts/grg/huebn
    er/grg312/graphics/wysections.jpg)
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
           don’t 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 weren’t careful
PLSS

   Tutorial:
       http://www.dnr.state.wi.us/org/land/fo
        restry/Private/PLSSTut/plsstut1.htm

								
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