# 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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