 Είμαι ο Ψηκόπηλος. Ευχαριστώ για να είμαι εδώ. Θα συνεχίσω πιο λίγο την προηγιότητα. Θα δείξω πώς δημιουργήσουμε η γεωγραφία για η μπροστιωματική. Μπορείτε να νομιστείτε από αυτήν την καλύτερη επίπεδο για την αντιμετωπική και γεωγραφική αντιμετωπία. Στο πιο λίγο την αυτοκλίνηση θα δημιουργήσουμε πιο σημαντικές δημιουργήσεις για τα γεωγραφία και τότε θα δώσω δυο παραδοσιασμούς πώς μπορείτε να συνεχίσετε τα γεωγραφία στην μπροστιωματική. Ο μπροστιωματικός μοδελ που μπορείτε να έχετε για εργαία είναι το εργαίο εργαίο. Το προβλήμα είναι ότι υπάρχουν κάποιοι που πιστεύουν και ότι είναι το μόνο που είναι το μόνο. Θα πάμε για κάτι πιο ρασσιανό. Θα πιστεύουμε ότι η αυτοκλίνηση έχει πιο μοδελ. Έτσι, έχουμε ένα μπροστιωματικό μοδελ, έχουμε ακρισία μόνο εργαία. Είναι εύκολη γιατί μπορείτε να χρησιμοποιήσετε ένα γεωγραφικό γεωγραφί. Είναι πολύ σύνδρο και σύνδρο αλγότητα. Αν θέλετε να πιστεύετε ένα πιο ακρισμό, μπορούμε να αυτοκλίνουμε ότι η αυτοκλίνηση είναι σφεία. Αυτή η αυτοκλίνηση που χρησιμοποιήσαμε, για παράδειγμα, στην Google Maps, δεν είναι πολύ ακριστή, αλλά είναι καλύτερη. Και έχουμε σύνδρο αλγότητα. Θα πιστεύω ένα μπροστιωματικό γεωγραφί. Έτσι, έχουμε ένα μπροστιωματικό μοδελ, που είναι η αυτοκλίνηση της αυτοκλίνησης. Αυτή η αυτοκλίνηση είναι αυτοκλίνησης στην Πόλη. Και αυτή η αυτοκλίνηση της αυτοκλίνησης στην GIS, σφαρδένου, και τώρα έχουμε περισσότερες αυτοκλίνησης για να αυτοκλίνονται. Και φυσικά, αν θέλουμε να πιστεύουμε πιο πιο, μπορείτε να έχετε γεωγραφίες, όπου έχετε also the center of the mass of the earth, και αυτή είναι για αυτοκλίνηση σφαρδένου και γεωγραφίδικου. Βλέπουμε εδώ, στην Βουστιωματική. Δεν κάνουμε γεωγραφίες. Βλέπουμε εδώ, κάποιες τύπες. Είναι ένας κορνέδος σύστημα στην Βουστιωματική. Έτσι έχουμε ένα καρτύσσιο τύπ, και έχουμε ένα σφαρδένου οικογραφίες, παραμετρικά από ένα δικοίτη ή ένα ρεδιάν, και έχουμε also the geographic. Το geographic είναι αυτοκλίνησης, στην τελευταία αυτοκλίνησης, και αυτή είναι η σφαρδένου, και αυτή είναι η σφαρδένου. Οι πιο πρώτοι αυτοκλίνησης στην Our Study είναι οι γεωδέσσεις, οι οποίες είναι οι καλύτερες πράγματα με 2 σπαρδένου. Στο βαθόνιο, το γεωδέσσιο είναι μία σπίτι. Στην αυτοκλίνηση είναι ένα καρτύπιο κέντρο, γίνετε το κέντρο, και ανταξύξετε με ένα κορνέδρο που μπαθάνε στο κέντρο, έχετε ένα καρτύπιο κέντρο. Στην αυτοκλίνηση, το γεωδέσσιο δεν είναι κλείδρο. Αυτό σαν να μην χρυματιστούν το γεωδέσσιο. Σε αυτό, τίποτα, δημιουργεί η καμπλεξίτητα και για τα αυτοκλίνηση. Εδώ, σαν να βρεις τι σοχεία στον αλυπσό της, αυτό είναι ο γεωδέσσιο. Δεν είναι κλείδρο, οπότε ούτε αυτοκλίνηση με ένα κλείδρο. Και δεν έχει δικαίωση. Από την μεραίδηση της εμφανίκης. Προσπαθείται την μεραίδηση της εμφανίκης, τελευταία σύγχρονη. Δηλαδή είναι πραγματική ελευταία. Αυτό είναι μόνο που υπάρχει also this loxodrome or ramp line that you can see in maps of if you use the marker projection so if you draw a line there in such a map then this is a curve of same angle that crosses the meridians in same angle or azimuth and the great circle there the shortest path is not of course that ramp line but goes around. So this is just not confused loxodromes with geodesics or loxodromes are easier to to compute so we want to compute geodesics the shortest paths. Okay so these are two main problems in geodesy and this will be used as core procedures to build more complicated algorithm so we have the direct problem when we are given a point the azimuth which is the angle and a distance and we want to compute the point which is that distance away and it's the given angle so it's like navigation and then we have the inverse problem when we are given two points and we want to compute the distance and also the angles. So having this assume that somebody gives us this implemented then we can start building some core geodesic algorithms like point to point distance area of a polygon or other objects intersection envelope all this kind of algorithms and then they are using this we can have higher level algorithms of like geometry distance like distance of a multipolygon with another crazy multipolygon and then we can have a set of set operations in geometry like union intersection etc. and relational operations and this can be used if we want to build the spatial index. So the main problem here as you maybe you can realize is the distance of two points so all of these algorithms need to compute the distance between two points so in flat you have you knew everything from primary school it's a Pythagorean theorem it's quite simple and also simple to implement it then for sphere okay again maybe high level not primary school but high school you have again we have a closed formula it's easy to implement using some trigonometric functions and then when you are in the ellipsoid there is no closed formula so this is the equations that connect the distance which is s here with a longitude latitude which are lambda and phi so I will not of course explain this format just have it here for illustration of how the difficulty goes and you can also two points here two comments you can realize that going down we increase accuracy also we increase complexity which means that our code our implementation will be slower okay so what we do in boost geometry we have different formulas which are selected internally by with respect to the coordinate system so as Adam shows the coordinate system comes together with a point so if you define a point with that is geographic and then start applying algorithms on this point or geometries that you construct with this point then internally the boost geometry will call either the more complicated algorithms for spheroid or if you have your point your point wants to be in on the sphere it will run the more easier faster but less accurate algorithms for the sphere for example and also this is in more detail we have for ellipsoid we have three different algorithms that are these three okay we call them strategies and there are three different ways of approximating the distance and there are some space some time accuracy trade-off so the first is less accurate but faster and so on and so forth and we follow the state of the art approach which is that in all these geodesic problems actually you solve a spherical problem which is easier and has a nice closed formula and then you do some ellipsoidal correction by solving some integrals that are similar to the ones I showed before either by series expansion or numerical regression we do series expansion which is enough to have a good accuracy if you want more details we can discuss it offline so I think this is enough for the details of the implementation so again I will give a similar example as my colleagues I'm from Athens so I calculate the distance from my home to the event I use boot geometry for this probably you have understand from Adam's talk what all these definitions are about here I also define a spheroid and I define the strategy which is let's say Thomas it's one of these three strategies that I show before and then I ask to compute the distance from the point which is Athens let's say Acropolis which is more fancy and the other point which is ULB Brussels here so and I say that I want to use this strategy and then I do the computation if I remove the strategy from here the default strategy will call and also if I go here and put instead of geographic into the point if I say that these points are spherical then the same code will compute the spherical distance ok so I run some examples with with boot geometry and these are the results so with spherical I have this distance spherical here I use a different radius so these these are both spherical but with different radius and as you can see the same as Google Maps so Google Maps give me this distance which is the same as this one so this means that we understand that Google Maps using uses the formula that I showed you before for spherical and also assumes that the radius of the earth is this one which is the mean as to reduce you can also use the WGS 84 major axis of the ellipsoid make a bit a bit larger radius and you have a more accurate result but I guess this happens only this example this does not mean that if you use this radius you have more accurate results the only way to have more accurate results if you want is to use the geographic so these are the three different strategies that we have implemented at Doyer, Thomas and Vincenti and as you can see the divergence is quite small between them and the result has something like two or three kilometers difference from spherical which is a Google earth ok so a normal question here is again what's the performance but the actual distance is no should be the actual distance how do you compute the actual distance no one knows a survey engineer should start from Athens and start shooting yes so ok another example so this is a Brussels Center it looks like a Pentagon so I just created a Pentagon and I asked for Google Maps to compute the area so Google Maps compute that area we'll show it the second slide yes before showing the area of Google Maps I use Google Maps to compute the area and then I put the points inside the Boost Geometry to compute the same area with Boost Geometry as you can see this is similar to the previous example I have again a strategy for area which is geographic and is parameterized by the Vincenti formula you can parameterize it with Andoyer or the Thomas and then you create one strategy and you plug it into the computation of the area if you leave it as it is so if you remove it this and you say area poly it will use the default the one that we believe is better but you can use your own ok and these are the results here again Google Maps looks at again it's using this mean Earth radius and gives this result and these are the results with geographic algorithms here the even in this smaller example you can see that again there is a difference in accuracy ok so performance what is expected is that the spherical is less accurate and then Andoyer is also is a bit more accurate Thomas a bit more accurate the most accurate one is Vincenti this we know from the theory how these algorithms are working ok because you you know how these algorithms operate so actually these algorithms create some series and this algorithm let's say very high level cuts the series in some small order this goes to a higher order and this is using the using more using a better algorithm for with Newton iteration so in theory this this computes a better a more accurate result so ok we don't have a detailed performance analysis and it seems that we also don't have a accuracy analysis that a lot of people seems that the question but yes I don't know if you have feedback or where is a database of accurate distances we could try to see the accuracy to make a study but also it's interesting to do a performance analysis of these of these implementations and yes we have some timings in github pull request so while we do the pull request of these features we also write some tests for in our computers but ok of course this is this not a detailed performance analysis ok so similar work there is this library which is called geographic lib it's it's again C++ and also implements similar similar algorithms it's robust and it's fast as far as I as far as I know and it's also used in post GIS from this version and on the thing is that it lacks a variety of algorithms so it only gives you the distance area and some special projections so we are going to we are even now have implemented more in boost geometry and yes just a tip of performance just testing in my computer geographic lib for distance and our algorithms for distance our implementation is faster but of course this is not a performance analysis but it seems that at some point we have a faster implementation ok that's it about the related work and let's go to future work we are ongoing this is work in progress we are implementing more algorithms for ellipsoid like a segment segment distance more projections convex hulls and droids and actually we also want to have all the relational operations that are going to be used for spatial indexing and yes this is an open problem I mean this is an open feature we don't support distance of near at the portal points in geographic algorithms so this is something that at some point we want to implement in boost geometry and also we are thinking that's why we have this face there of Google summer of code proposals but we don't think we are not sure if we propose something but we are thinking of some project so yes keep in touch if you are interested and yes that's all Can you repeat the question? Ah yes the question is if one can implement its own coordinate system but what does it mean its own coordinate system I mean something, I mean how do you define it? There is an internal number of metric with overflow exploding overflow for calculation distance on coordinate and it's quite specific and it differs from any Cartesian or spherical or spherical system it differs Ok so it's a special coordinate system so I guess if you want to do this you have to at least implement how the distance between two points is computed at least this and then you can probably use all the other algorithms that use the distance as a main procedure but then for example for area you have to area cannot directly call distance it's a different algorithm so for area for example it depends on what you want to use for example if you want perimeter you can implement distance and then you will have perimeter for free but for area for example you have to implement again an algorithm that computes the area of a polygon but I guess this is you have to implement at least how do you measure the distance between two in this coordinate system Yes yes yes union and yes I think I had yes yes there is so the operations are there you can perform union of polygons for example if the polygons are Cartesian so the algorithms will remain the same but somehow we have to do the work of plugging in the the geographic system for example if you want to do in unions of geographic polygons Yeah just go one more question it's good We have also the comparable distance which is supposed to be much faster Ah yes Yes So this has to do with square roots you don't compute the square roots and yes I haven't think of it but it's not straightforward because you don't have a square root you have some iterative method that approximates some integral actually so I don't know how can you gain from this but maybe maybe you can do something not end the computation at some point when you know that the one computation is the one result is larger than the other maybe you can stop it something like this yes this could be useful Thank you Thank you