Flattening-the-Earth-with-map-projections.pdf

Info about GIS

1. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 1
Purpose of map projections Purpose of map projections

2. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 2
Selecting a coordinate system

3. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 3
Globe:

Three‐dimensional (3D)

Expensive, cumbersome, no detail, but no distortion
Map:

Two‐dimensional (2D)

Easier to measure distance, area, direction

Can show more detail

Easy to work with, portable, cheaper

Distortion.
Globe vs. Map

4. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 4
http://thetruesize.com

5. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 5
How projections work How projections work

6. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 6
What is a map projection?
Transformation of 3D Earth to a 2D map

7. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 7
Goal: Flat map with scale of 1:100,000,000
1:100,000,000
Principal scale
Full‐sized Earth
First, imagine that the Earth 
has been shrunk to the desired scale
Can use either 
sphere or ellipsoid
Can use either 
sphere or ellipsoid
Reference 
Globe

8. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 8
Hypothetical
Still 3D
Principal scale =
6.378 cm
Reference globe
1:100,000,000
Reference globe radius
Earth’s radius
637813700 cm
or 1:100,000,000
= 0.00000001

9. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 9
Transfer all points from 3D globe to 2D map…
1:100,000,000 1:100,000,000
Reference Globe (3D) Flat Map (2D)

10. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 10
Distances are distorted
1:100,000,000 1:100,000,000
Reference Globe (3D) Flat Map (2D)

11. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 11
From curved surface to flat
To flatten globe, must stretch, tear, or distort…
reference
globe
paper
Map distance
(shrunk, or distorted)
Actual distance
(based on Melita and Kopp, 2004)

12. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 12
"Not only is it easy to lie with maps, it's 
essential.  To portray meaningful relationships 
for a complex, three‐dimensional world on a flat 
sheet of paper or video screen, a map must 
distort reality" 
Mark Monmonier, How to Lie with Maps, 1996 
How to lie with maps...

13. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 13
Associating points from 3D to 2D Associating points from 3D to 2D

14. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 14 Associating points

15. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 15
Equator
Draw a line from the equator to the pole…

16. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 16
Rotate 90°
Equator

17. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 17
Rotate 90°

18. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 18
Projection from sphere to plane

19. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 19
Cylindrical Equal Area
Should be equal,
but are not
Should not be equal,
but are

20. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 20
Adjust distances

21. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 21
Equirectangular projection

22. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 22

We adjusted points along a meridian

Same can be done along a parallel 

Can adjust both, and in different ways.
Two perpendicular directions

23. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 23
Graticule: indicates how projections work

24. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 24
Projection class
(developable surfaces)
Projection class
(developable surfaces)

25. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 25 Projection classes
Cylindrical
ConicPlanar
(adapted from Lo and Yeung, 2006)

26. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 26
Standard line and scale factor Standard line and scale factor

27. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 27
Standard point or line
Cylindrical
ConicPlanar
(Melita and Kopp, 2004; Lo and Yeung, 2006)
Where globe 
touches 
developable surface

28. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 28
Hypothetical
Still 3D
Principal scale =
6.378 cm
Reference globe
1:100,000,000
Reference globe radius
Earth’s radius
637813700 cm
or 1:100,000,000
= 0.00000001

29. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 29
Scale factor
Local scale
Principal scale
1/100,000,000
1/100,000,000
SF =
SF =
SF = 1 at the standard line
Where will the local scale
be the same as the principal scale? 
i.e., where is there no distortion?
What happens to the SF as you 
move away from the standard line?
Scale on
ref. globe
Scale on
map
SF=2
(Kennedy et al., 2004)
SF > 1
SF > 1
SF=1
SF > 1
SF > 1

30. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 30
Scale factor
No distortion (3D)
Scale factor varies
1:100,000,000
(at all locations)
1:100,000,000 
(only along standard line)
2D Projected Map
3D Reference Globe
1/50,000,000
1/100,000,000
SF =
0.00000002
0.00000001
=
2.0 =
(Kimerlinget al., 2009)

31. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 31
Variable scale bar
http://en.wikipedia.org/wiki/Mercator_projection

32. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 32
1:500,000
Representative fraction
(absolute scale) 
Bar scale
“one inch to one mile”
Verbal scale
0 50 100 200 300 400 Km
> 1:250,000

33. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 33
Projection case Projection case

34. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 34
Tangent case
Cylindrical
ConicPlanar
(adapted from Lo and Yeung, 2006)Scale factor = 1
at standard lines
How can we 
reduce distortion?

35. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 35
Secant case
(adapted from Lo and Yeung, 2006)
Cylindrical
ConicPlanar
Scale factor = 1
at standard lines

36. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 36
Scale factor with two standard lines
Scale factor > 1
Scale factor = 1
Scale factor < 1

37. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 37
Visualizing the change in scale factor
(adapted from Kimerlinget al., 2009)
Reference globe
Mercator projection

38. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 38
Rule of thumb: match region to projection
where distortion is minimized
(adapted from Lo and Yeung, 2006)
Cylindrical
ConicPlanar
Tropical
Temperate
Polar

39. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 39
Projection aspect Projection aspect

40. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 40
Aspect
(adapted from Lo and Yeung, 2006)
Normal
SF > 1
SF > 1
SF > 1
SF > 1 Transverse Oblique

41. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 41
Aspect
(adapted from Kimerlinget al., 2006)
Polar
Equatorial Oblique

42. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 42
http://www.nrcan.gc.ca/earth‐sciences/geography/atlas‐canada/wa ll‐maps/16870

43. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 43
http://www.nrcan.gc.ca/earth‐sciences/geography/atlas‐canada/wa ll‐maps/16870k

44. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 44
http://www.nrcan.gc.ca/earth‐sciences/geography/atlas‐canada/wa ll‐maps/16870

45. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 45
http://www.nrcan.gc.ca/earth‐sciences/geography/atlas‐canada/wa ll‐maps/16870k

46. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 46
Let’s all hate Toronto

47. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 47
Projection central meridian Projection central meridian

48. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 48
Central meridian
(adapted from Lo and Yeung, 2006)

49. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 49
Central meridian: ‐100
Central meridian

50. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 50
Central meridian: ‐75
Poor choice of central meridian…

51. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 51
Poor choice of central meridian…
http://www.greenlandsc.com/en/World_America.aspx`

52. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 52
Central meridian set to China

53. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 53
Tissot’s indicatrix Tissot’s indicatrix

54. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 54
Observing distortion

55. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 55
Visualizing the change in scale factor
(adapted from Kimerlinget al., 2009)
Reference globe
Mercator projection

56. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 56

Draw infinitely small circles on 
reference globe

On globe, they are circles

When projected, they are 
distorted in shape and/or size
Tissot’s Indicatrix

57. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 57
Tissot’s Indicatrix: Cylindrical Equal Area

58. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 58
Meridians straightened out
SF < 1
SF > 1

59. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 59
Area = πr
2
= πab
Tissot’sIndicatrix
a
b
a= 1
b= 1
a
x
b= 1
This is on the original
reference globe (3D)
aandb
arescale factors (SF)

60. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 60
Shapes distorted
a
b
Area = πr
2
= πab
a= 1.25  
b= 0.8   
a
x
b= 1     
Since ax b= 1, 
2D map is equal‐area
but a≠ b
so map is not conformal

61. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 61
Tissot’s Indicatrix: Mercator

62. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 62
Horizontal and vertical scale factors
SF > 1
SF > 1

63. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 63
Sizes distorted
a
b
Area = πr
2
= πab
a= 1.25  
b= 1.25  
a
x
b= 1.5625
Since ax b≠ 1, 
2D map is not equal‐area
but a= b
so map is conformal

64. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 64
Area vs. Shape
You can preserve area or shape, but not both
a = bor a
x
b= 1
You can preserve area or shape, but not both
a = bor a
x
b= 1
Area
Shape

65. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 65
SF = 1
SF = 1
Standard 
line
SF > 1
Equidistant projection Equidistant projection

66. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 66
Equidistant projection
SF = 1
SF = 1
Standard 
line
SF > 1

67. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 67
Great circles and rhumblines Great circles and rhumblines

68. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 68
Equidistant projection
SF = 1
SF = 1

69. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 69
Azimuth
N
45°
A
B

70. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 70
Great circle route
New York
London

71. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 71
Great circle route
New York
London

72. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 72
Angles are all different

73. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 73
Great circle vs. Rhumb line
New York New York
London London

74. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 74
Rhumb lines are longer (really!)

75. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 75
Series of rhumb lines
New York New York
London London

76. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 76
Projection distance and direction Projection distance and direction

77. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 77
Distance, direction, and projection
B
A
B A
B
A
Straight line and shortest distance (i.e., great circle)?
Rhumb line, 
not great circle
Rhumb line, 
not great circle

78. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 78
Direction and projection
B A
B
A
B A
B
A

79. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 79
Azimuthal projection centered on New York
New York New York
B A
B
A
B A
B
A

80. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 80
None of the above: Compromise

81. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 81
Grid coordinate systems and UTM Grid coordinate systems and UTM

82. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 82 Mercator projection

83. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 83
Transverse Mercator
(adapted from Lo and Yeung, 2006)
Normal
SF > 1
SF > 1
SF > 1
SF > 1 Transverse Oblique

84. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 84
One UTM zone (6°wide)

85. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 85
Rotate cylinder, create new zone
(adapted from Lo and Yeung, 2006)

86. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 86
UTM zones

87. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 87
Projections and coordinate systems
Projected
coordinate system
(2D: UTM)
Projected
coordinate system
(2D: UTM)

88. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 88
Plane rectangular coordinate system
X
Y
How do we describe
where this is?
12345
1
2
3
4
5
0
x
y
(4,3)
Origin
(‐4,‐3)(‐4,3)
(4,‐3)
‐5
‐4
‐3
‐2
‐1
‐5 ‐4 ‐3 ‐2 ‐1
First 
Quadrant
Second 
Quadrant
Third 
Quadrant
Fourth 
Quadrant

89. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 89
Staying positive!
X
Y
12345
1
2
3
4
5
0
‐5
‐4
‐3
‐2
‐1
‐5 ‐4 ‐3 ‐2 ‐1

90. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 90
UTM coordinate grid

91. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 91
Where origin would normally be…
0,0

92. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 92
False easting
500000mE, 0mN
0mE, 0mN

93. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 93
North vs. South hemispheres
0mE, 0mN
0mE, 10,000,000mN
Easting
Northing
0mE, 0mN
0mE, 10,000,000mN
Easting
Northing

94. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 94
Geographic vs. UTM coordinate systems
Grid north

95. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 95
National Topographic System (NTS) map
http://geogratis.gc.ca/api/en/nrcan‐rncan/ess‐sst/cb864dc7‐25a1 ‐5136‐57f4‐c095ce1c6a6d.html

96. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 96
UTM coordinates
621,000mE 4,845,000mN, Zone 17 North 621,000mE 4,845,000mN, Zone 17 North
EastingNorthing Northing

97. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 97
UTM coordinates
CN Tower:
630,084mE, 4,833,438mN, Zone 17 North
CN Tower:
630,084mE, 4,833,438mN, Zone 17 North

98. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 98

99. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 99
Projections and ArcGIS Projections and ArcGIS

100. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 100
Projected Coordinate System

101. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 101
ArcMap and coordinate systems
Menu and toolbars
Table of 
contents
ArcCatalog
Geographic
Geographic
and
Projected
Mercator
Data frame
Mercator
Robinson
“on the fly”

102. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 102

Data can be saved to new file in new 
projection

Coordinates in the data file are recalculated 
and adjusted to match the new projection

You can’t always transform it “back” to exactly 
the same coordinates due to rounding errors.
Projection changes to data files

103. © Donald Boyes, Department of Geography and Planning, Universit y of Toronto 103

Exchange GIS data in a 3D geographic 
coordinate system, not a 2D projected 
coordinate system
To avoid errors…