Map Projections

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Map Projections
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3D-2D Transformation Process

• Geoid

• Reference Ellipsoid

• Sphere of Equal Area

• Reference Globe

• Transformation

• Map Projection

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Earth as Sphere

• Most commonly used in cartography
• Primarily for small-scale maps

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Earth as Sphere Size

• 40 million meters in circumference
• 20 million meters from Pole-to-Pole
• 10 million meters from Equator to Pole
• 1 Meter 1/10-million of the distance from the
  equator to the north pole as a line of longitude
  running through Paris France

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3D-2D Transformation Process

• Geoid

• Reference Ellipsoid

• Sphere of Equal Area

• Reference Globe

• Transformation

• Map Projection

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Reference Globe
A reference globe is where the size of the sphere
(or other shape) is reduced until it matches the
final scale desired for map making
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3D-2D Transformation Process

• Geoid

• Reference Ellipsoid

• Sphere of Equal Area

• Reference Globe

• Transformation

• Map Projection

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3D-2D Transformation Process
Projection selection and final transformation
from 3d-to-2d
Distorting inevitable
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Tearing, Shearing Compression
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Uncertainty Errors

• Need to know where error is and how to control it
• Which map is more accurate?
• Conformal vs. Equal Area

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• Conformal correct proportions shape
• Equal Area area relationships of all parts are
  maintained property of equivalence

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Map Error Projections can Preserve

• Shape
• Scale preserved in all directions around a point
• Preserves local angular relationship
• Conformal Maps
• Area (size)
• Equal Area Maps
• Distance (scale)
• Equidistance Maps
• Direction (azimuth)
• Equidirection Maps

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No map can be both Conformal Equal Area
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3D-2D Transformation Process

• Geoid

• Reference Ellipsoid

• Sphere of Equal Area

• Reference Globe

• Transformation

• Map Projection

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Flattening the globe Map Projections
Transforming a globe to a flat map requires
distortion of one kind or another
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Map projectionsclassification by surface

• Earth features are projected on to three kinds of
  developable surfaces
• plane a flat plane touches the globe
• cylinder a cylinder is wrapped around the globe,
  and unwrapped to a flat map
• cone a cone is wrapped around the globe, and
  unwrapped to a flat map

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Common planar projections
Light source
Gnomonic
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Common projections
Cylindrical
Mercator
Peters
Equal-area cylindrical
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Common projections
Conic
Albers equal-area
Lambert conformal
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Projection surfaces
conic
planar (or azimuthal)
cylindrical
Can construct the appearance of each with geometry
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Some mathematical projections
pseudocylindrical
pseudoconic
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Why all the projections?
All have distortion of some kind, but each
serves a unique purpose or has a unique property
that is useful for a certain application
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Distortion and accuracy on projections

• Strategy compare the projection to a globe (an
  unprojected map)

Meridians converge
90
All distances are equal
Distances decrease
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Some questions accuracy distortion on
projections

• Can we compare one area to another accurately?
• Are shapes correct?
• Are directions from one place to another
  accurate?
• Can we compare distances between places?
• Which parts of the map suffer the most distortion?

• Property aspects of the globe that aprojection
  preserves
• Distortion aspects that the projection doesnt
  preserve

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Some equal-area projections
Equal-area cylindrical
Albers equal-area
Mollweide pseudocylindrical
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Conformal angles shapes (sort of) preserved
90

• Shapes, in small areas near the middle of the
  projection, look the same on the map as they do
  on the globe
• As on globe, all meridians parallels intersect
  at right angles

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Directions are preserved azimuthal
Gnomonic

• Angular measurement from one location to another
  is accurate
• Straight lines are great circles

Not a great circle!
GOOD FOR Air navigation, radio signals or
radiation patterns
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Distances are preserved equidistant
2000 miles
If measured on a line that passes through center
Not this line!
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Area preserved equivalent or equal-area
1 million sq. miles
Important for maps where land area is important
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Conformal projections and navigation
But the line isnt a great circle!
Mercator projection

• With Mercator You can use a straight line from
  origin to destination to determine your compass
  bearing to the destination

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Some examples of map types
Map purpose Critical Properties for the purpose Critical Properties for the purpose Critical Properties for the purpose Critical Properties for the purpose
Map purpose Conformal Equal-Area Equidistant Azimuthal
Navigation ü ü ü
Show distances ü
Large-scale reference ü
Teaching ü ü
Comparing regions ü ü
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Four Key Projection Properties
Property of a projection how its true to the
globe

• Equivalence Equal areas
• Conformality Correct angles (rhumb lines
  straight lines)
• Azimuthality Correct directions
  (great circles straight lines)
• Equidistance Correct distances

Cant occur together!
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Aspect

• The orientation of the developable surface to the
  reference globe
• Three aspects equatorial, transverse, and oblique

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Equatorial Aspect

• Globe axis parallel to developable surfaces axis
• A normal aspect simple map graticule

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Oblique Aspect

• Non-great circle or meridian used

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Transverse Aspect

• Rotates the world 90 degrees

• Good for showing areas with large N-S Extent

• Where a meridian of longitude touches the the
  developable surface

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Secant Map Projections
Developable surfaces cuts through the reference
globe
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Secant Variations
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Meeting several needsCompromise projections
Miller
Robinson
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The projection matrix
Projection properties
equivalent conformal equidistant azimuthal
cylindrical Peters Mercator Plate-Carree Mercator
conic Albers equal area Lambert conformal conic Simple conic Lambert conformal conic
planar Lambert azimuthal Stereographic Azimuthal equidistant Azimuthal equidistant
Projection surfaces
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What part of a map has the least amount of
distortion?
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What part of a map has the least amount of
distortion?

• Where the projection surface touches the globe

Standard line
Standard point
Standard line
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For what country or region would you choose these
surfaces?
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Guidelines
Projection surfaces and orientations with areas
for which theyre suited
Planar roundish area, anywhere on earth
Conic or pseudoconic area in middle latitudes,
extensive east west
Cylindrical or pseudocylindrical area
extending along equator or a meridian
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Keys to Choosing Projections

• Whats the purpose of your map?
• What types of accuracy (properties) are most
  important?
• For maps of portions of the world What surface
  and orientation will best fit the area of the
  world youre mapping?

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Quick Reading

• Claudius Ptolemaeus (Ptolemy) Representation,
  Understanding, and Mathematical Labeling of the
  Spherical Earth