MAP Projections

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MAP Projections

• Alan A. Lew
• WWW READING Map Projection Basics at
• http//everest.hunter.cuny.edu/mp/mpbasics.html

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The Globe

• Advantages
• most accurate map
• Distances, Directions, Areas (sizes), and Angles
• Latitude Longitude Lines
• Latitude are Parallel
• Longitude Meridians Converge at the Poles
• Parallels Meridians meet at Right Angles
• Parallels become shorter toward the poles
• Disadvantages
• expensive to make
• cumbersome to handle store
• difficult to measure
• not fully visible at once

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Flat Paper or Screen Map

• Advantage
• has none of the Globes disadvantages
• Disadvantage
• must transform the spherical surface into a flat
  surface
• not able to maintain all forms of accuracy
• Area/Size OR Angles OR Distance OR
  Direction
• Projection
• How the Earths Spherical Surface is Transformed
  into a Flat Plane Surface
• only able to maintain one or two forms of
  accuracy only
• Correct Projection more useful than a globe
• Wrong Projection major problems deceptions

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Application Examples

• Mercator Projection
• 1569 - most well known
• designed for sea navigation
• angles are accurate throughout
• enabled use of compasses to determine their
  bearing
• grossly over-exaggerates Area at the poles
• and under- exaggerates area at the Equator
• Angle and Direction are maintained Distance
  and Area are lost
• Computer Cartography GIS
• Often require converting from one projection to
  another
• error can result in misalignment of lines and
  points
• Software easily does this

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Major Projection Factors

• infinite number of map projections possible
• Factor 1 SCALE
• RF Representative Fraction
• same at all points on a globe
• divide the Radius of the Earth by the Radius of
  the Globe
• When transforming the globe to a plane surface
• the RF will vary from one point to another
• caused by Stretching Shrinking
• Principal Scale base scale of the map
• the scale if the map were wrapped around a globe
• Actual Scale at any point
• larger or small than the Principal Scale

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SF Scale Factor

• Actual Scale - transformed scale at any point
• Divided by the Principal Scale - base scale
• Degree Direction of Variation from Principal
  Scale
• 1.0 no variation, gt1.0 Larger, scale lt1.0
  smaller scale
• SF varies at different rates over a flat map
• same line can appear longer based on
• (1) Location on the Map
• (2) Alignment on the Map
• E/W direction may be scaled different from N/S
• Basis of Tissots Theorem
• on next overhead

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Tissots Theorem (a and b)

• On a Globe
• Infinite Number of Paired Orthogonal (right
  angled) Directions at an Point
• eg., N-S/E-W and NE-SW/NW-SE
• On a flat map - they may no longer be orthogonal
• Tissots Theorem
• for any Transformation/Projection
• At Least one Pair of Directions remains
  Orthogonal
• known as the PRINCIPAL DIRECTIONS
• SF is GREATEST in the two Principal Directions
• Direction greatest above 1.0 - and is referred
  to as a
• Direction greatest below 1.0 - and is referred to
  as b
• vis. the axes of an OVAL

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Angular Area Distortion

• Most Common Projection Issues
• assessed by a b
• on a Globe a b SF 1.0 and
  a b SF 1.0
• on the Paper Map
• if a b - then there is No Angular
  Distortion
• if a ltgt b - then there Angular Distortion is
  Present
• if ab 1.0 - then there is No Area
  Distortion
• though there may be SHAPE distortions
• if ab ltgt 1.0 - then Area Distortion is
  Present
• INDICATRIX
• Mean Distortion average for entire map

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Conformal Map Projections

• Maintains Correct ANGLES
• on a Globe Cardinal Directions (N/S E/W)
• 90 degrees apart everywhere, except poles
• Conformal or Orthomorphic Projections
• correct form or shape - for a point
• Mercator Projection
• Tissots Theorem
• a b at all points No
  Angular Deformations
• a b ltgt 1 at all points Area
  Deformations
• Poles are Stretched Equator is Shrunk
• Does not affect Direction

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Equal Area Map Projections

• Area Sizes are correct everywhere on the map
• although SHAPES are not
• Equal Area or Equivalent Projections
• Tissots Theorem
• a b 1.0 everywhere
  No Area Deformation
• a b at only at One or Two Points or Lines
• known as Standard Lines No Deformity
• everywhere else, a ltgt b
  Angles Are Deformed
• No Projection Can Be BOTH Conformal
  and Equal
  Area at All Points
• Standard Lines where Globe intersects the Flat
  Map
• Distortion measure away from Standard Lines

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Distance Direction Distortions

• for Distance -- Scale Must be Uniform at All
  Points
• Physically Cannot occur for All Points in All
  Directions
• 1 - Scale is the same on Standard Lines
• 2 - Scale may be maintained FROM One or Two
  Points
• Equidistant Projections
• Directionltgt Bearing Angle
• on Flat Plane - Bearing Angle is based on the
  Grid (North)
• a RHUMB line -- using a Compass
• on a Globe - Direction is measured as a GREAT
  CIRCLE
• along a plane that cuts through the centre of the
  globe
• Azimuthal/Planar Projections
• Great Circles as Straight Lines from One or Two
  Points

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COMPARING PROJECTIONS

• The Graticule
• uniformly the same in all directions
• normally based on the Cardinal Directions -
  N,S,E,W
• Distortion Pattern
• 1. Are Parallels Parallel ? (a b)
• 2. Are Parallels Equally Spaced ? (ab 1
• 3. Do Parallels and Meridians intersect a Right
  Angles ? ab
• 4. Do Meridians converge at the Poles ? (ab 1
• 5. Are Meridians equally spaced on any given
  Parallel ? (ab 1
• 6. Do Meridians and Parallels form Squares at the
  Equator ? ab
• 7. At 60 deg Latitude, are Meridians about ½ as
  far apart as Parallels ? ab1
• 8. Is the surface area bounded by two Parallels
  and two Meridians the same along the same
  line of Latitude? (ab 1

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Factor 2 - Map Plane

• relationship of the plane surface to the globe
• All of the Scale Distortions may appear on
• Cyclindircal Projections
• Conic Projections
• Azimuthal and Oval Projections
• 1 - Cylindrical
• paper wrapped like a cylinder around the globe
• 90 deg angle graticule
• One or Two Standard Line - either Parallels or
  Meridians
• Tangent to Equator Regular or Normal
• Tangent to Meridian (Long) Transverse
• Tangent to any other line Oblique

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Conic Azimuthal Projections

• 2 - Conic Projections
• Paper forms a cone
• One or Two Standard Lines
• usually Parallels
• Pseudo-conic heart-shaped projections
• 3 - Azimuthal / Planar
• Paper touches one point on the globe
• Circular graticule around the point
• One Standard Point, or One or Two Standard Line
  (circles)
• 3.1- Oval
• stretched or rolled version of Azimuthal

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Aspect Perspective

• Factor 3 - Aspect
• Map can be Centred on Any Point or Line
• Equatorial Projection - Polar (N or S
  Pole) - Oblique
• Factor 4 - Perspective
• Normal Perspective
• shows the whole earth on a piece of paper
• normal does not appear in projection name
• Vertical Perspective
• as if in a plane of space craft above the earth
  at a specific height
• Orthographic Persepctive (Azimuthal
  Projection)
• lines from globe are orthogonal to paper
• Stereographic Perspective (Azimuthal
  Projection)
• as if viewed from inside the globe

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Combinations Interruptions

• Factor 5 - Interrupted Projections
• Interrupted Projection (common)
• Factor 6 - Combination Projections
• Same Projections used Twice
• Two Different Projections
• Magnified Projection

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CHOOSING a MAP PROJECTION

• Step 1 - What are you mapping?
• Location, Size/Area, Shape
  (linear, oval)
• Step 2 - Traditional Aspect Rule
• Tropics Cylindrical
  Equatorial Aspect
• Temperate Zone Conic Oblique
  Aspect
• Polar Azimuthal Polar Aspect
• Step 3 - What do you want to Emphasise?
• True Angle - Conformal
• True Area - Equal Area
• True Direction - Azimuthal
• True Distance - Equidistant

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Continued

• Step 4 - Further Modifications
• Common
• Aspect Changes
• Limit the Boundary
• Do not show entire globe - just a small area
  (I.e., a Large Scale)
• Rare
• Use Interrupted Projections
• use same projection centred on two different
  points or lines
• Combine more than one type of Projection
• EXERCISE
• How would you Classify the Projections ?
• Distortion Conformal, Equal Area,
  Azimuthal, Equidistant ?
• Plane Cylindrical, Conic,
  Azimuthal ?
• Aspect Equatorial, Polar, Oblique ?