Search Results (7)

If we take a distance from a map without considering the distortion, we will not get the correct value. We will get the correct value if we take the distortion into account, that is, if we use an expression for distance that does not contain distortion. The article shows how to determine the distance measured along a loxodrome between two points on a sphere if we have an image of the loxodrome in a cylindrical projection. Using the examples of Mercator, equidistant and equal-area projections, a formula was derived that does not depend on the map projection used. This was done with the aim of achieving two goals. The first was that when calculating the distance between two places shown on a map, the distortion inherent in the map due to the applied map projection should be considered. The second goal was to show how, using the theory of map projections, a formula can be derived that does not depend on the applied map projection. Full article

This paper recasts four geodetic curves—the great ellipse, the normal section, the loxodrome, and the curve of alignment—into a parametric form of vector-algebra formula. These formulas allow these curves to be drawn using simple, efficient, and robust algorithms. The curve of alignment, which seems to be quite obscure, ought not to be. Like the great ellipse and the loxodrome, and unlike the normal section, the curve of alignment from point A to point B (both on the same ellipsoid) is the same as the curve of alignment from point B to point A. The algorithm used to draw the curve of alignment is much simpler than any of the others and its shape is quite similar to that of the geodesic, which suggests it would be a practical surrogate when drawing these curves. Full article

In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants $A,B$ and Gauss’s constant G, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results. Full article

This paper investigates, as a first step, the four branches of BMS transformations, motivated by the classification into elliptic, parabolic, hyperbolic and loxodromic proposed a few years ago in the literature. We first prove that to each normal elliptic transformation of the complex variable $\zeta$ used in the metric for cuts of null infinity, there is a corresponding BMS supertranslation. We then study the conformal factor in the BMS transformation of the u variable as a function of the squared modulus of $\zeta$. In the loxodromic and hyperbolic cases, this conformal factor is either monotonically increasing or monotonically decreasing as a function of the real variable given by the modulus of $\zeta$. The Killing vector field of the Bondi metric is also studied in correspondence with the four admissible families of BMS transformations. Eventually, all BMS transformations are re-expressed in the homogeneous coordinates suggested by projective geometry. It is then found that BMS transformations are the restriction to a pair of unit circles of a more general set of transformations. Within this broader framework, the geometry of such transformations is studied by means of its Segre manifold. Full article

Route planning procedures for ocean-going vessels depend significantly on prevailing weather conditions, the ship’s design characteristics and the current operational state of the vessel. The operational status considers hull and propeller fouling, which significantly affects fuel oil consumption coupled with route selection. The current paper examines the effect of the fouling level on the selection of the optimized route compared with the clean hull/propeller as well as the orthodrome/loxodrome route. A developed weather routing tool is utilized, which is based on a physics-based model for the calculation of the main engine’s fuel oil consumption enriched to account for different fouling levels of the hull and the propeller. A genetic algorithm is employed to solve the optimization problem. A case regarding a containership in trans-Atlantic transit using forecasted weather data is presented. The effect of ocean currents is also examined as it was derived that they greatly affect route selection, revealing a strong dependence on the level of fouling. Ignoring the fouling impact can result in miscalculations regarding the estimated fuel oil consumption for a transit. Similarly, when ocean currents are ignored in the route planning process, the resulting optimal paths do not ensure energy saving. Full article

Whale migrations are poorly understood. Two competing hypotheses dominate the literature: 1. moving between feeding and breeding grounds increases population fitness, 2. migration is driven by dynamic environmental gradients, without consideration of fitness. Other hypotheses invoke communication and learned behaviors. In this article, their migration was investigated with a minimal individual-based model at the scale of the Global Ocean. Our aim is to test if global migration patterns can emerge from only the local, individual perception of environmental change. The humpback whale (Megaptera novaeangliae) meta-population is used as a case study. This species reproduces in 14 zones spread across tropical latitudes. From these breeding areas, humpback whales are observed to move to higher latitudes seasonally, where they feed, storing energy in their blubber, before returning to lower latitudes. For the model, we developed a simplified ethogram that conditions the individual activity. Then trajectories of 420 whales (30 per DPS) were simulated in two oceanic configurations. The first is a homogeneous ocean basin without landmasses and a constant depth of −1000 m. The second configuration used the actual Earth topography and coastlines. Results show that a global migration pattern can emerge from the movements of a set of individuals which perceive their environment only locally and without a pre-determined destination. This emerging property is the conjunction of individual behaviors and the bathymetric configuration of the Earth’s oceanic basins. Topographic constraints also maintain a limited connectivity between the 14 DPSs. An important consequence of invoking a local perception of environmental change is that the predicted routes are loxodromic and not orthodromic. In an ocean without landmasses, ecophysiological processes tended to over-estimate individual weights. With the actual ocean configuration, the excess weight gain was mitigated and also produced increased heterogeneity among the individuals. Developing a model of individual whale dynamics has also highlighted where the understanding of whales’ individual behaviors and population dynamic processes is incomplete. Our new simulation framework is a step toward being able to anticipate migration events and trajectories to minimize negative interactions and could facilitate improved data collection on these movements. Full article

In this paper, we present a new reference model that approximates the actual shape of the Earth, based on the concept of the deformed spheres with the deformation parameter $\lambda$. These surfaces, which are called $\lambda$-spheres, were introduced in another setting by Faridi and Schucking as an alternative to the spheroids (i.e., ellipsoids of revolution). Using their explicit parametrizations that we have derived in our previous papers, here we have defined the corresponding isothermal (conformal) coordinates as well as obtained and solved the differential equation describing the loxodromes (or rhumb lines) on such surfaces. Next, the direct and inverse problems for loxodromes have been formulated and the explicit solutions for azimuths and arc lengths have been presented. Using these explicit solutions, we have assessed the value of the deformation parameter $\lambda$ for our reference model on the basis of the values for the semi-major axis of the Earth a and the quarter-meridian $m_p$ (i.e., the distance between the Equator and the North or South Pole) for the current best ellipsoidal reference model for the geoid, i.e., WGS 84 (World Geodetic System 1984). The latter is designed for use as the reference system for the GPS (Global Positioning System). Finally, we have compared the results obtained with the use of the newly proposed reference model for the geoid with the corresponding results for the ellipsoidal (WGS 84) and spherical reference models used in the literature. Full article