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Projection

The Earth is Not Round

  • First the Earth was flat
  • 500 BC Pythagoras declared it was a sphere
  • 1600's Sir Issac Newton hypothesized that the true shape of the earth was really closer to an ellipse
    • More precisely, an oblate ellipsoid 

 

Eratosthenes

  • Greek Mathematician
  • Lived 276-194 BC
  • Responsible for
    • Establishing geography
    • First accurate measure of Earth's circumference
      • Noticed how the Sun shone directly down a well in Syene at noon on the Solstice. Made a second observation at Alexandria with a pole and noticed a shadow. He measured the angle of the Shadow and inferred the circumference of Earth, which was already known to be spherical
    • Calculating the tilt of Earth's axis
    • Calculating the distance of Earth to the Sun
    • Invention of the leap day
    • First world map of the (spherical) Earth

 

 

Geodesy

  • The science that studies and determines the shape and dimensions of the Earth
  • Different models have been defined to represent Earth's shape, varying in complexity and accuracy.

 

Geoid

  • Physical approximation of the figure of the Earth
  • Shape of the surface of calmed oceans, in the absence of other influences such as winds and tides
  • Computed using complex physical models and gravity readings of Earth's surface
  • Used to measure surface elevations with a high degree of accuracy

 

Ellipsoid

  • Mathematical approximation of the shape of the earth
  • The earth is flattened at the poles and bulges at the equator due to its revolution
  • Ellipsoid geodesy is uniquely defined by two numbers: semi-major (wide axis) and semi-minor axis (shorter axis)
    • A>B

 

Sphere

  • Simplest (and least accurate) approximation of the shape of the Earth
  • Radius is constant
  • a=b

 

Spheroid is an ellipsoid that approximates the shape of a sphere

  • Earth can be approximated by an oblate ellipsoid, but it's major and minor axes do not vary greatly
  • Shape is so close to a sphere that is it often called a spheroid rather than an ellipsoid

 

 

 

Why do we need more than one ellipsoid?

  • The Earth's surface is not perfectly symmetrical
  • The semi-major and semi-minor axes that fit one geographical region do not necessarily fit another

 

Datum

  • A reference ellipsoid for locating points on Earth's surface
    • Defines origin and orientation of latitude/longitude line
    • Defined by ellipsoid and ellipsoid's position relative to Earth's center
    • Two Types
      • Earth-centered
      • Local

 

 

Latitude

  • Angle from center Earth that describes the North-South position

Longitude

  • Angle from center Earth that describes the east-west position

 

Latitude/longitude

  • Not uniform units of measure
  • Meridians converge near poles

 

 

Prime meridian

  • Arbitrary origin of Longitude lines
  • Usually greenwich, England
  • Others include Paris, Bogota, etc.

 

Degrees Minutes Seconds (DMS)

  • Degrees of latitude and longitude are further broken down into Minutes and Seconds

 

Decimal Degrees (DD)

  • Decimal degrees are similar to degree/minutes/seconds (DMS) except that minute and seconds are expressed as decimal values
  • Decimal degrees make digital storage of coordinates easier and computations faster

 

 

Conversion from DMS to DD:

  • Example coordinate is 37 (deg) 36' 30" (dms)
  • Divide each value by the number of minutes or seconds in a degree:
    • 36 minutes = 0.6 degrees (36 /60)
    • 30 seconds = 0.0833 degrees (30/3600)
    • Add up the degrees to get the answer
    • 37 (deg) + .60 (deg) + 0.0833 (deg) = 37.60833 DD

 

 

 

Basic Geodesy Facts

  • Great circle - arc formed by the intersection of the earth with a plane passing through any two surface points and the center of the Earth
  • Magnetic directions must take into account the compass variation (magnetic declination)
  • Rhumb line - loxodrome or constant azimuth - line which makes a fixed angle with all meridians; spirals to pole

 

Geographic Coordinate System

  • The Equator and Prime Meridian are the reference points
  • Latitude / Longitude measure angles
    • Latitude (parallels) 0 deg - 90 deg
    • Longitude (meridians) 0 deg - 180 deg
  • Defines locations on 3-D surface
  • Units are degrees (or grads)
  • Not a map projection

 

 

Map projection

  • Curved surface (3D) -> 2D flat surface
  • Approaches to transfer the spherical Earth on a two dimensional plane
  • Some distortions will always occur
  • Visualize a light shine through the Earth onto a surface
  • Distortions
    • Fitting sphere to plane causes stretching or shrinking of features
    • Types
      • Shape
      • Area
      • Distance
      • Direction

 

Projection Properties

  • Conformal
    • Maintains shape
  • Equal-area
    • Maintains area
  • Equidistant
    • Maintains distance
  • Azimuthal (Planar)
    • Maintains some directions

 

 

 

How to measure distortion from map projections?

  • Tissot's Indicatrix
    • Measures and visualize distortions of shape and area at a single location on a projected map relative to a reference globe

 

 

Developable Surfaces

  1. Cylindrical projections
    1. The earth is projected on a cylinder
    2. Whole-world maps are rectangular
    3. Distortion on the poles
  2. Conic projections
    1. The Earth is projected on a cone
    2. Good for representing parts of the Earth
  3. Planar projections
    1. The Earth is projected on a plane
    2. Lots of distortion towards the edges

 

Developable surfaces contacting spheres

  • Tangent
    • Projection surface touches sphere
  • Secant
    • Surface cuts through sphere
  • No distortion at contact points
  • Increases away from contact points

 

 

Cylindrical projection

  • Longitudes equally spaced
  • Latitudes unequally spaced
  • Scale is true along equator
  • Shape and scale distortions increase near poles
  • Best for equatorial or low latitudes

 

 

 

Common projections

  • Mercator
    • Projected on a cylinder
    • Any straight line is a line of constant direction
    • Used for navigation
    • True directions
    • Conformal (angles and shapes true in small areas) but not equal area or equidistant
    • Cylindrical
  • Universal transverse Mercator
    • Divides the earth from latitudes 84N to 80S in 60 vertical zones that are 6 deg wide
    • Zones are numbered starting at 180th meridian in eastward direction
    • Each zone is divided into sections of 8 deg latitude each
    • Eastings (from Central meridian) and Northings (from equator) can be designated for each zone
    • UTM preserves area, Distance, and Shape well
  • Albers equal area
    • Conic (secant case)
    • Well suited for areas that are mainly east-west in extent
    • Areas - True
    • Drections - Reasonably accurate in limited regions
    • Distances and Scale true only along standard parallels
    • Map - not conformal
    • Used for Thematic maps
  • Lambert's conformal conic
    • Conic (Secant case)
    • Distances - True only along standard parallels
    • Map - conformal but not equal area or equidistant
    • Area and Shape - Distortion minimal at std. parallels
    • Directions - Reasonably accurate
    • Shape - True for small areas
    • To map large ocean Areas and regions in E-W extent
  • Azimuthal equidistant
    • Extent - World; Eq/mid-lat/Polar
    • Distances measured from centre are true; Distortion of other properties increases from centre point
    • Useful for showing airline distances from centre point
    • Useful for seismic and radio work