PRÁCTICA GEODESIA (2017)

Pràctica Inglés
Universidad 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
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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 flat-Earth formula ....................................................................6 The Earth as a spherical-surface .................................................................................8 3.1 Tunnel distance .......................................................................................................8 3.2 Great-circle distance ...............................................................................................9 The Earth as an ellipsoidal-surface ..........................................................................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) Barcelona-Hong Kong 10063,41 10776,86 7,09 0,000133 Buenos Aires-Canberra 11746,94 18914,8 61,02 0,000121 Barcelona-Canberra 17089,61 18410,33 7,73 0,000156 Barcelona-Llançà 135,27 135,14 0,10 0,000114 Barcelona-Bakú 3962,4 4010,16 1,21 0,000134 Longyearbyen-Artigas 16382,14 17638,09 7,67 0,000138 Scientific Base Barcelona-Pacific 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 Barcelona-Llançà 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 − ⁡0.56605⁡cos(2φ𝑚 ) ⁡ + ⁡0.00120⁡cos(4φ𝑚 ) K2⁡ = ⁡111.41513⁡cos(φ𝑚 ) − ⁡0.09455⁡cos(3φ𝑚 ) ⁡ + ⁡0.00012⁡cos(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) Barcelona-Hong Kong 10063,41 10797,22 7,29 0,000444 Buenos Aires-Canberra 11746,94 18957,22 61,38 0,000273 Barcelona-Canberra 17089,61 18404,7 7,70 0,000341 Barcelona-Llançà 135,27 135,17 0,07 0,000261 Barcelona-Bakú 3962,4 4020,52 1,47 0,000308 Longyearbyen-Artigas 16382,14 17567,41 7,24 0,000314 Scientific Base Barcelona-Pacific 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 Barcelona-Llançà 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 flat-Earth formula Polar coordinate flat-Earth 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 FLAT-EARTH FORMULA Set of Points Barcelona-Hong Kong Buenos Aires-Canberra Barcelona-Canberra Distance Real computed Table 4distance Polar coordinate flat-Earth 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 Barcelona-Llançà 135,27 141,51 4,61 0,000103 Barcelona-Bakú Longyearbyen-Artigas Scientific Base Barcelona-Pacific 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 flat-earth formula distances For Polar coordinate the result for Buenos Aires-Canberra 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 pro-point for this method could be the fact that the execution time is smaller than in the cases before.
3.
The Earth as a spherical-surface 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 great-circle containing both points.
3.1 Tunnel distance The tunnel between two points on the Earth is defined by a line through three-dimensional space between the points of interest. The great circle chord length may be calculated as follows: 𝐷 = 𝑅𝑐𝐻 , 𝑓𝑜𝑟⁡⁡𝐶ℎ = √∆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 Barcelona-Hong Kong Buenos Aires-Canberra Barcelona-Canberra Barcelona-Llançà Barcelona-Bakú Longyearbyen-Artigas Scientific Base Barcelona-Pacific 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 Barcelona-Llançà, 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 Great-circle 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 great-circles 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, 𝑓𝑜𝑟⁡𝜎 = 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) Barcelona-Hong Kong 10063,41 10053,74 0,09609 Buenos Aires-Canberra 11746,94 11735,65 0,09611 Barcelona-Canberra 17089,61 17073,19 0,09608 Barcelona-Llançà 135,27 135,137 0,09832 Barcelona-Bakú 3962,4 3958,59 0,09615 Longyearbyen-Artigas 16382,14 16366,4 0,09608 Scientific Base Barcelona-Pacific 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 Great-circle distances Surprisingly, the error has drop down within a maximum value of 0,09832% for Barcelona-Llançà 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 (Barcelona-Pacific 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 ellipsoidal-surface 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) Barcelona-Hong Kong 10063,41 10071,41 0,0795 Buenos Aires-Canberra 11746,94 11758,81 0,1010 Barcelona-Canberra 17089,61 17074,49 0,0885 Barcelona-Llançà 135,27 135,17 0,0739 Barcelona-Bakú 3962,4 3968,7 0,1590 Longyearbyen-Artigas 16382,14 16342,11 0,2444 Scientific Base Barcelona-Pacific 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 Barcelona-LLancà or Barcelona-Bakú, 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) Barcelona-Hong Kong 10063,41 0,1188 10075,37 Buenos Aires-Canberra 11746,94 11735,68 0,0959 Barcelona-Canberra 17089,61 17096,07 0,0378 Barcelona-Llançà 135,27 135,3 0,0222 Barcelona-Bakú 3962,4 3968,76 0,1605 Longyearbyen-Artigas 16382,14 16434,78 0,3213 Scientific Base Barcelona-Pacific 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 step-by-step see reference [1].
Set of Points Barcelona-Hong Kong Buenos Aires-Canberra Barcelona-Canberra Barcelona-Llançà Barcelona-Bakú Longyearbyen-Artigas Scientific Base Barcelona-Pacific 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 north-south 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 sub-method, however in the rest of cases, we would use the great-circle 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 ...

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