PRÁCTICA GEODESIA (2017)
Pràctica InglésUniversidad  Universidad Politécnica de Cataluña (UPC) 
Grado  Ingeniería de Aeronavegación  3º curso 
Asignatura  Navegación, cartografía y cosmografía 
Año del apunte  2017 
Páginas  14 
Fecha de subida  03/07/2017 
Descargas  1 
Subido por  areig 
Vista previa del texto
Geographical distances
Air Navigation, Cartography and Cosmology
Alba Martín
Ferran Morales
Anna Reig
6GX31
17.04.2017
Table of contents
1.
Introduction ......................................................................................................................3
2.
The Earth as a flat surface formulas ..........................................................................5
3.
4.
5.
2.1
Spherical Earth projected to a plane ..................................................................5
2.2
Ellipsoidal Earth projected to a plane ................................................................6
2.3
Polar coordinate flatEarth formula ....................................................................6
The Earth as a sphericalsurface .................................................................................8
3.1
Tunnel distance .......................................................................................................8
3.2
Greatcircle distance ...............................................................................................9
The Earth as an ellipsoidalsurface ..........................................................................10
4.1
Lambert's formula for long lines ......................................................................10
4.2
Bowring's method for short lines .....................................................................11
4.3
Vicenty's formulas .................................................................................................13
Conclusions ....................................................................................................................14
6. References ………………………………………………………………………………………………………. 14
1.
Introduction
The main objective of this project is to compute and compare different
methods for calculating distances between two points of the Earth. The
mentioned points can be defined by their latitude and longitude.
This problem arises from the irregularity of the Earth surface that can be
considered in three different shapes analysed through the project. These
three ways of representing the Earth are a flat surface, a sphere and an
ellipsoid. Algorithms will be compared in terms of relative error (directly
related to precision) and time of execution of the program elaborated with
the corresponding equations of each method.
Chosen points and consequently distances in between can be divided as
follows:
•
Long distance in the North Hemisphere
Barcelona, 41°23′19″ N, 2°09′32″ E
Hong Kong, 22°17′07″ N, 114°09′27″ E
Distance: 10063.41km
•
Long distance in the South Hemisphere
Buenos Aires, 34°36′13.264″ S, 58°22′53.612″ W
Canberra, 35°17′00″ S, 149°07′41″ E
Distance: 11746.94km
•
Long distance between the North and South Hemisphere
Barcelona, 41°23′19″ N, 2°09′32″ E
Canberra, 35°17′00″ S, 149°07′41″ E
Distance: 17089.61km
•
Short distance
Barcelona, 41°23′19″ N, 2°09′32″ E
Llançà, 42°21′44″ N, 3°09′07″ E
Distance: 135.27km
•
Medium distance
Barcelona, 41°23′19″ N, 2°09′32″ E
Bakú, 40°22′39″ N, 49°53′31″ E
Distance: 3962.40 km
•
Distance from pol to pol
Longyearbyen, 78°13'26.1"N, 15°37'23.4"E
Artigas scientific base (Antarctic), 62°11'05.1"S, 58°54'20.1"W
Distance: 16382.14km
•
Distance between a point and its antipodal
Barcelona, 41°23′19″ N, 2°09′32″ E
Pacific Ocean, 41° 23' 6.187" S, 177° 49' 28.9453" W
Distance: 20034.20km
Table 1 Chosen locations
2.
The Earth as a flat surface formulas
The simpler approximation for the earth surface is the planar one. It may
be very useful for small distances but it has increasing error on accuracy.
As separation between two points become greater and a point becomes
closer to a geographic pole the accuracy on the approximation becomes
smaller. Even over short distances, the accuracy can be considered great,
it also depends on the method used to project latitude and longitude onto
the plane.
2.1
Spherical Earth projected to a plane
This method takes into account the variation in distance between
meridians with latitude:
𝐷 = 𝑅√(∆𝜙)2 + (cos(𝜙𝑚 )Δ𝜆)^2
Once the procedure has been computed, the distances between points
we obtained are the following:
Algorithm: SPHERICAL EARTH PROJECTED TO A PLANE
Distance
Time to
Real distance computed by
Committed
perform
Set of Points
(km)
the algorithm
error (%)
computations
(km)
(s)
BarcelonaHong Kong
10063,41
10776,86
7,09
0,000133
Buenos AiresCanberra
11746,94
18914,8
61,02
0,000121
BarcelonaCanberra
17089,61
18410,33
7,73
0,000156
BarcelonaLlançà
135,27
135,14
0,10
0,000114
BarcelonaBakú
3962,4
4010,16
1,21
0,000134
LongyearbyenArtigas
16382,14
17638,09
7,67
0,000138
Scientific Base
BarcelonaPacific Ocean
20034,2
22029,6
9,96
0,000135
(Antipodal)
Table 2 Spherical Earth projected to a plane distances
As it can be seen, the execution time is very small for all the distances
without taking into account if it is bigger or smaller. Apart from this
vantage, it is surprising that the error between two points separated a
large distance in the north is much smaller (7,09%) than the one for the
south (61,02%). This can be related to the fact that trajectory between
Canberra and Buenos Aires is always pretty close to the South Pole,
location where the error committed by the algorithm is bigger. The best
calculation goes for BarcelonaLlançà that are very close and this method
is thought to be used for short distances.
2.2
Ellipsoidal Earth projected to a plane
For
distances
not
exceeding
475km/295miles
the
Federal
Communications Commission (FCC) prescribes the following formula:
𝐷 = √(𝐾1 Δ𝜙)2 + (𝐾2 Δ𝜆)2
Where:
K1 = 111.13209 − 0.56605cos(2φ𝑚 ) + 0.00120cos(4φ𝑚 )
K2 = 111.41513cos(φ𝑚 ) − 0.09455cos(3φ𝑚 ) + 0.00012cos(5φ𝑚 )
When applying the algorithm to the set of points stated in section two,
the obtained results are the next ones:
Algorithm: ELLIPSOIDAL EARTH PROJECTED TO A PLANE
Distance
Time to
Real distance computed by
Committed
perform
Set of Points
(km)
the algorithm
error (%)
computations
(km)
(s)
BarcelonaHong Kong
10063,41
10797,22
7,29
0,000444
Buenos AiresCanberra
11746,94
18957,22
61,38
0,000273
BarcelonaCanberra
17089,61
18404,7
7,70
0,000341
BarcelonaLlançà
135,27
135,17
0,07
0,000261
BarcelonaBakú
3962,4
4020,52
1,47
0,000308
LongyearbyenArtigas
16382,14
17567,41
7,24
0,000314
Scientific Base
BarcelonaPacific Ocean
20034,2
22028,5
9,95
0,000434
(Antipodal)
Table 3 Ellipsoidal Earth projected to a plane distances
The results are more or less the same as the ones for spherical earth
projected to a plane but within a little more error in general. As well as
before, the smaller error goes for BarcelonaLlançà but this time within a
0,03% less. The same can be observed for the distance between Buenos
Aires and Canberra and has the same explanation as before. However, the
time to perform the algorithm has increased a little bit. Consequently, it
can be said that the previous algorithm works better than this one for the
selected distances.
2.3
Polar coordinate flatEarth formula
Polar coordinate flatEarth formula takes into account colatitude values in
radians as follows:
𝐷 = 𝑅√𝜃12 + 𝜃22 − 2𝜃1 𝜃2 cos(∆𝜆)
When applying the algorithm to the set of points stated in section three,
these are the obtained results:
Algorithm: POLAR COORDINATE FLATEARTH FORMULA
Set of Points
BarcelonaHong Kong
Buenos AiresCanberra
BarcelonaCanberra
Distance
Real
computed
Table 4distance
Polar coordinate flatEarth
formulaby
distances Committed
(km)
the algorithm
error (%)
(km)
10063,41
10774,555
7,07
Time to
perform
computations
(s)
0,000112
11746,94
17089,61
26989
18695,85
129,75
9,40
0,000087
0,000114
BarcelonaLlançà
135,27
141,51
4,61
0,000103
BarcelonaBakú
LongyearbyenArtigas
Scientific Base
BarcelonaPacific Ocean
(Antipodal)
3962,4
4415,99
11,45
0,000099
16382,14
16620,92
1,46
0,0001
20034,2
20015,11
0,10
0,000155
Table 4 Polar coordinate flatearth formula distances
For Polar coordinate the result for Buenos AiresCanberra has goes to
129,75% surprisingly which is due to the fact, as explained before, that
the distance in between always pretty close to the South Pole, location
where the error committed by the algorithm is bigger. Despite this value,
it seems to be a good method for antipodal distances (0,10% error), but
in general the precision has decreased for the other distances. However,
this method is also good for short distances. A propoint for this method
could be the fact that the execution time is smaller than in the cases
before.
3.
The Earth as a sphericalsurface
A possibility for considering Earth shape is using formulas of spherical
trigonometry, willing to accept an approximate error of 0,5%. The shortest
distance along the surface of a sphere between two points is a greatcircle
containing both points.
3.1
Tunnel distance
The tunnel between two points on the Earth is defined by a line through
threedimensional space between the points of interest. The great circle
chord length may be calculated as follows:
𝐷 = 𝑅𝑐𝐻 ,
𝑓𝑜𝑟𝐶ℎ = √∆𝑋2 + ∆𝑌 2 + ∆𝑍 2
Where:
∆X = cos(φ2)cos(λ2) − cos(φ1)cos(λ1)
∆Y = cos(φ2)sin(λ2) − cos(φ1)sin(λ1)
∆Z = sin(φ2) − sin(φ1)
Once the procedure has computed, the distances between points we
obtained are the following:
Set of Points
BarcelonaHong Kong
Buenos AiresCanberra
BarcelonaCanberra
BarcelonaLlançà
BarcelonaBakú
LongyearbyenArtigas
Scientific Base
BarcelonaPacific Ocean
(Antipodal)
Algorithm: TUNNEL DISTANCE
Distance
Real distance computed by
Committed
(km)
the algorithm
error (%)
(km)
10063,41
9042,57
10,14
11746,94
10145,44
13,63
17089,61
12403,9
27,42
135,27
135,135
0,10
3962,4
3895,22
1,70
Time to
perform
computations
(s)
0,000956
0,001161
0,000958
0,000958
0,001001
16382,14
12223,17
25,39
0,002042
20034,2
12742,02
36,40
0,000985
Table 5 Tunnel distances
For this new surface shape consideration, the tunnel distance results to
be very precise for all distances without exceeding a 37% of error. The
weakest point to consider is that execution time has increased by a factor
of ten in some cases compared to the time execution of the methods that
consider the Earth as a flat surface. For short distances, this method is
really precise since for instance for BarcelonaLlançà, it is equal to 0.10%.
However, using this method, it is easy to see that the greater the distance
the greater the error committed is.
3.2
Greatcircle distance
Also known as orthodromic distance, it is the shortest distance between
two points on the surface of a sphere. As a sphere is not an Euclidean
space, the length between two points cannot be considered a straight line.
For antipodal distances, there are infinite greatcircles but all the arcs
have the same length. As the Earth is nearly a sphere, the error of this
method will be close to 0,5% or so.
The formula used to compute all distances in this case takes into account
the central angle between the two points of interest:
𝐷 = 𝑟∆𝜎,
𝑓𝑜𝑟𝜎 = arccos(𝑠𝑖𝑛𝜙1 𝑠𝑖𝑛𝜙2 + 𝑐𝑜𝑠𝜙1 𝑐𝑜𝑠𝜙2 𝑐𝑜𝑠Δ𝜆)
When applying the algorithm to the set of points stated in section two,
the following results are obtained:
Algorithm: GREAT CIRCLE DISTANCE
Distance
Real distance computed by
Committed
Set of Points
(km)
the algorithm
error (%)
(km)
BarcelonaHong Kong
10063,41
10053,74
0,09609
Buenos AiresCanberra
11746,94
11735,65
0,09611
BarcelonaCanberra
17089,61
17073,19
0,09608
BarcelonaLlançà
135,27
135,137
0,09832
BarcelonaBakú
3962,4
3958,59
0,09615
LongyearbyenArtigas
16382,14
16366,4
0,09608
Scientific Base
BarcelonaPacific Ocean
20034,2
20014,96
0,09604
(Antipodal)
Time to
perform
computations
(s)
0,0005
0,000719
0,000898
0,000514
0,000588
0,000314
0,001071
Table 6 Greatcircle distances
Surprisingly, the error has drop down within a maximum value of
0,09832% for BarcelonaLlançà which represents an accuracy of 1km over
thousands of kilometers. This fact could be caused by the proximity of
the points. For the antipodal points (BarcelonaPacific Ocean) the error
becomes minimum. About the execution time, one can see that the
biggest is for the antipodal distance, this could be caused by the infinite
possible arcs between them, as has been mentioned before. Despite that,
the time to perform computations is very small, what places this method
in one of the best positions compared to the ones already mentioned.
4.
The Earth as an ellipsoidalsurface
Supposedly, an ellipsoid approximates the surface of the earth much
better than the two previous methods as the earth, which is a geoid, is
theoretically much more similar to an ellipsoid rather than to a sphere or
a plane. In this section, we are going to find whether this statement is
true or not.
Even though there are several algorithms that approach the earth well,
nowadays the most commonly used algorithm is the Vicenty’s formula.
Before we get started, it is important to mention that the shortest distance
between two points along the surface of an ellipsoid is called the geodesic
distance and all of them follow more complicated paths than the greatcircle lines.
4.1
Lambert's formula for long lines
This first method gives an accuracy of about 10 meters over thousands
of kilometres.
To apply the algorithm the following steps must be followed:
 Convert the latitudes of both points to reduced latitudes using
tan 𝛽 = (1 − 𝑓 ) tan 𝜙
Where:
β is the reduced latitude
ϕ is the true latitude
f is the flattening
Done this, we proceed as follows:
When applying the algorithm to the set of points stated in section one,
the following results are obtained:
Algorithm: LAMBERT’S FORMULA FOR LONG LINES
Distance
Real distance computed by
Committed
Set of Points
(km)
the algorithm
error (%)
(km)
BarcelonaHong Kong
10063,41
10071,41
0,0795
Buenos AiresCanberra
11746,94
11758,81
0,1010
BarcelonaCanberra
17089,61
17074,49
0,0885
BarcelonaLlançà
135,27
135,17
0,0739
BarcelonaBakú
3962,4
3968,7
0,1590
LongyearbyenArtigas
16382,14
16342,11
0,2444
Scientific Base
BarcelonaPacific Ocean
20034,2
20022,72
0,0573
(Antipodal)
Time to
perform
computations
(s)
0,000816
0,000666
0,000827
0,001092
0,000839
0,000752
0,001648
As seen in the table above the algorithm is fast when computing all
distances.
When computing long distances the algorithm has little error, however it
can be seen that when the computation occurs during the course of a
meridian (approximately) the accuracy can be a significantly lower.
For short distances, such as BarcelonaLLancà or BarcelonaBakú, the
accuracy is unstable it can give very small errors for very short distances
and then increase rapidly when the distance increase one order of
magnitude.
Therefore, we can conclude that the relative position of points within the
earth surface must be taken into account as well as the geodesic distance
between them.
4.2
Bowring's method for short lines
This algorithm uses a sphere of radius R’ with longitude (λ’) and latitude
(ϕ’) to map the points.
The procedure that needs to be used is the following:

Define A and B as follows:
Once the procedure has computed the distances between points we
obtained are the following:
Algorithm: BOWRING’S METHOD FOR SHORT LINES
Distance
Real distance computed by
Committed
Set of Points
(km)
the algorithm
error (%)
(km)
BarcelonaHong Kong
10063,41
0,1188
10075,37
Buenos AiresCanberra
11746,94
11735,68
0,0959
BarcelonaCanberra
17089,61
17096,07
0,0378
BarcelonaLlançà
135,27
135,3
0,0222
BarcelonaBakú
3962,4
3968,76
0,1605
LongyearbyenArtigas
16382,14
16434,78
0,3213
Scientific Base
BarcelonaPacific Ocean
20034,2
20013,04
0,1056
(Antipodal)
Time to
perform
computations
(s)
0,000582
0,00056
0,000575
0,000874
0,000699
0,000527
0,002107
It can be observed that all computation times are similar and fast however
it can be realized that when the set of points are antipodal the
computation takes more or less 4 times what it takes to compute any of
the others.
Concerning the accuracy of the algorithm, we note that for very short
distances the method is highly accurate whereas if distances are longer
the accuracy becomes worse. Particularly, when the geodesic distance
follows more or less the course of a meridian the error becomes
surprisingly bigger. However, globally, the computed errors are
significantly small.
4.3
Vicenty's formula
This is the last algorithm to be analysed and it is the most widely used
nowadays as it is the most efficient method and gives great accuracies
except when the geodesic distance is calculated between antipodal
points.
In order to watch the algorithm stepbystep see reference [1].
Set of Points
BarcelonaHong Kong
Buenos AiresCanberra
BarcelonaCanberra
BarcelonaLlançà
BarcelonaBakú
LongyearbyenArtigas
Scientific Base
BarcelonaPacific Ocean
(Antipodal)
Algorithm: VICENTY’S FORMULA
Distance
Real distance computed by
Committed
(km)
the algorithm
error (%)
(km)
10063,41
0,0794
10071,4
11746,94
11758,81
0,1010
17089,61
17074,49
0,0885
135,27
135,17
0,0739
3962,4
3968,7
0,1590
Time to
perform
computations
(s)
0,004413
0,0005166
0,006072
0,004306
0,005164
16382,14
16342,12
0,2443
0,005321
20034,2
19947,36
0,4335
0,004943
Note that as stated before when computing an antipodal distance the
algorithm fails to be as accurate as in the other cases.
For all kind of distances the error committed is more or less the same as
the Lambert Method and the smallest of all the methods studied up to
here, even though, as happened in both previous methods, when the
distance is straight northsouth the algorithm losses accuracy.
Concerning the computation times, note that they are greater than in the
previous methods, that is so because in order to reach better accuracy in
all any set of points (antipodal not included) a loop needs to be
implemented.
5.
Conclusions
The objective of this project was to learn more about how to compute
distances along the earth’s surface. To do so, we have learned three kinds
of methods: Earth as a flat surface, Earth as a spherical surface and
eventually we have approximate the Earth’s surface as an ellipsoid.
In the first approximation, we have obtained that computations were
realised very quickly, however when distances increased so did the error.
As seen in results above, we can only take this method (Earth as a flat
surface) as a good one when distances are extremely short. We should
adjust this limiting distances depending on the error we accept.
In the second approximation, we got that accuracy in all set of points was
increased a lot, thus we could use this approximation a lot more than the
previous one. In any case, the computation times were increased but not
significantly. Specifically, if two points are extremely close to each other
(order of a few kilometres) we would use the tunnel distance as it’s the
most accurate submethod, however in the rest of cases, we would use
the greatcircle line method as the error committed is similar for all the
set of points.
Last but not least, we took into account the ellipsoidal approximation,
this resulted to be the best one as the errors committed were significantly
lower than in the other two approximations. When paying attention to the
methods we obtained that for long distances we could choose either the
Lambert’s formula or the Vicenty’s because results were similar. When
considering short distances the Lambert’s approximation lost effectivity
so we should use the Bowring’s one.
For all three algorithms, the accuracy when computing distances between
antipodal points was low compared to other points, thus we would use
the great circle line method when computing these kinds of distances.
If we had to compute all kinds of distances we would definitely use
Vicenty’s algorithm as its performance is similar in all cases, however it
is the slowest algorithm so, in order to consider which algorithm needs
to be used in a certain application we must take into account the accuracy
as well as the computation time.
6.
References
[1] Atenea Documentation, NACC, Marc Pérez
...