Transcript Projections - Mohawk College

Why Project?
• Make accurate measurements
• Want to preserve one of these:
– Area, shape, distance or direction
• Small scale map
– Change overall appearance
– Longitude straight or curved
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Projections
• Ellipsoid
– Mathematical figure generated by a revolution
of an ellipse around one of its axes
– Ellipsoid approximates the spheroid which's
approximates the earth
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Common Spheroids
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What is a Geoid?
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The geoid is theoretical only. You can't see it, touch it or even dig down to find it.
Simply put, the geoid is the natural extension of the mean sea level surface under the
landmass.
We could illustrate this idea by digging an imaginary trench across the country linking the
Atlantic and Pacific oceans. If we allowed the trench to fill with seawater, the surface of the
water in the trench would represent the geoid.
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Spherical Measurements
• Latitudes (parallels)-horizontal lines
• Longitudes (meridians)-Vertical lines
• Graticule-complete gridded network
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Spherical Measurement
• Latitude and longitude measured in degrees
• 0° lat at equator, 90° at north pole, -90° at south pole
• Longitudes-positive 0° to 180° travel east from
Greenwich, negative going west
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Map Projections
• Manner in which the spherical surface of the
earth is represented on a two-dimensional surface
• No such thing as an ideal map (distortion free)
• Must satisfy the following:
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Conformality- preserves the shape
Equivalence-Equal Area
Equidistance- Equidistant between two points
True direction- lines of constant direction
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Ideal Map
• Conformality
– Projection preserves the shape of any small
geographical area
– At a point relative local angles are preserved
– Meridians intersect parallels at right angles
– Areas are enlarged or reduced
– Important-used analyzing, guiding, or
recording motion, as in navigation
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Ideal Map
• Equal area
– If you put a dime on the map, it always covers
the same area but the angles and shapes are
distorted
– Important-comparing density and distribution
of data, as in population
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Example of Equal Area
Left not projected; right view Equal-area Cylindrical
Area is preserved in projection
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Ideal Map
• Equidistance
– Equidistant between two points and rest of
map only, or along meridians
– Scale of distance is constant over entire map
– Important-analyzing velocity, such as ocean
currents
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Ideal Map
• True Direction
– All rhumblines (lines of constant direction) are
straight
– On a spherical surface, the shortest surface
distance between two points is a great circle
along which azimuths change constantly
– All meridians are great circles but the only
parallel that is a great circle is at the equator
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Projection Types
• Conformal
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Preserve local shape
Gradicule lines on the globe are perpendicular
Area greatly distorted
Good for wide states
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Two types of Conformal
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Cylindrical
• Most common
• Can also be secant (UTM)
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Also know as Planar Projection
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Planar or Azimuthal
• Accomplished from drawing lines from a given
perspective point through the globe onto a tangent plane.
• Origin of projection lines
– Earth (gnomonic)
– Infinite distance away (orthographic)
– Earth’s surface opposite the projection plane
(stereographic)
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Planar from point in space
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Problems creating Ideal Map
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Small Scale Examples
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Mollweide Homolgraphic
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Goode’s Interrupted Homolosine
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Sinusoidal
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Mercator
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UTM Zones in Canada
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UTM 6° Zone
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UTM
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UTM-Cross-section
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Convergence
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Elevations
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The Datum and Ellipsoid
• Datum - A system
for anchoring an
ellipsoid to known
locations (surveyed
control points) on
the earth.
Geocentric vs. Local
• A reference ellipsoid that
closely fits the local geoid is
tied to an ‘initial point’. This
initial point is assigned a
long/lat value and from there
all other coordinates are
calculated upon the reference
ellipsoid.
• This process produces a
Datum and a common
reference system for a region.
• Thus, local Datums are fitted
to a particular region of the
earth such as NAD27 or
NAD83 using a reference
ellipsoid that best fits that part
of the globe. Global datums,
like WGS84 attempt to provide36
a best fit to the entire geoid.
• The ellipsoid
approximates the shape of
the earth.
• A datum defines the
position of the spheroid
on the earth.
• Once the ellipsoid is
anchored, long/lat are
defined.
• Therefore, two different
datums mean two
different long/lat values
for the same geographic
feature.
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Datums Important
• The problem is that a given point location can vary by
several hundred meters when projections are referenced
to different datums using different ellipsoids.
– For example, the NAD27 Datum was used in Canada and the US
for the past 50 or so years for making maps. NAD27 is based on
the Clark 1866 reference ellipsoid.
– Recently, all new maps use NAD83 which is based on the GRS80
ellipsoid. WGS84 also uses GRS80.
• By doing so, what was once the long/lat of your house
under NAD27 has changed by as much as 300 meters
under NAD83.
• Referencing geographic coordinates to the wrong datum
can result in positional errors of hundreds of meters.
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Important Datums
• Because of this change in Datums from
NAD27 to NAD83, the long/lat of almost
every point in north America has changed
its longitude/latitude.
• There are defined conversion parameters
(mapping equations) for transferring from
one datum to another.
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DATUM IMPORTANT
• Some projection equations use reference
ellipsoids. If so then for large-scale maps ensure
that the projection equations use the same
ellipsoid parameters as those used to define the
local datum, otherwise projections will not be inline with the control points.
• Changing ellipsoids = changing Datums =
changing coordinate systems.
• This is a source of possible error.
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1:2,000,000
1:50,000
1:500,000
1:5000
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