Search

Fuel consumption is an important factor to be considered in the process of weather routing. How to choose an appropriate route according to the requirements is particularly important. This paper proposes a method to optimize the ship weather routing. Based on the original genetic algorithm, the trigonometric function selection operator is introduced, the mutation operator is improved to increase the search range in the initial stage of the algorithm, and gradually narrow the search range in the middle and late stages of the algorithm, thus the convergence is sped up and the running time of the algorithm is reduced. Aiming at the incomplete ship speed fuel consumption comparison table, this paper uses SPSS (Statistical Product and Service Solutions) software to perform curve fitting and a curve with the best fitting degree is found. Then the curve equation is used to calculate the total fuel consumption of the case ship (S175 container ship) sailing through between the two ports Yokohama (35° N, 141° E) and San Francisco (37° N, 123° W), to verify the performance of the improved algorithm. Aiming at the minimum fuel consumption, the fuel consumption of the optimized route is reduced by 7.84% compared with that of the Rhumb Line.

Identification of when and where moving areas intersect is an important problem in maritime operations and air traffic control. This problem can become particularly complicated when considering large numbers of objects, and when taking into account the curvature of the earth. In this paper, we present an approach to conflict identification as a series of stages where the earlier stages are fast, but may result in a false detection of a conflict. These early stages are used to reduce the number of potential conflict pairs for the later stages, which are slower, but more precise. Our approach is generally applicable to objects moving in piece-wise straight lines on a 2D plane, and we present a specific case where the Mercator Projection is used to transform objects moving along rhumb lines on the earth into straight lines to fit in our approach. We present several examples to demonstrate our methods, as well as to quantify the empirical time complexity by using randomly generated areas.

This article aims at generalizing loxodromic (rhumb line) navigation to conformally flat Riemannian manifolds. We admit space-time dependence of both perturbing vector field and ship’s self-speed. Thereafter, the findings are applied to time-efficient navigation by a variational approach, referring to the local solutions of Zermelo’s navigation problem under arbitrary wind. This yields the corresponding conditions for loxodromic time-minimal and time-maximal navigation in relation to the navigation data. Our research is also illustrated by a two-dimensional example (an oblate ellipsoid), which distinguishes perturbations of different force: weak, critical and strong. It includes numerical simulations and discussion, emphasizing and comparing loxodromic solutions among the minimizing, maximizing and anomalous time extremals.

The article proposes a method of improving the accuracy of determining the current coordinates of a moving object moving along a loxodromic trajectory by analytical three-dimensional projection of its coordinates determined by navigation measurements in conditions of intense interference on the trajectory of its true motion. The results of testing the efficiency of the proposed method, which was carried out by numerical simulation of the algorithms described in the work of determining the coordinates of the object on the loxodromic trajectory on noisy navigation measurements. The results indicates the possibility of effective use of the proposed approach. Keywords: analytical three-dimensional projection coordinates, geographic information systems, interference, known as a loxodrome path, navigation measurement.

If two linear surfaces move one after another without sliding, then they can be considered as axoids of a solid body that performs a corresponding motion in space. If the axoids are cylindrical surfaces, then the study of their rolling can be replaced by the study of the rolling of centroids - curves of the orthogonal cross section of these cylindrical surfaces. Usually the rolling of a moving centroid on a stationary one is considered. However, there are cases when centroids roll one after another, while rotating around fixed centers. Examples of round centroids are circles, non-round - congruent ellipses, in which the centers of rotation are foci. In both cases, the center-to-center distance is constant. The point of contact of the circles is located at the center distance and is stationary during their rotation, and for ellipses it "floats" on this segment. In the article [1] the congruent centroids formed by symmetric arcs of a logarithmic spiral are considered. The centers of rotation of the centroid are the poles of the spirals. A characteristic feature of logarithmic spirals is that they intersect all radius vectors emanating from the pole at a constant angle. For a sphere, the prototype of a logarithmic spiral is a loxodrome, which crosses all the meridians at a constant angle and twists around the pole of the sphere. The paper hypothesizes that closed spherical curves formed from arcs of loxodrome, like centroids from arcs of a logarithmic spiral on a plane, can also roll around axes intersecting in the center of the sphere. If these closed curves are connected by rectilinear segments with the center of the sphere, then two cones are formed - axoids of non- circular conical wheels. This hypothesis is based on the fact that with an infinite increase in the radius of the sphere, its surface around the pole turns into a plane, and the meridians - in straight lines emanating from the pole. Accordingly, the loxodrome is transformed into a logarithmic spiral. The article shows that the hypothesis is confirmed. Axoids of non- circular wheels are built in it, the axes of which intersect at right angles. The expression of the arc length of the loxodrome is found and it is shown that when the conical axoids rotate at appropriate angles around their axes, the contact curves pass straight paths. This means that the rolling of the axis occurs without sliding., Якщо дві лінійчаті поверхні перекочуються одна по одній без ковзання, то їх можна розглядати, як аксоїди твердого тіла, що здійснює відповідний рух у просторі. Якщо аксоїди є циліндричними поверхнями, то дослідження їх кочення можна замінити дослідженням кочення центроїд – кривих ортогонального перерізу цих циліндричних поверхонь. Зазвичай розглядається кочення рухомої центроїди по нерухомій. Однак існують випадки, коли центроїди котяться одна по одній, одночасно обертаючись навколо нерухомих центрів. Прикладом круглих центроїд є кола, некруглих – конгруентні еліпси, у яких центрами обертання є фокуси. У обох випадках міжцентрова відстань є сталою. Точка контакту кіл розташована на міжцентровій відстані і є нерухомою під час їх обертання, а для еліпсів вона «плаває» на цьому відрізку. У статті [1] розглянуті конгруентні центроїди, утворені симетричними дугами логарифмічної спіралі. Центрами обертання центроїд є полюси спіралей. Характерною особливістю логарифмічних спіралей є те, що вони перетинають всі радіус-вектори, які виходять із полюса, під сталим кутом. Для кулі прообразом логарифмічної спіралі є локсодрома, яка перетинає всі меридіани під сталим кутом і закручується навколо полюса кулі. В статті висунута гіпотеза, що замкнені сферичні криві, утворені із дуг локсодроми подібно до центроїд із дуг логарифмічної спіралі на площині, теж можуть обкочуватися навколо осей, що перетинаються в центрі сфери. Якщо ці замкнені криві сполучити прямолінійними відрізками із центром сфери, то утворяться два конуси – аксоїди не круглих конічних коліс. Ця гіпотеза ґрунтується на тому, що при нескінченному зростанні радіуса кулі її поверхня в околі полюса перетворюється у площину, а меридіани – у прямі лінії, що виходять із полюса. Відповідно, локсодрома перетворюється у логарифмічну спіраль.В статті показано, що висунута гіпотеза підтверджується. В ній побудовано аксоїди некруглих коліс, осі яких перетинаються під прямим кутом. Знайдено вираз довжини дуги локсодроми і показано, що при повороті конічних аксоїдів на відповідні кути навколо своїх осей криві дотику проходять рівні шляхи. Це означає, що обкочування аксоїдів відбувається без ковзання.

Η παρούσα ερευνητική εργασία αποτελεί μια πιλοτική εφαρμογή που πραγματοποιήθηκε στα πλαίσια του Μεταπτυχιακού Προγράμματος Σπουδών με τίτλο «Επιστήμες της Αγωγής: Διδακτικές Πρακτικές και Διαδικασίες Μάθησης» και κατεύθυνση «Ψηφιακές Εφαρμογές στην Εκπαίδευση των Φυσικών Επιστημών, της Τεχνολογίας, της Μηχανικής και των Μαθηματικών», του ΠΤΔΕ, ΑΠΘ. Η έρευνα αποτελεί μια μελέτη περίπτωσης, η οποία διεξήχθη σε δεκατέσσερις μαθητές/τριες της Στ’ τάξης του Δημοτικού Σχολείου της Νέας Απολλωνίας Θεσσαλονίκης. Το διδακτικό αντικείμενο είναι η Γεωγραφία με έμφαση στη χαρτογραφία. Μέσα από καθοδηγούμενη διερεύνηση και βιωματικές δράσεις, οι μαθητές/τριες έμαθαν για τις αλλοιώσεις που υπάρχουν αναφορικά με τις αποστάσεις μεταξύ των χωρών πάνω σε χάρτες μικρής κλίμακας και πειραματίστηκαν σε παγκόσμιους ψηφιακούς χάρτες Μερκατορικής προβολής. Κύριος στόχος της έρευνας ήταν η ανάδειξη των παρανοήσεων των μαθητών/τριών για τις αποστάσεις που μετρούνται σε παγκόσμιους χάρτες και στη συνέχεια η εξάλειψη αυτών. Ένας επιπλέον στόχος ήταν η ανάδειξη της σημαντικότητας του μαθήματος της Γεωγραφίας που συνεχώς περιθωριοποιείται και ως επί το πλείστον διδάσκεται με παραδοσιακό τρόπο. Οι διδακτικές παρεμβάσεις ήταν διάρκειας τεσσάρων εβδομάδων και περιλάμβαναν συνολικά επτά δραστηριότητες. Τα ερευνητικά εργαλεία που αξιοποιήθηκαν για τη συλλογή δεδομένων ήταν: α) pre tests & post tests (ερωτηματολόγια με ερωτήσεις ανοικτού τύπου), β) φύλλα εργασίας, γ) ηχογραφήσεις. Τα εργαλεία που αξιοποιήθηκαν στην τάξη ήταν: α) βιντεοπροβολέας, β) ηλεκτρονικοί υπολογιστές, γ) online λογισμικά με ψηφιακούς χάρτες και υδρογείους δ) έντυποι παγκόσμιοι χάρτες διαφορετικών προβολών, ε) χάρακας, στ) πλαστελίνη, ζ) φουσκωτή και ξεφούσκωτη υδρόγειος σφαίρα (μεγάλων διαστάσεων). Οι μαθητές/τριες γνώρισαν τις προβολές χαρτών και διαπίστωσαν ότι η μικρότερη απόσταση μεταξύ δύο σημείων στη Μερκατορική προβολή μπορεί να είναι μια καμπύλη γραμμή (ορθοδρόμος) και όχι μια ευθεία (λοξοδρόμος) που ενώνει τα δύο αυτά σημεία. Αυτό συμβαίνει λόγω του τρόπου κατασκευής του συγκεκριμένου χάρτη και λόγω των στρεβλώσεων που δημιουργούνται κατά την προσπάθεια αποτύπωσης της γήινης σφαίρας σε μια επίπεδη επιφάνεια. Η Μερκατορική προβολή επιλέχθηκε καθώς χρησιμοποιείται σε παγκόσμιο επίπεδο κι έχει επικρατήσει στους ψηφιακούς χάρτες (π.χ.Google Maps). Επιπλέον, στη συγκεκριμένη προβολή παρουσιάζονται σημαντικές αλλοιώσεις ως προς τις αποστάσεις πλησιάζοντας τους δύο Πόλους της Γης έχοντας ως επακόλουθο τη δημιουργία παρανοήσεων στα παιδιά του Δημοτικού. Τα αποτελέσματα της έρευνας έδειξαν ότι η διδασκαλία των χαρτογραφικών προβολών και των παραμορφώσεων της επιφάνειας της Γης μέσα από τους χάρτες μπορούν και πρέπει να ενταχθούν στο Αναλυτικό Πρόγραμμα της Γεωγραφίας της Πρωτοβάθμιας Εκπαίδευσης. Οι μαθητές/τριες κατά τη διάρκεια της σχολικής τους ζωής συναντούν συνήθως χάρτες Μερκατορικής προβολής και θεωρούν λανθασμένα πως οι αποστάσεις, τα σχήματα και τα μεγέθη των ηπείρων αποτυπώνονται σωστά. Η άποψή τους αυτή τροποποιήθηκε μέσα από βιωματικές δραστηριότητες που πραγματοποιήθηκαν στην τάξη. Στο τέλος οι μαθητές/τριες κατέληξαν να ελέγχουν την ορθότητα των αποστάσεων που μετρούνται στους χάρτες. Τα συμπεράσματα στα οποία κατέληξε η παρούσα έρευνα είναι ότι το μάθημα της Γεωγραφίας πρέπει να διδάσκεται με τη συνοδεία ψηφιακών εργαλείων (GIS) κι εναλλακτικών δραστηριοτήτων. Τα παιδιά, μέσω καθοδηγούμενης διερεύνησης μπορούν να κατανοήσουν πως η μετατροπή της γήινης σφαίρας σε οποιαδήποτε προβολή, είναι δυνατό να προκαλέσει σφάλματα. Είναι σημαντικό οι χάρτες να ενσωματωθούν στο σχολικό πρόγραμμα με τρόπο ώστε οι μαθητές/τριες να μπορούν να διακρίνουν τα σφάλματα και τις παραμορφώσεις που υπάρχουν σε αυτούς., This research is based on a pilot project that took place within the Master’s Program “Education Sciences: Teaching Methods and Learning Processes” with a focus on “Digital Applications in Science, Technology, Engineering and Mathematics Education (STEM)”of the Faculty of Primary Education, AUTH. The research was addressed to fourteen students of the sixth grade of the Elementary School at Nea Apollonia, Thessaloniki. The subject is Geography with an emphasis in cartography. Through guided investigation and hands-on activities, students learn about the distance distortions among countries in small scale maps and experiment in digital world maps based on the Mercator projection. The main aim of this study was to highlight and eliminate the misconceptions the students had concerning distances that are measured in digital maps. In addition, another aim was to showcase the importance of the subject of Geography that is constantly marginalized and is often taught through traditional methods. The duration of the didactic intervention was four weeks and included seven activities. The research tools that were used for the data collection were a) pre test & post test (open questionnaires), b) worksheets, c) recording. The tools that were utilized in the classroom were a) projector, b) computers, c) online software with digital maps and globes, d) printed world maps with different projections, e) ruler, f) plasticine, g) inflatable and deflatable globes (with large dimensions). The students learnt about map projections and found out that the shortest distance within two points in Mercator projection might be a curved line (great circle) rather than a straight line (rhumb line). This is caused due to the way this map is constructed and because of the distortions that are caused through the attempt of transforming the spherical shape of the earth into a flat surface. The Mercator projection has been chosen due to the fact that it is globally used and has prevailed in digital maps (e.g. Google Maps). Furthermore, in this projection distances are distorted when approaching the North and South Pole, causing misconceptions to the children of the Elementary School. The results of the study showed that teaching about map projections and the transformations of the shape of the earth through maps can and must be integrated in the curriculum of Geography of Primary Education. The students during the entirety of their school life come across Mercator projection maps and wrongly believe that the distances, the shapes and the sizes of the continents are presented properly. This belief was altered through ‘hands-on’ activities that conducted into the classroom. By the end, students were able to check the correctness of the distances that are measured in maps. In conclusion, data analysis showed that the subject of Geography should be taught with the presence of digital tools (GIS) and alternative activities. Through guided investigation children can understand that the alteration of the earth in any map projection would prone to errors. The maps must be embedded in the curriculum such a way that students can distinguish the errors and the distortions.

We develop an energetic model that captures the twisting behavior of spindle-shaped polymer microparticles with nematic ordering, which display remarkably different twisting behavior to ordinary nematics confined to spindles. We have previously developed a geometric model of the twisting, based on experimental observations, in which we showed that the twist pattern follows loxodrome spirals [Ansell et. al., Phys. Rev. Lett., 123, 157801 (2019)]. In this study, we first consider a spindle-shaped surface and show that the loxodrome twisting behavior of our system can be captured by the Frank free energy of the nematic with an additional term constraining the length of the integral curves of the system. We then extend the ideas of this model to the bulk and explore the parameter space for which the twisted loxodrome solution is energetically favorable., 10 pages, 7 figures

The article proposes the implementation of the algorithm for calculating the course along the rhumb line in MATLAB, as well as the implementation of new empirical formulas for constructing the ship's turning circle in MATLAB. This elaboration will help in creating software for control devices in terms of programming the calculation of the parameters of the ship's motion. The elaboration was based on the algorithm proposed by the author for calculating the course (bearing) of the rhumb line, as well as the empirical formulas proposed by the author for constructing the trajectory of the vessel's turning circle. The article also focuses on the patterns of calculating the course (bearing) on the rhumb line and constructing the turning circle of the vessel, which were not mentioned earlier

Despite the wealth of studies on seasonal movements of birds between southern nonbreeding locations and High Arctic breeding locations, the key mechanisms of navigation during these migrations remain elusive. A flight along the shortest possible route between pairs of points on a sphere (‘orthodrome’) requires a bird to be able to assess its current location in relation to its migration goal and to make continuous adjustment of heading to reach that goal. Alternatively, birds may navigate along a vector with a fixed orientation (‘loxodrome’) based on magnetic and/or celestial compass mechanisms. Compass navigation is considered especially challenging for summer migrations in Polar regions, as continuous daylight and complexity in the geomagnetic field may complicate the use of both celestial and magnetic compasses here. We examine the possible use of orientation mechanisms during migratory flights across the Greenland Icecap. Using a novel 2 g solar‐powered satellite transmitter, we documented the flight paths travelled by a female red knot Calidris canutus islandica during two northward and two southward migrations. The geometry of the paths suggests that red knots can migrate across the Greenland Icecap along the shortest‐, orthodrome‐like, path instead of the previously suggested loxodrome path. This particular bird's ability to return to locations visited in a previous year, together with its sudden course changes (which would be appropriate responses to ambient wind fields), suggest a map sense that enables red knots to determine location, so that they can tailor their route depending on local conditions.

A design method is proposed for a ball end mill with constant inclination of the cutting edge to the spherical section. The tooth profile specified in the normal cross section is the generatrix, while the curved cutting edge at the sphere in the form of a loxodrome is the directrix.

This paper describes George Biddell Airy's almost completely unknown method of approximating an orthodromic arc (great circle arc) using a circular arc in the normal aspect Mercator projection of a sphere. In addition, it is demonstrated that the centre of the circle can be defined in at least two different ways, which yields slightly different results. Airy's approach is built upon in this paper. The method of computing coordinates of Airy's circle arc centre is described. The formulae derived in the paper can be used to calculate the length of Airy's approximation of the orthodromic arc connecting two points on the sphere and on the Mercator chart. Moreover, the actual length of the orthodromic arc on the sphere and on the Mercator chart can be computed using the formulae derived in this paper. The purpose of the paper is not to suggest an application of Airy's method in navigation, but to analyse Airy's proposal and to show that a great circle arc on a Mercator chart is close to a circular arc for distances which are not too great. This property can be useful in education, having in mind that the stereographic projection is the only one that maps any circle on a sphere onto a circle in the projection plane.

This work tackles the problem of misconception when using sophisticated mathematical tools, nonlinear optimization in this particular case, to solve a navigational problem. Namely, to reach the Great Circle vertex with two rhumb line legs ensuing the optimized distance, an initial rhumb line course equal to the orthodromic course at middle latitude may be used. The initial course is thereupon optimized by the incremental value steps. The optimized distance is achieved if the rhumb line course is altered towards the vertex at the orthodrome-loxodrome intersection point. As determination of this point cannot be formulated in a closed form, an iterative solution is to be applied. The derived transcendental equation forms a basis for an iterative solution of intersection using the Newton-Raphson method. To the contrary, finding solutions to a system of nonlinear equations can mislead a researcher unable to comprehend and grasp the mathematical meanings of the algorithm. The gist of this essay is a novel concept showing an intrinsic property i.e. orthodrome-loxodrome correlation using a well-known formula.

The Great Elliptic Sailing (GES), which can reduce sailing distance, is important to navigators. Whether a Great Ellipse (GE) is worth using depends on whether the distance saved is significant. Otherwise, the Rhumb Line (RL) is easier to steer. We propose a simple criterion to evaluate the difference in distance between the GE and the RL. The criterion is that the GE is worth using when the vertex lies between the departure and destination. In order to take the advantage of shorter distance, the GE is usually approximated as a series of waypoints. Unlike currently practised methods, we propose the Longitude Bisection Method (LBM) which determines waypoints with varying intervals. This approach can establish the appropriate number of waypoints to approximate the GE effectively. The proposed criterion and the LBM are demonstrated in practical examples.

Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references.

In the present paper, we investigate the differential equations of the space-like loxodromes on the helicoidal surfaces having space-like meridians and time-like meridians, respectively in Minkowski 3-space. Also we illustrate our main results by using Mathematica.

In this paper, we prove some results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, a part of which generalizes classical results about loxodromes on rotational surfaces in $${{\mathbb {R}}}^3$$. In particular, we show how to parametrize a loxodrome on an invariant surface of $${\mathbb {H}}^2\times {{\mathbb {R}}}$$ and $${\mathbb {H}}_3$$, and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.

We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected geometrical examples illustrate the usage of parametrisation. Our work extends the set of objects in Lie sphere geometry—circle, lines and points—to the natural maximal conformally-invariant family, which also includes loxodromes.

The contribution is part of a research that aims to address the problems of knowledge, interpretation and documentation of coffered domes geometry. The main question is to define the relationships between the coffer shape, the layout used for coffers distribution on dome surface and different kind of surfaces. With regard to coffered domes we have analyzed the methods illustrated by Francesco Milizia, Giuseppe Vannini and some historical surveys. We have grouped coffered domes in relation to grid geometry and to coffer shape. We have defined three different ways to distribute the coffers in relation to different grid layout: grid composed by 2D lines (meridians and parallels), grid composed by 3D lines on surface (lattice of rhumb lines) or coffers distribution between ribs. We have analyzed each of them and we have defined algorithmic models in relation to spherical domes. The main goal of our research is to study what's different in not spherical domes, such as policentrical domes, ellipsoidal domes or ovoidal domes, generated using curves network. We have compared computational models based on treatises rules with particular case studies. This comparison allows us to do a critical analysis based on geometric rules. From a methodological point of view we have built a parametric model able to connect the different processes, using the same parameters. By comparing this model with point clouds, it is possible to evaluate analogies or identify new rules that will be used to develop a more complex parametric model based on surveys.

The Mercator map or projection is one of the map projections treated in Social Studies of junior high and high school courses. The most important feature is that the projection is conformal meaning that it preserves angles. In this article, we first explain this conformal property elementarily but rigorously in order to propose that the topic is suitable for undergraduates as a STEM topic. Also, we give a parametrization of the plane curve obtained by projecting a great circle on the earth onto the Mercator map. Finally, we examine a rhumb line and calculate the distance along it., 本研究は，JSPS科研費JP17H00820の助成を受けたものである。

Ortodroma in loksodroma sta krivulji na Zemlji, katere obliko obravnavamo kot kroglo. Ortodroma je del velikega kroga in predstavlja najkrajšo razdaljo med točkama. Loksodroma je od ortodrome daljša, vendar meridiane seka pod enakim kotom, zato pri gibanju ni potrebno spreminjati smeri gibajočega se objekta. V diplomski nalogi predstavljamo značilnosti krivulj in postopke numeričnega reševanja situacij na posameznih krivuljah. Kot rezultat naloge predstavljamo program v okolju MATLAB, ki ob vnosu geografskih koordinat dveh poljubnih krajev na Zemlji izračuna koordinate krajev na ortodromi in loksodromi ter omogoča vizualizacijo rešitev na karti Mercatorjeve projekcije in v Google Earth-u. Great circle and rhumb line are the curves on Earth, whose shape is treated as a sphere. Great circle is a part of a large circle and represents the shortest distance between the points. Rhumb line is longer than the great circle, but it crosses every meridian under the same angle, so meanwhile moving, direction change of the moving object is not needed. In the graduation thesis we present the characteristics of the curves and procedures of numerical solving for the situations on individual curves. As a result of the task, we present a program in the MATLAB environment, which, when entering the geographical coordinates of any two places on Earth, calculates the coordinates of places on the great circle and rhumb line, and allows the visualization of results on the Mercator projection map and in Google Earth.

A loxodrome is a curve on a parametrized surface that intersects one family of parametric lines at a constant angle. In this paper, we investigate loxodromes on rotational surfaces in the 3-dimensional simply isotropic space \({I^3}\) and give an example of loxodromes. Also, we completely classify geodesics on rotational surfaces in \({I^3}\).

This unit presents an overview of projection issues that are likely to be encountered by a GIS user. It discusses common map projections and associated parameters, how to determine appropriate projections for certain GIS applications and issues specific to raster data including the choice of grid cell size and resampling methods.

We investigate the parametrizations of loxodromes on the time-like rotational surfaces by using suitable Lorentzian angles in Minkowski space-time. Also, some examples and corresponding graphs are given by using Mathematica.

Helixes on Clifford surfaces in a hyperbolic space H^3 of positive curvature are investigated. We formulate the following facts for a Clifford surface of every type in the space H^3. A curve on a Clifford surface of the space H^3 is a helix if and only if it is a loxodrome. If a helix ξ on a Clifford surface of the space H^3 contains opposite vertices of the coordinate rectangle of the basic coordinate network, then the helix ξ divides this coordinate rectangle into two parts of equal areas. The absolute points of the hyperbolic axis of a Clifford surface in the space H^3 are poles of each loxodrome on this surface.

In nature, science and engineering, we often come across helicoidal surfaces. A curve on a helicoidal surface in Euclidean 3-space is called a loxodrome if the curve intersects all meridians at a constant azimuth angle. Thus loxodromes are important in navigation. In this paper, we find the differential equation of the loxodrome on a helicoidal surface in Euclidean 3-space. Also we give some examples and draw the corresponding pictures via the Mathematica computer program to aid understanding of the mathematics of navigation.

Normal circular-arc spiral bevel gear is a new type gear based on Bertrand conjugation principle. Normal circular-arc bevel gear selects directrix on the basis of paying attention to machining planning process. When the directrix serves as loxodrome, the formed tooth profile surface not only has fine transmissibility like general arc bevel gear, but also is convenient to machining. Starting with differential geometry, the geometry nature and inner relationship of Bertrand conjugation surface are studied. Then the normal circular-arc surface id put forward which is a typical Bertrand surface; the conjugation theory of loxodrome normal circular-arc bevel gears is studied. The characters of loxodrome line are analyzed. The tooth profile of loxodrome normal circular-arc bevel gears, the primary condition the directrix line must be satisfied, the relative curvature of the conjugating surfaces, the relations between non-interventional condition and curvature axle of the directrix line is provided; the NC machining theory of normal circular-arc bevel gears is studied, including the designing and manufacture of the tool, the movement of the machine, the adjustment of the tool under the condition of tangency and non-interventional condition and so on.

With the aim of extracting environmental data around routes, as the basis of ship routing optimization and other related studies, this paper, taking wind grid data as an example, proposes an algorithm that can effectively extract the grid data around rhumb lines. According to different ship courses, the algorithm calculates the wind grid index values in eight different situations, and a common computational formula is summarised. The wind grids around a ship route can be classified into ‘best-fitting’ grids and ‘additional’ grids, which are stored in such a way that, for example, when the data has a high-spacing resolution, only the ‘best-fitting’ grids around ship routes are extracted. Finally, the algorithm was implemented and simulated with MATLAB programming. As the simulation results indicate, the algorithm designed in this paper achieved wind grid data extraction in different situations and further resolved the extraction problem of meteorological and hydrogeological field grids around ship routes efficiently. Thus, it can provide a great support for optimal ship routing related to meteorological factors.

In this paper we show that, for historical reasons, the most likely way for Mercator to have drawn his map was by using a table of rhumbs. This is argued by examining in some detail the development of the concept of a rhumb line from its initial articulation by Pedro Nunes in 1537 to its advanced treatment in 1566. We also describe the intellectual debate that took place around this topic in Europe during that period. A hitherto neglected Portuguese source of about 1540, where tables of rhumbs are discussed for the first time, has been found to be an important link in the chain connecting the initial idea of a rhumb line to Mercator’s achievement of 1569.

We find all space-like loxodromes on rotational surfaces which have space-like meridians or time-like meridians, respectively by using a relevant Lorentzian angle in Minkowski 3-space. To understand loxodromes better, we draw some pictures of them via Mathematica computer program. (c) 2013 Elsevier Inc. All rights reserved.

Some information geometric structures concerning the Mercator projections are studied. It is known that a loxodrome on the surface of the globe is related to the straight line on a Mercator map by the Mercator projection. It is not well known that an affine connection with torsion plays a fundamental role to describe an auto-parallel path on the surface. Based on these information geometric structures, Gauss distribution is reconsidered from the view point of the affine connection with a torsion. Some relations with deformed functions are also pointed out., 17pages, 3 figures, resubmitted to Physica A

In this study we assess the accuracy of the cartographic projection introduced by Mercator in his world map of 1569 aiming to identify its construction method. An error distribution of the latitude scale in the Organum directorium abacus was first determined in such a way as to be independent of the physical distortions that affected the sheets after printing. This distribution, which we have called the ‘error signature’ of the map, was then compared with the errors associated with various methods that Mercator might have used to calculate the projection. A new construction method not hitherto considered in the literature, based on the use of a table of rhumbs, is proposed and tested. The method is intuitive, simple to apply and historically plausible, given the fact that tables of rhumbs were already known in Europe at the time Mercator was engraving his world map. A particular type of table, based on the iterative resolution of a set of right-angle triangles placed along a rhumb line on the sphe...

In this note the Middle Latitude Rule is derived. Namely, to reach Great Circle vertex in two steps such that the total distance is a minimum, an initial rhumb line course equal to the orthodromic course at Middle latitude is to be used. The shortest distance is achieved if the rhumb line course is altered towards the vertex at the orthodrome-loxodrome intersection point.

Proposals of algorithms for solving Navigation faculty’s problems regarding navigation along rhumb line (loxodrome), for officer trainees and civil students of Faculty of Navigation and Naval Weapons and for deck seamen on operational level, are presented in this paper. Proposals of calculation algorithms given in the article cover simple mathematical totting based on plane sailing (navigational) triangle and Mercator triangle (Mercator sailing). The algorithms refer to traditional methods of solving problems and they do not cover solutions applicable in automated (integrated) navigational systems. The formulas enable to solve the tasks with a use of electronic calculator and application of traditional methods on plane and sphere, and they also take the Earth’s ellipsoidal shape into consideration.

In order to solve problems that deal with rhumb lines solution based on different reference ellipsoids in maritime navigation, the rhumb lines basic equations were established. After the symbolical formula of meridian arcs inverse solution being introduced, rhumb lines forward and inverse expressions which are non-iterative were derived. Finally, numerical examples were given to testify the accuracy of the forward and inverse expansions derived in this paper. Theory analysis and numerical results show that these new formulas have compact and symbolical form, and the precision of the inverse formulas is up to , which can satisfy practical utilization.

Navigation at sea is based on Electronic Chart Display and Information Systems (ECDIS) that allow for the use of a limited number of projections. As navigation in the Arctic region becomes a reality due to the progressive melting of the polar ice cap, a re-examination of the most suitable projections for navigation in the Arctic becomes timely. Several projections are proposed in the literature for this area. In our approach, the selection is based on an analytical study utilizing three criteria: the control of the magnitude of distortions within acceptable limits, the shape of great circles (GCs) and rhumb lines/loxodromes, and the shape of the graticule lines portrayed on the chart. The analysis carried out shows that to fulfill the set criteria, the arctic area should be divided into Arctic and sub-Arctic regions. More specifically, for the Arctic region the Azimuthal Polar Equidistant projection and the Azimuthal Polar Stereographic projection are the most suitable ones. For the sub-Arctic region, the...

If you sail in a constant, oblique direction with respect to north, you follow a so-called rhumb line. You would be excused to think that you were following a straight line, which would bring you to your target in the shortest possible time. But then you would have forgotten that the Earth is round.

There have been a number of slew algorithms in literature such as half-cone and rhumb line that allow attitude control of a prolate spinning spacecraft using only a single attitude-control thruster. Recently, several novel slew algorithms have been further introduced to the field: extended half-cone, dual cone, and spin sync, which offer a number of advantages to the existing algorithms. The ultimate output of this paper is a slew algorithm selection tradeoff, which is performed in the context of a case study of theMoonLightweight Interior andTelecomExperiment interplanetary penetrator mission. To support this final quantitative tradeoff, the following analyses are presented beforehand: 1) a summary of the state-of-the-art slew algorithms, 2) a thorough generic parametric analysis of relevant characteristic performances (slew accuracy, slew duration, energy consumption, and torque capacity) as influenced by relevant mission variables, and 3) a comprehensive slew algorithm selection tradeoff methodology. As the parametric analyses and tradeoffmethodology are generic in nature, this work could be reused to support similar tradeoffs for other mission scenarios; this represents a major added value of this study. The numerical results presented are validated using a newly developed software simulator called the Attitude Control Simulator for Spinning Spacecraft with Single Thruster.

The attitude dynamics and manoeuvre survey in this paper is performed for a mission scenario involving a penetrator-type spacecraft: an axisymmetric prolate spacecraft spinning around its minor axis of inertia performing a 90 spin axis reorientation manoeuvre. In contrast to most existing spacecraft only one attitude control thruster is available, providing a control torque perpendicular to the spin axis. Having only one attitude thruster on a spinning spacecraft could be preferred for spacecraft simplicity (lower mass, lower power consumption etc), or it could be imposed in the context of redundancy/ contingency operations. This constraint does yield restrictions on the thruster timings, depending on the ratio of minor to major moments of inertia among other parameters. The Japanese Lunar-A penetrator spacecraft proposal is a good example of such a single-thruster spin-stabilised prolate spacecraft. The attitude dynamics of a spinning rigid body are first investigated analytically, then expanded for the specific case of a prolate and axisymmetric rigid body and finally a cursory exploration of non-rigid body dynamics is made. Next two well-known techniques for manoeuvring a spin-stabilised spacecraft, the Half-cone/Multiple Half- cone and the Rhumb line slew, are compared with two new techniques, the Sector-Arc Slew developed by Astrium Satellites and the Dual-cone developed at Surrey Space Centre. Each technique is introduced and characterised by means of simulation results and illustrations based on the penetrator mission scenario and a brief robustness analysis is performed against errors in moments of inertia and spin rate. Also, the relative benefits of each slew algorithm are discussed in terms of slew accuracy, energy (propellant) efficiency and time efficiency. For example, a sequence of half-cone manoeuvres (a Multi-half-cone manoeuvre) tends to be more energy-efficient than one half-cone for the same final slew angle, but more time-consuming. As another example, the new Sector-Arc Slew and Dual-cone techniques are designed to overcome a specific restriction on attainable slew angle that is associated with the half-cone manoeuvre, giving one additional degree of freedom for designers to fine-tune. © 2012 Elsevier Ltd.