CHAPTER 4
Mapping and Map Projection
CE 316
January 2012
66
4.1 Introduction





Coordinates


Great Circles
Distances
 km of ‘arc
Scales
 A map is identifies by its scale
 1 : 10,000 large scale
 1 : 125,000 small scale
Direction
 North, Magnetic North, Bearing,
Azimuth, Grid North
Contours

Contour Intervals




Natural – water lines
Vertical interval
Pictorial or Graphic Relief
Isopach ???
Airborne Mapping


Photogrammetry
Lidar
REFERENCE
Understanding Map Projections
GIS by ESRI®
67
68
4.2 Valued Map Properties
 Only a globe can give a valid picture with
nearly a true scale.
 However, a globe has practical disadvantages
 Valuable properties of a map are:
 Shape
 Area
 Distance
 Direction
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4.2 Valued Map Properties
Shape: When a map preserves shape over a small area it is said to be
CONFORMAL (Orthomorphic) “right” in Greek. (preferred by Engineers
and Military Strategists)
Area: want to see the area pictured to be in definite proportion to the area it
represents.
• If a map passes “the penny test” it is called an equal-area map.
(equivalent or proportional)
• Good for maps of population per square meter or other distribution
maps, such as for trees, sheep, tons of coal, etc.
• The equal-area map is therefore primarily one for the strict
illustration of statistics rather than giving us a pictorial impression of
the size of geographic features.
Distance: We can’t keep the scale distance constant all over the map at
any price
Direction: compass directions, everywhere on earth - curve the way the
earth curves
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4.3 Valued Map Projections
(General)
 Plane
 Cone
 Cylinder
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4.3.1 Map Projections onto a Plane
 The easiest geometric map projection to visualize is when the
projection plane is a plane tangent to the sphere at a point.
 The three possibilities for the projection center are Gnomonic,
Stereographic and Orthographic projections.
Gnomonic Projection:
Projection centre at centre of earth.
All great circles show up as straight lines.
(Including meridians and the equator)
Considered to be the oldest map
projection (Thales, 6th century BC)
Gnomonic projection of earth
centered on the geographic north
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pole
4.3.1 Map Projections onto a Plane
C”
Stereographic
Projection
Projection center
opposite to the point of
tangency
B”
B
C
A
C’
B’
Conformal (preserves angles)
Projection center at
infinity
a
A
D”
D’ A
P
Orthographic
Projection
T
C
c
D
b
B
T
de
D
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4.3.2 Conical Map Projections



Standard Parallel: The circle of latitude where a cone touches a
sphere that the cone is put over. This parallel will be at true
scale with increasing distortions to the north and south.
As the height of the cone increases, the standard parallel moves closer to
the equator. When the standard parallel reaches the equator, the cone
becomes a cylinder.
As the height of the cone decreases the standard parallel moves to higher
latitudes, and the cone becomes a plane when the standard parallel is on
the pole.
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4.3.3 Lambert Conformal Conic
Projection (Johann Heinrich Lambert)
 A conic projection with two standard parallels that intersect
the zone of interest at 1/6 of the zone width from the North
and South zone limits.
1728 - 1777
 Parallels of latitude are arcs of concentric circles having their center at
the apex.
 Meridians are straight lines converging at the cone.
The direction of
the central meridian establishes grid North.
 Scale is true at both standard parallels.
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4.3.3 Lambert Conformal Conic
Projection
Graphical representation of coordinates on the Lambert conformal
projection
Y
R cos θ
Z
c
Rb
θ
R
P
Standard
Parallels
R sin θ
Xp
M
Central Meridian
O
Yp
76 X
4.3.4 Mercator Map Projection


Created to show a line of constant bearing on the globe appear
as a straight line on the map. This is called a rhumb line or
loxodrome.
1512 - 1594
The meridians are equally spaced vertical lines and the
parallels are horizontal lines whose spacing increases towards
the poles.
Note: Vertical axis modified so that within a small area vertical and
horizontal scales the same (trick ensure directional vectors all the
same.
Rhumb line is obviously
easier to manually steer,
than the constantly
changing heading of the
shorter great circle route.
Boston to Cape Town
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Rhumb line
Compass Rose
32 point for
32 rhumb lines
(11 ¼ degrees)
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4.3.4 Mercator Map Projection
1o at  = 60o, is twice
as large as 1o at the
equator (because
cos(60) = 0.5
 The scale at
any parallel is
the same in all
directions and
can be
calculated by:
S = Scos 
where S=the scale
factor at the
equator
Scale Factor
(SF) = 1.0


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4.3.5 Universal Transverse Mercator Map
Projections
•
Similar to a Mercator Projection except it is
turned 90° so that it is related to a central
meridian and it does not retain the straight
rhumb line property of the Mercator projection
Normal Mercator
Central
Meridian
Transverse Mercator
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4.3.5 Universal Transverse Mercator Map
Projections
Central
Meridian N
S
Line of
Intersection
Single Meridian
with true scale
N
Cylinder
Lines of
Intersection
(N-S Direction)
A
B
A’
B’
Scale 1:1
Central
Meridian
S
Two Meridians
with true scale
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4.3.5 Universal Transverse Mercator Map
Projections
Short Vertical Section of Map
Zone Width
Scale Factor < 1
1/6
1/3
1/3
1/6
A
B
A’
Scale factor >1
Scale factor =1
Scale factor =0.9996
Central Meridian
Scale factor =1
Scale factor >1
 = 80oN
N. Pole
A
B
B’
Lines of Intersection
(N – S Direction)
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4.3.5 Universal Transverse Mercator Map
 = 84 N
Projections
o
• UTM projection zones are 6° wide
• Reference Ellipsoid is Clark’s 1866
(NAD 27) or NAD 83
Origin of Zone
• Basis for 1:50,000 topographic Maps
--- 6o Zone ---
• Unit of measure is the meter
O m North
Equator
500,000 m East
10,000,00O m North
• For the Southern hemisphere, a false
northing of 10,000,000 is given to the
equator
• Origin for longitude (Easting) is
• at the Central Meridian
Meridian 3o East of
Control Meridian
Central Meridian
• Origin for latitude (Northing)
• is the equator
X and Y
Coordinates
of the UTM Zone
Meridian 3o West of
Control Meridian
• A false easting of 500,000 is given to the
Central Meridian for each zone
• A 30’ overlap is provided between zones
 = 80oS
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4.3.5 Universal Transverse Mercator Map
Projections
Zone width @ 50o ≈ RE (cos) 2π 6/360 = 430.18862 km
Max Easting ≈ 715,094.310 m
NORTH
Approx.
64 km
Approx. 180 km
Rm/RN
500,000 E
Central Meridian
Secant Projection
3o
Rm/RN
6o
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4.3.5 Universal Transverse Mercator Map
Projections
Scale Factor for the Central
Meridian is 0.9996
NORTH
S.F. = 0.9996
(1:2,500)
S.F. = 1.00033
S.F. = 1.0000
S.F. = 1.0000
3o
Rm/RN
S.F. = 1.00033
Rm/RN
6o
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4.3.5 Universal Transverse Mercator Map
Projections
NORTH
1/6
1/3
1/3
1/6
≈ 2o
≈ 3o
Rm/RN
Rm/RN
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4.3.5 Universal Transverse Mercator Map
Projections
S.F. = 1.00033
S.F. = 0.9996
≈ 2o
Scale Factor @ Central Meridian
S.F.
S.F.
S.F.
S.F.
S.F.
= Map Distance/Ground Distance
= Map Radius/Ground Radius
= 0.9996
≈ [RE cos 2o] / RE
≈ 0.99939
Scale Factor at Zone Boundary
S.F.
S.F.
S.F.
= 1.00033
≈ ([RE (cos 2o / cos 3o)]/RE
≈ 1.00076
≈ 3o
Rm/RN
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4.3.5 Universal Transverse Mercator Map
Projections
Zone 2
Zone 3
UTM zones are numbered beginning
with 1 for the zone between 180° W
and 174° W to zone 60 between 174°
E to 180° E
Zone 1
N. Pole
 = 80oN
Equator
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4.3.5 Universal Transverse Mercator Map
Projections
13
Sa s k
a too
n
89
4.3.5 Universal Transverse Mercator Map
Projections
67A
1:250,000
NTS map
73M
1:250,000
NTS map
90
4.3 Map Projections
Modified Transverse Mercator

Transverse Mercator projection is in zones that are 3 wide
 The reference ellipsoid is Clarke 1866 or NAD 83
 The origin of longitude is at the central meridian
The origin of latitude is at the equator
 The unit of measurement is the foot
 A false easting of 1,000,000 ft. (304,800 m) is given to the central
meridian of each zone.
 Scale factor at central meridian is 0.9999 (1:10,000).
 Arbitrary Canadian zones numbered east to west from = 51o 30’E
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4.3 Map Projections
Modified Transverse Mercator
The scale factor for the 3o UTM projection is 0.9999 and the false easting is 304800.0 meters.92