CHAPTER 4 Mapping and Map Projection - U of S Engineering 316 Ch 4A 24 -01-11... · Orthographic...

66January 2012

CE 316

Mapping and Map Projection

CHAPTER 4

67

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 Natural – water lines

Contour Intervals Vertical interval

Pictorial or Graphic Relief Isopach ??? Airborne Mapping Photogrammetry Lidar

REFERENCEUnderstanding Map Projections GIS by ESRI®

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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

Direction

Distance

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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)

Cone

Cylinder

Plane

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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 of earth centered on the geographic north pole

Considered to be the oldest map projection (Thales, 6th century BC)

Gnomonic Projection: Projection centre at centre of earth. All great circles show up as straight lines.(Including meridians and the equator)

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A

T

P

A

Stereographic Projection

Projection center opposite to the point of tangency

Orthographic Projection

Projection center at infinity

B

B’

B”

D’

D”

D

C”

C

C’

Ad

D

B

b

C

c

T

a

e

4.3.1 Map Projections onto a Plane

Conformal (preserves angles)

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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.

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.

4.3.2 Conical Map Projections

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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.

Meridians are straight lines converging at the cone. The direction of the central meridian establishes grid North.

Scale is true at both standard parallels.

Parallels of latitude are arcs of concentric circles having their center at the apex.

4.3.3 Lambert Conformal Conic Projection (Johann Heinrich Lambert)

1728 - 1777

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Graphical representation of coordinates on the Lambert conformal projection

Y

XO

R c

os θ

R sin θ

P

Yp

Rθ

Xp

Z

M

Rb

c

Standard Parallels

Central Meridian

4.3.3 Lambert Conformal Conic Projection

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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.

4.3.4 Mercator Map Projection

Rhumb line is obviously easier to manually steer, than the constantly changing heading of the shorter great circle route.

Note: Vertical axis modified so that within a small area vertical and horizontal scales the same (trick ensure directional vectors all the same.

1512 - 1594

Boston to Cape Town

The meridians are equally spaced vertical lines and the parallels are horizontal lines whose spacing increases towards the poles.

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Compass Rose32 point for32 rhumb lines(11 ¼ degrees)

Rhumb line

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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

1o at = 60o, is twice as large as 1o at the equator (because cos(60) = 0.5

4.3.4 Mercator Map Projection

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• 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

4.3.5 Universal Transverse Mercator Map Projections

Normal Mercator Transverse Mercator

CentralMeridian

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Single Meridianwith true scale

N

SLine ofIntersection

CentralMeridian Cylinder


Two Meridianswith true scale

A’

A BN

SLines ofIntersection(N-S Direction)

Scale 1:1

CentralMeridianB’

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Scal

e fa

ctor

=1

Scal

e fa

ctor

=1

Scal

e fa

ctor

>1

Scal

e fa

ctor

>1

Scal

e fa

ctor

=0.

9996

Cen

tral

Mer

idia

nA B

A’ B’

Lines of Intersection(N – S Direction)

Zone Width

1/6 1/3 1/3 1/6

Scale Factor < 1

A B

= 80oN

N. Pole


Short Vertical Section of Map

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Cen

tral

Mer

idia

n

500,

000

m E

ast Equator10,000,00O m North

Origin of ZoneX and YCoordinatesof the UTM Zone

= 84oN

= 80oS

Meridian 3o East ofControl Meridian

Meridian 3o West ofControl Meridian

O m North

--- 6o Zone ---

• UTM projection zones are 6° wide

• Reference Ellipsoid is Clark’s 1866 (NAD 27) or NAD 83

• Basis for 1:50,000 topographic Maps

• Origin for latitude (Northing)• is the equator

• Origin for longitude (Easting) is• at the Central Meridian

• Unit of measure is the meter

• For the Southern hemisphere, a false northing of 10,000,000 is given to the equator

• A false easting of 500,000 is given to the Central Meridian for each zone

• A 30’ overlap is provided between zones


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6o

NORTH

Cen

tral

Mer

idia

n

500,

000

E

Rm/RN Rm/RN3o

Zone width @ 50o ≈ RE (cos) 2π 6/360 = 430.18862 kmMax Easting ≈ 715,094.310 m

Secant ProjectionApprox. 180 km

Approx.64 km


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NORTH

6oRm/RN Rm/RN

3o

S.F. = 0.9996(1:2,500)S.F. = 1.00033

S.F. = 1.00033

Scale Factor for the Central Meridian is 0.9996

S.F. = 1.0000S.F. = 1.0000


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NORTH

Rm/RN Rm/RN≈ 3o

≈ 2o

1/6 1/3 1/3 1/6


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Scale Factor @ Central MeridianS.F. = Map Distance/Ground DistanceS.F. = Map Radius/Ground RadiusS.F. = 0.9996S.F. ≈ [RE cos 2o] / RES.F. ≈ 0.99939

Scale Factor at Zone BoundaryS.F. = 1.00033S.F. ≈ ([RE (cos 2o / cos 3o)]/RES.F. ≈ 1.00076

Rm/RN≈ 3o

≈ 2o

S.F. = 0.9996S.F. = 1.00033


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Zone 3

Zone 2

Zone 1

= 80oNN. Pole

Equator

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


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13Saskatoon


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73M1:250,000NTS map

67A1:250,000NTS map


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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

Modified Transverse Mercator

4.3 Map Projections

92The scale factor for the 3o UTM projection is 0.9999 and the false easting is 304800.0 meters.

Modified Transverse Mercator

4.3 Map Projections

CHAPTER 4 Mapping and Map Projection - U of S Engineering 316 Ch 4A 24 -01-11... · Orthographic...

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