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Stereographic Projections

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					Stereographic
Projections
  Jacqueline Hess
    History

• Earliest known uses were in Greece for map making
• Earliest references in literature (Roman, ~100 B.C.)
   – Vitruvius ~ Ten Books on Architecture
   – Ptolemy’s ~ Representation of the Sphere in the plane
       • Stereographic projection was used for a horoscopic instrument for
         determining time
       • Horoscope ~ the point of intersection of the ecliptic & eastern part of
         the horizon determined by means of this instrument
• The ancients knew how to prove
   – Circles through the pole are mapped onto straight lines
   – Other circles are mapped onto circles
• “Stereographic projection”
   – Introduced by Francois D’Aguillon (1566-1617) ~ Six Books of Optics
• 10th Century Scholar al-Saghani suggested Gnomic Projection
    – Projection form an arbitrary point on the axis – if that point is the center
      of the sphere
Stereographic Projections
• How projected
  – It is projected from a center point
• Advantages
  – All circles on sphere plot as circles on plane
• Drawbacks
  – Radial Distortion
• Uses
  – Most widely used projection in mineralogy and
    structural geology
Stereographic Projections

• “Family of
  Projections” that
  map from the sphere
  directly onto the
  plane
• Maps everything
  except (0,0,1) onto
  whole plane z = -1
 Stereographic Projections

It is projected from the point on the
opposite side of the Earth:
Stereographic Projections
Orthographic Projections
• How projected
  – From sphere perpendicular to plane
• Advantages
  – True visual view
  – All circles plot as ellipses or straight lines
• Drawbacks
  – Distortion near edges
• Uses
  – Mostly used in structural geology for drawing block
    diagrams
Orthographic Projections

• Think Perpendicular
• Maps southern
  hemisphere onto
  unit circle
Orthographic Projections
•   It is a parallel projection that   •
    shows the Earth as it would be         Polar Orthographic projection.
    viewed from deep space. The            The projection plane is tangent
    parallel projection lines are          to pole.
    perpendicular to the projection
    plane. In the Polar Orthographic
    projection, the projection plane
    is tangent to the earth's pole
    (see diagram at right).
    The Orthographic projection has
    become popular since the first
    photographs were taken in
    space in the 1960s, and it is
    excellent for showing what the
    earth looks like from outer
    space. But it is not widely used
    to display scientific data
    because distortion is very high
    near the edges, and only half
    the earth can be shown.
Orthographic Projections
Gnomic Projections

• How projected
  – From center of sphere
• Advantages
  – Great circles always plot as straight lines
• Drawbacks
  – Radial distortion
• Uses
  – Mineralogy
Gnomic Projections

• Center is projected
  out radially
• Only 1 hemisphere
• Maps southern
  hemisphere were
  plane z = -1
Gnomic Projections
Gnomic Projections
      It is projected from the center of the earth.
      It is also called the “Central Projection,” or
      simply “Gnomic.” This projection displays
      less than half of the earth:




The Gnomonic projection is very distorted in all but one respect: Great
Circles are always displayed as straight lines. That's right, any straight line
drawn on a Gnomonic map will be the shortest distance between two points.
The scale will be badly distorted along the line, but the route will be precise
for the sphere.
Equal Area Projections
• How projected
  – Draw an arc from point on sphere to plane
• Advantages
  – Area conserved & moderate distortion
• Drawbacks
  – Curves are complex
• Uses
  – Structural geology
  – For statistical analysis of spatial data
Equal Area Projections

• Arc length lies flat
Citations

  • http://www.3dsoftware.com/Cartography
    /USGS/MapProjections/
  • http://www.uwgb.edu/dutchs/structge/sp
    hproj.htm
  • http://www.technion.ac.il/guides/matlab/t
    oolbox/map/proj43.html

				
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