Any systematic arrangement of meridians and parallels portraying the curved surface of the spheroid

1

Lecture 6 Map Projections

• Any systematic arrangement of meridians and
  parallels portraying the curved surface of the
  spheroid upon a plane

2
Introduction

• The aim of a map projection is to represent the
  Earths surface or mathematical representation of
  the Earths surface on a flat piece of paper
  with a minimum of distortion.
• Recall
• Spheroidal Earth can be approximated to a plane
  over small areas with minimal distortion
• As the area of the spheroid becomes greater then
  the distortion becomes greater

3
The Problem
P
Q
P
Y
?
?
Q
X
We can say that x f1(???) y f2(???)
Therefore, the coordinates on the plane have a
direct functional relationship with latitude and
longitude. It follows that should be a one to
one correspondence between the earth and the
map. However 1) some projections may not be able
to show the whole surface of the Earth. 2) some
points may be represented by lines instead of
points This is because the spheroid has a
continuous surface whereas a plane map must have
a boundary.
4
Projection surfaces

• developable surfaces
• geometric or mathematical
• gnomonic, stereographic, orthographic

5
Map projections are attempts to portray the
surface of the earth or a portion of the earth on
a flat surface. Some distortions of conformality,
distance, direction, scale, and area always
result from this process. Some projections
minimize distortions in some of these properties
at the expense of maximizing errors in others.
Some projection are attempts to only moderately
distort all of these properties.
Conformality When the scale of a
map at any point on the map is the same in any
direction, the projection is conformal.
Meridians (lines of longitude) and parallels
(lines of latitude) intersect at right angles.
Shape is preserved locally on conformal or
orthomorphic maps. Distance -
equidistant A map is equidistant
when it portrays distances from the center of the
projection to any other place on the map.
Direction - azimuthal A map preserves
direction when azimuths (angles from a point on a
line to another point) areportrayed correctly in
all directions. Scale
Scale is the relationship between a distance
portrayed on a map and the same distance on the
Earth. Area - equal-area When a map
portrays areas over the entire map so that all
mapped areas have the same proportional
relationship to the areas on the Earth that they
represent, the map is an equal-area map.
6
Map projections fall into four general classes.
Cylindrical projections result from projecting
a spherical surface onto a cylinder.
In the secant case, the cylinder touches the
sphere along two lines, both small circles.
When the cylinder is tangent to the sphere
contact is along a great circle
7
When the cylinder upon which the sphere is
projected is at right angles to the poles, the
cylinder and resulting projection are
transverse.
When the cylinder is at some other,
non-orthogonal, angle with respect to the
poles, the cylinder and resulting projection is
oblique.
8
Conic projections result from projecting a
spherical surface onto a cone.
When the cone is tangent to the sphere contact
is along a small circle.
In the secant case, the cone touches the sphere
along two lines, one a great circle, the other a
small circle.
9
Azimuthal projections result from projecting a
spherical surface onto a plane.
In the secant case, the plane touches the sphere
along a small circle if the plane does not pass
through the center of the earth, when it will
touch along a great circle.
When the plane is tangent to the sphere contact
is at a single point on the surface of the Earth.
10
The Universal Transverse Mercator UTM
The Universal Transverse Mercator projection is
actually a family of projections, each having in
common the fact that they are Transverse Mercator
projections produced by folding a horizontal
cylinder around the earth. The term transverse
arises from the fact that the axis of the
cylinder is perpendicular or transverse to the
axis of rotation of the earth. In the Universal
Transverse Mercator coordinate system, the earth
is divided into 60 zones, each 6 of longitude in
width, and the Transverse Mercator projection is
applied to each zone along its centerline, that
is, the cylinder touches the earth's surface
along the midline of each zone so that no point
in a given zone is more than 3 from the location
where earth distance is truly preserved.

• unit of length is the metre
• an ellipsoid is adopted as the shape and size of
  the earth
• coord obtained by a TM of f and l of points on
  the ellipsoid
• the true origin of coords is the intersection of
  the equator and the central meridian of a zone
• a central scale factor of 0.9996 is superimposed
  on the central meridian
• for points in the northern hemisphere, E and N
  coords are related to a false origin 500,000m W
  if the true origin and for points in the southern
  hemisphere, E and N are related to a false origin
  500,000m W and 10,000,000m S of the true origin
• the projection has 60 zones, 6o wide in
  longitude, beginning with zone 1 having a central
  meridian of 177oW, numbered consecutively
  eastwards, ending with zone 60 with a central
  meridian of 177oE
• the latitude extent of each zone is 80oS and 84oN

11
The Universal Transverse Mercator UTM
12
AMG and MGA

• The AMG and MGA are both systems of rectangular
  coordinates based on TM projections of f and l
  related to the AGD and GDA.
• closely corresponds with the UTM grid used
  globally
• coordinates in metres
• zones are 6 wide (1/2 degree overlap)
• zones numbered from zone 49 with central
  meridian 111E to zone 57 with central meridian
  159E
• central scale factor k0 0.9996
• origin of each zone is the intersection of
  central meridian with the equator

• false origin S 10 000 000m, W 500 000
• coordinates described in Easting (E) and
  Northing (N)

13
The TM graticule and the AMG/MGA
Projection Transverse Mercator Ellipsoid GRS
80 Central Meridian 141.00000 Reference
Latitude 0.00000 Scale Factor 0.99960 False
Easting 500000 False Northing 10 000 000.00000
These parameters mean that the Central Meridian
of Zone 55 is at 141E so that it covers from
138E to 144E the Reference Latitude is 0.0000
(the equator, which is 0N) the origin of the
coordinate system is at the intersection of
the Central Meridion with the Reference Latitude
and thus is at (0N,144E), where the coordinates
are (x, y) (500 000,10 000 000) m. The false
Easting of 500,000m is to ensure that all points
in the zone have positive x coordinates. The
y-coordinates are always positive in the Northern
hemisphere because 0 is at the equator. In the
Southern Hemisphere, a false Northing of
10,000,000m is applied to ensure that the
y-coordinate is always positive. The Scale
Factor of 0.9996 means that along the Central
Meridian, the true scale of 1.0 is reduced
slightly so that at locations off the true
meridian the scale factor will be more nearly 1.0
(the Transverse Mercator projection distorts
distance positively as you move away from the
Central Meridian).
14
Converting spheroidal coordinates to grid (AMG)
coordinates

• E (K0nwCosf)1 term1 term2 term3
• Term1 (w2/6)Cos2f(y-t2)
• Term2 (w4/120)Cos4f4y3(1-6t2)y2(18t2)-y2t2t4
  
• Term3 (w6/5040)Cos6f(61-479t2179t4-t6)
• E E False Easting
• N K0m Term1 Term2 Term3 Term4
• Term1 (w2/2)nSinf Cosf
• Term2 (w4/24)nSinf Cos3f(4y2y-t2)
• Term3 (w6/720)nSinf Cos5f8y4(11-24t2)-28y3(1-6t
  2)y2(1-32t2)-y(2t2)t4
• Term4 (w8/40320)nSinf Cos7f(1385-3111t2543t4-t6
  )
• N N False Northing

• Ko central scale factor 0.9996
• ? as defined in previous lecture
• ???????????
• ? geodetic latitude
• t tan?
• ??? geodetic longitude measured from central
  meridian ?0,
• positive eastwards ? - ?0
• m meridian distance
• To translate to false origin of AMG
• 10 000 000 m to northing
• 500 000 m to easting

15
Example from AGD to AMG
Data Station BUNINYONG Latitude f -37o 39
15.557 Longitude l c Zone 54 lo 141o
Computations w l - lo 143o 55 30.633 -
141o 2o 55 30.63
meridian distance
16
Example from AGD to AMG
m -4184650.83515514.577-8.259-.016
-4169144.533
Radii of curvature
Easting E .9996(258127.64828.736-.031
-.000036 258053.090 Northing N
.9996(-4169144.533-4025.327-2.435-.001.00000024
-4171503.027
E E False origin 500000.000 258053.090
758053.090 N N False origin
10000000.000 -4171503.027 5828496.973
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Converting grid (AMG) coordinates to spheroidal
coordinates

• E E - False Easting x E/(K0n')
• f f - Term1 Term2 - Term3 Term4
• Term1 (t/K0r)(xE/2)
• Term2 (t/K0r)(Ex3/24)-4y29y(1-t2)12t2
  
• Term3 (t/(K0r)(Ex5)/720)8y4(11-24t2)-12y3
  (21-71t2)15y2(15-98t215t4)180y(5t2-3t4)
  360t4
• Term4 t/K0r)(Ex7/40320)(13853633t24095t4
  1575t6)
• w Term1 - Term2 Term3 - Term4
• Term1 x Secf Term2 (x3/6)Secf(y2t2)
• Term3 (x5/120)Secf-4y 3(1-6t2)y2(9-68t2)
  72yt224t4
• Term4 (x7/5040)Secf(61662t21320t4720t6)
• l l0 w

18
Meridian Distance and Foot-point Latitude
m aA0f -A2Sin2fA4Sin4f -A6Sin6f where A0
1-(e2/4)-(3e4/64)-(5e6/256) A2
(3/8)(e2e4/415e6/128) A4 (15/256)(e43e6/4) A6
35e6/3072
The foot-point latitude (f) is the latitude for
which the meridian distance equals the true
northing divided by the central scale factor
(mN/k0). This value can be calculated directly,
once three other values are available. n
(a-b)/(ab) f/(2-f) G a(1-n)(1-n2)(1(9/4)n2(
225/64)n4)(p/180) s (mp)/(180G) The foot point
latitude (in radians) is then calculated by f
s((3n/2)-(27n3/32))Sin2s ((21n2/16)-(55n4/32))
Sin4s(151n3/96) Sin6s(1097n4/512)Sin8s
19
Example from AMG to AGD
Data Station BUNINYONG Zone 54 k0 0.9996 E
758053.090 N 5828496.973 False origin
-500000.000 False Origin -10000000.000
Computations E 258053.090 N
-4171503.027 m N/k0 -4173172.296 foot
point latitude
20
Example from AGD to AMG
f -37o 41 26.198 0o 2 10.7616 -
0.1206.0001-.00000015 -37o 39 15.557 l
141o 0o 55 36.9341 - 6.3081.0071-.0000099
143o 55 30.633
Redfearns formulae To transform to MGA use
parameters of the GRS80 ellipsoid
http//www.anzlic.org.au/icsm/gdatm/gdatm.htm
http//www.osg.vic.gov.au/mgadoc.htm
21
Which AMG zone are we in

• AMG zones are 6 wide with 1/2 overlaps into
  adjoining zones
• AMG zones are numbered from zone 49 with central
  meridian 111E to zone 57 with central meridian
  159E
• Given the longitude of a point,??, it is simple
  to work out which zone we are in

zone number z INT (? / 6 31) 0
??????180 z INT (? / 6 - 29) 180 ??????360
eg Victoria ? 147, zone number INT(24.5 31)
55
22
Defining Central Meridian for a Zone

• Given the zone number, we can work out the
  position of the central meridian, ?0
• the terms ? and ?0 are required in spheroid to
  grid conversion formulae (Redfearns Formulae)
• In these equations they are expressed as ? ? -
  ?0 (measured ve eastwards)

?0 6z - 183 31 ??z???60 ?0 6z - 3 1
??z???30
eg zone 55, ?0 6 x 55 - 183 147
23
Computation of coordinates on the AMG/MGA

• The AMG/MGA is a plane and if the bearing and
  distance between two points on the AMG/MGA plane
  is known, simple trigonometric relationships can
  be used to compute AMG/MGA coordinates of points.
  Bearings and distances on the AMG/MGA plane are
  plane bearings and plane distances
• Points on the ground are not on the AMG/MGA plane
  and measurements between them are not AMG/MGA
  plane bearings or distnaces. Using field
  measurements directly to compute coordinate
  differences which are then added or subtracted
  from an AMG/MGA coordinate of one of the points
  will not give AMG/MGA coordinates of the other
  points in the survye.
• horizontal distances measured by a total station
  are local plane distances not AMG/MGA plane
  distances.
• The bearing and distance obtained by simple
  geometric and trigonometric relationships between
  two points with known AMG/MGA coords will be the
  plane bearing and distance . These will not be
  the same as the field observations made between
  these two points

24
Geometry on the AMG

• plane cartesian coordinate system so to first
  approximation
• simple plane geometry works on grid

• l is known as plane distance. It is the length
  of the line between two points on the grid
• ? is known as plane bearing. It is the angle
  between grid north and the plane distance, l
• Using plane distances the formulae of plane
  trigonometry hold completely.
• However, we observe angles and bearings on the
  spheroid which are different to l and ?

ie given point A with Easting and Northing EA
and NA, by measuring the bearing ???and
distance l to point B, we can work out the
coordinates of B, EB and NB
grid north
B
?
l
A
EB EA l.sin? ? NB NA l.cos?
25
Geometry on the AMG

• Points A and B on the spheroid are projected to
  points a and b on the grid.
• The shortest distance on the spheroid between A
  and B (the geodesic) is projected onto the curved
  dotted line ab
• Similarly, the meridian, which points towards
  geodetic north, when projected onto the grid,
  becomes a curved line
• It is evident that the plane bearing and plane
  distance is different to those that would be
  measured, in practice, on the spheroid

insert pic maria lecture5 slide 6
26
Reduction of distances to the spheroid

• The reduction of the wave path chord distance
  (d2), to the ellipsoidal chord distance (d3), can
  be given as a single rigorous formula (Clarke,
  1966, p299)
• d3(d22 - (hA - hB)2) / (1 hA/Ra) (1 hB/Ra
  )½
• The ellipsoidal chord distance (d3) is then
  easily reduced to the ellipsoidal distance
• s d31 (d32/24Ra2 3d34/640Ra4 ...
• Where Ra is the radius of curvature in the
  azimuth of the line.
• For a distance of 30 kilometres in the Australian
  region the chord-to-arc correction is 0.028
  metres. For a distance of 50 kilometres, the
  correction reaches about 0.l3 metre and it is
  more than 1 metre at 100 km. The second term in
  the chord-to-arc correction is less than 1 mm for
  lines up to 100 km, anywhere in Australia and
  usually can be ignored.

• The formulae given in this chapter use
  ellipsoidal heights (h). If the geoid-ellipsoid
  separation (N value) is ignored and only the
  height above the geoid (H - the orthometric or
  AHD height) is used, an error of 1 part per
  million (ppm) will be introduced for every 6½
  metres of N value (plus any error due to the
  change in N value along the line). As the N value
  in terms of GDA varies from -35 metres in south
  west Australia, to about 70 metres in northern
  Queensland, errors of from -5 to almost 11 ppm
  could be expected. Of course there are areas
  where the N value is small and the error would
  also be small.

27
Conversion of azimuth, ?, to plane bearing, ?

• Convergence ? at a point is the angle between
  grid north and the projected geodetic north

• If we measure an azimuth on the spheroid and wish
  to plot it on a plane grid,???must be converted
  to a plane bearing??
• Correction for
• difference between grid north and geodetic north
• difference between plane bearing and grid bearing

28
Conversion of azimuth, ?, to plane bearing, ?

• Arc-to-chord correction???is the angle between
  the plane bearing and the grid (arc) bearing

• Grid (or Arc) Bearing ? from a to b is the angle
  between the projected arc and grid north

grid bearing azimuth grid convergence
plane bearing grid bearing arc to chord
?????????
?????????
29
Conversion of azimuth, ?, to plane bearing, ?
plane bearing azimuth grid convergence arc
to chord correction
q a g d

• Computing grid convergence

• Computing arc to chord correction

Rigorous formulae Simplified formulae
30
PointScale Factor
Simplified formula for line scale factor
k is the ratio of an infinitesimal distance at a
point on the grid, dL, to the corresponding
distance on the spheroid, ds

• For short baselines (lt16km), the line scale
  factor well is represented by the mean of the
  point scale factors at either end of the
  baseline For short baselines (lt16km), the line
  scale factor well is represented by the mean of
  the point scale factors at either end of the
  baseline
• This is accurate to 1ppm over baselines up to
  16km in length
• (ie 1mm/km)

k dL / ds
Even though k might be defined for a particular
projection, it will change for points furthest
away from the line(s) of zero distortion
Line Scale Factor

• From point to point along a line on the grid the
  point scale factor will, in general, vary.
• The line scale factor K is the ratio of the plane
  distance l to the
• corresponding spherical distance, s.

Rigorous equation for line scale factor (AGD
Technical Manual 5.6.1,5.6.2)
This approximation is independent of the
location in the zone
31
AMG Calculation Example

• A solar observation is made to determine the
  azimuth from point A to B. The observed azimuth
  aAB is 77o 22 43. Given the AMG coordinates of
  the point A and the sea level distance of the
  line AB 5400.00m, determine the AMG coordinates
  of the point B.

CM

Pt A E 758 053.090 N 5 828 496.973 f
-37o 39 15.5571 l 143o 55 30.6330 Zone
54 CM 141o
GN
TN
B
b
g
aAB
A
32
AMG Calculation Example

• Determine the grid convergence
• w 143o 55 30.6330 - 141o 10530.633
• sin f -0.61089605
• approx grid convergence g sinf w
• 1o 47 13.122

• Determine the arc to chord correction
• d -(5829512 - 5828496.973)(763356 758053.090
  - 106)(0.127 8 10-8)
• -0.66
• grid bearing 79o 09 59.67 - 0.66
• 79o 09 59.01
• Determine the point and line scale factors
• E 758053.090 E0 500 000
• PSFA 0.9996 1.23(E-E0)2 10-14
• 1.000419074
• PSFB 0.9996 1.23(E-E0)2 10-14
• 1.000453084
• LSF (PSFA PSFB)/2 1.000436079
• Grid Distance SL Distance LSF
• 5402.354

Determine approx grid bearing and dist approx
grid bearing b a g 79o 09 59.67 approx
grid distance SL distance 5400. DE
5400sinb 5303.741 DN 5400cosb
1015.047 Approx coords of new point B EB EA
DE 763356 NB NA DN 5829512
33
AMG Calculation Example

• Calculation of final AMG coordinates
• DE 5402.354 sinb 5306.068
• DN 5402.354 cosb 1015.413
• Approx coords of new point B
• EB EA DE 763359.158
• NB NA DN 5829512.385

34
3 dimensional transformations
We have 2 problems
1. Transform GPS positions given in cartesian X,
Y, Z to geodetic ????, h (WGS84) to project onto
UTM
2. Convert GPS geodetic coordinates from WGS84
spheroid to AGD spheroid and hence to AMG.
With the advent of GDA2000 we are faced with the
further problem of converting AMG coordinates on
the AGD (ANS66) to MGA coordinates on the GDA
(GRS80).
35
1. Cartesian to Geodetic Coordinates
Recall
3 dimensional cartesian coordinates
spheroidal geodetic coordinates
As we measure on surface of the Earth, each
spheroidal geodetic coordinate has an associated
ellipsoidal height, h
h is measured positive outwards from the surface
of the spheroid along the surface normal
Remember ellipsoidal height orthometric height
geoid height
36
Bowrings Forward Transformation
(x, y, z)
(????, h)
The forward transformation is simple because it
is a closed formula
For GPS applications, the inverse transformation
is more important
Bowrings formulae work for transformations on
any spheroid
37
Example Bowrings Forward Transformation
Given the following ANS geodetic coordinates,
compute ANS cartesian coordinates
ANS cartesian coordinates have their origin at
centre of origin of ANS66. This origin is NOT
geocentric
? -32.000 ? 11554? h 30.0m
ANS66 parameters a 6378 160m e2 0.006
694 542
radius of prime vertical ? 6384163.694m
38
X (6 384 163.694 30.0). cos(-32). cos(115.9)
-2 364 890.007m
Y (6 384 163.694 30.0). cos(-32). sin(115.9)
4 870 298.747m
Z (6 384 163.694(1- 0.006 694 542) 30.0).
sin(-32) -3 336 458.978m
39
Bowrings Reverse Transformation
(x, y, z)
(????, h)
The reverse formula is more complicated because
after we rearrange the previous equation set we
get
which has ? on each side of the equation (in???
equation has to be solved iteratively
40
Bowrings Inverse Transformation - Iterative
Solution
1. Compute longitude, ?. This can be done
directly
? tan-1(y/x)
2. Compute p (x2 y2)1/2
3. Compute an approximate value ?0 of the
latitude from
41
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42
Example - reverse of previous example
x -2 364 890.008 y 4 870 298.747 z -3 360
458.976
all on ANS66 but method holds for other spheroids
eg WGS84
1. computed longitude, ? tan-1(y/x)
tan-1 (4 870 298.747/-2 364 890.008) -6406?
or 11554?
Note ambiguity in tan-1 term. However, if you are
in Australia it is simple enough to work out
which is the correct longitude as one longitude
puts you on the other side of the world.
43
Example cont....
44
4. Compute an approximate value for the radius of
curvature in the prime vertical, ?0 from
6 384 163.694 (from program)
5. Compute the ellipsoidal height from
h p / cos?o - ?o
5414103.309/cos(-32.0000) - 6 384 163.694
30.00m
45
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46
2. Change of Spheroid
We now know how to transform cartesian (X, Y, Z)
coordinates to geodetic (????? h) coordinates
(and vice versa)
Because GPS coordinates are computed in WGS84 and
AGD is on the ANS66 spheroid we must be able to
transform from one spheroid to another
This can be done in a number of ways but the
easiest is using a 7-parameter transformation of
cartesian coordinates
47
A 7-parameter transformation is essentially a
helmert transformation in 3 dimensions
Z
z?
x?
O?
y?
Y
O
X
We wish to transform one cartesian coordinate
system to another
48
7 parameter transformation
where
X, Y, Z and x?, y?, z? are cartesian coordinates
centred on O and O? respectively
1 ds represents the scale factor to be applied
between the two coordinate systems
dx, dy, dz is the vector representing the origin
shift between O? and O
49
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50
The changes induced by the rotations can be
expressed in terms of the following rotation
matrices
When fully multiplied out the full rotation
matrix becomes
51
When the values of ?x, ?y, ?z are very small
(usually the case in change of spheroid
computations) the cosines can be assumed equal to
1 and the sines equal to the numerical values of
the angles themselves expressed in radians.
R becomes
where rx, ry, rz are the small rotations in
radians
The matrix R is known as a Rodrigues matrix
52
So the 7-parameter transformation can be written
as
To transform from one spheroid to another we need
to know the transformation parameters, dx, dy,
dz, ds, rx, ry, rz
Note for the reverse transformation we need only
change the signs of the seven transformation
parameters
53
Transformation Parameters in Australia

• A set of transformation parameters from
  WGS84 to AGD84 have been computed by Higgins
  (1987)
• dx -116.00m
• dy -50.47m
• dz 141.69
• rx -0.23
• ry -0.39
• rz 0.344
• ds -0.0983ppm

• A set of transformation parameters from
  AGD84 to GDA94 have been computed by Auslig
• dx -117.763m
• dy -51.510m
• dz 139.061
• rx -0.292
• ry -.443
• rz -.277
• ds -.191ppm
• These supersede Higgins 1987 parameters

54
Example
convert WGS84 coordinates
to AGD84.
1. Form R matrix (convert angles to radians)
55
2. Apply matrix and origin shift
X -2 364 759.300 Y 4 870 348.935 Z
-3 360 610.238AGD
56
WGS84 cartesian to AMG
7 parameter
Bowring
Redfearn
Note height transformed by hWGS84 - N HAHD