Convergence Of Meridians Research Articles

Copying-lineages of portolan chart metrics and implications of their pre-medieval origin

ABSTRACT Seven portolan charts and atlases (including the anonymous Rex Tholomeus chart) made between 1311 and 1538 were cartometrically analysed. The findings indicate that their geometry was not improved chronologically and that the accuracy of Pietro Vesconte's charts (1311–13) was never surpassed. Portolan charts and atlases appear to be mosaics composed of subsections, each with variable metrics and accuracy levels that are, on average, twice as great in comparison to their Mediterranean and Black Sea areas (as a whole), and it is highly unlikely that they are genuine late medieval products, which is also supported by historical evidence. Their anticlockwise tilts fit well with the tilt of the Gibraltar-Antioch line on modern maps that closely approximate Ptolemy I and Ptolemy II projections centred near Gibraltar and with their meridian tilts for areas around Genoa. If late medieval Italian cartographers had acquired a collection of maps or charts from before the Middle Ages, it is theoretically feasible that they could have achieved the typical geometry of portolan charts by graphically combining their manuscript copies of large-scale regional maps in Mercator or Mercator-like projection, using a small-scale map in conic or pseudoconic projection (on which the convergence of meridians is displayed) as a template.

Analysis of Polar Region Navigation Algorithm of Laser Gyro Single-axis Rotation Modulation Inertial Navigation

In the polar regions, the local geographic frame will become invalid due to the convergence of meridians and other reasons. In response to this problem, the inertial navigation algorithm was reorganized in the transversal frame, and the error equation was also analyzed. The simulation experiment and actual experiment are carried out to verify the effectiveness of the inertial navigation algorithm in the transversal frame based on the single-axis rotation modulation inertial navigation system. Experimental results show that the transversal INS mechanization based on single-axis rotation modulation can solve the computational problems caused by the convergence of meridians in polar regions, achieving high-precision navigation.

A Barotropic Tide Model for Global Ocean Based on Rotated Spherical Longitude-Latitude Grids

Ocean modeling and simulation are important for understanding the dynamic processes in the geophysical system, and the simulation of tidal dynamics is of great significance for understanding the dynamic evolution of the ocean. However, there are some problems in existing simulations, including lack of specific standards to produce a desirable discrete spherical mesh for global ocean modelling. Many global ocean numerical models based on conventional longitude-latitude (LL) coordinates suffer from the “pole problem” in regions adjacent to the North Pole due to the convergence of meridians, which seriously hinders global ocean simulations. In this paper, a new longitude-latitude spherical grid coupled with rotated coordinate mapping is proposed to overcome the problem. In the design of the numerical model, for spatial approximation, the finite volume method on staggered C grid is proposed to solve the two-dimensional tidal wave equations for the global ocean. For temporal integration, the third-order Adams-Bashforth method is used to explicitly extrapolate the value on the next time interval half layer, and then the fourth-order implicit Adams-Moulton method is used to correct the water level. Finally, the constructed model is used to simulate the dynamics of two-dimensional tidal waves in the global ocean, and the co-tidal maps of two major diurnal tide and semidiurnal tide components are shown. The results demonstrate that the proposed model can support the simulation of tidal dynamics in the global ocean, especially for the Arctic Ocean.

A global geographic grid system for visualizing bathymetry

Abstract. A global geographic grid system (Global GGS) is here introduced to support the display of gridded bathymetric data at whatever resolution is available in a visually seamless manner. The Global GGS combines a quadtree metagrid hierarchy with a system of compatible data grids. Metagrid nodes define the boundaries of data grids. Data grids are regular grids of depth values, coarse grids are used to represent sparse data and finer grids are used to represent high-resolution data. Both metagrids and data grids are defined in geographic coordinates to allow broad compatibility with the widest range of geospatial software packages. An important goal of the Global GGS is to support the meshing of adjacent tiles with different resolutions so as to create a seamless surface. This is accomplished by ensuring that abutting data grids either match exactly with respect to their grid-cell size or only differ by powers of 2. The oversampling of geographic data grids, which occurs towards the poles due to the convergence of meridians, is addressed by reducing the number of columns (longitude sampling) by powers of 2 at appropriate lines of latitude. In addition to the specification of the Global GGS, this paper describes a proof-of-concept implementation and some possible variants.

Research on Polar-region Coarse Alignment Algorithm Based on Grid System

Due to the convergence of meridians in polar regions, the geography-based northern-azimuth mechanics arrangement has problems such as increase of error and computational overflow in polar navigation, and the traditional compass alignment scheme is unable to operate normally. In this paper, the problem in polar navigation is solved by using the mechanics arrangement scheme, and a simulated analysis has been carried out on the application of compass alignment in high latitudes through static data, which verifies that it is unable to operate normally at the pole. Then compass alignment has been replaced by anti-sloshing coarse alignment and a simulated verification has been conducted on its correctness. After that, the latitude increment method is used to convert the sports car data at low-latitude regions to high-latitude regions and carry out a semi-physical simulation, which further verifies the correctness of the grid inertial navigation and the coarse alignment method in anti-sloshing polar region.

Rapid alignment for SINS based on expansion of observation vectors in polar regions

In polar regions, the rapid convergence of meridians and the small horizontal component of Earth's rotation make traditional North-pointed initial Alignment algorithm (NA) ineffective. While Alignment algorithm based on Pseudo-Earth frame (PA) can solve the problems caused by the rapid convergence of meridians, the observability of Strap-down Inertial Navigation System (SINS) on stationary base is so poor that it takes much time to complete initial alignment. According to this case, in this paper, a novel alignment algorithm based on expansion of observation vectors is proposed to implement rapid initial alignment in polar regions. This paper mainly completes the following works: firstly, a rapid Alignment algorithm based on Expansion of Observation vectors for Pseudo-earth frame (EOPA) is proposed, then the reason for its rapid convergence is explicated by analysis of the error equations; secondly, two observability analysis methods: Singular Value Decomposition (SVD) and the Attitude Dilution Of Precision (ADOP) are adopted to compare the observability of PA with that of EOPA. Finally, semi-physical simulation shows that in polar regions, even in the case of latitude of 89° and initial misalignment angle of 3°, the azimuth misalignment error can be reduced within 0.25° in 40s. It has significant engineering application value.

Математические модели проекции Гаусса – Крюгера для вычисления сближения меридианов на плоскости и масштаба изображения

This work contains continued research carried out on improving mathematical models of the Gauss-Krueger projection in accordance with the parameters of any ellipsoid with the removal of points from the axial meridian to l ≤ 6° . In terms of formulae earlier derived by the authors with improved convergence for the calculation of planar rectangular coordinates by geodesic coordinates, the algorithms for determining the convergence of meridians on the plane and the scale of the image are obtained. The improvement of the formulae represented in the form of series in powers of the difference in longitudes was accomplished by separating spherical terms in series and then replacing their approximate sums by exact expressions using the formulae of spherical trigonometry. As in previous works published in this journal [7, 8], determining the sums of the spherical terms was carried out according to the laws of the transverse-cylindrical projection of the sphere on the plane. Theoretical studies are given and formulae are proposed for estimating the observational errors in the results of the derived algorithms. The maximum of observational errors of convergence of meridians and scale, proceeding from the specified accuracy of the determined quantities was established through analytical methods.

Initial Alignment for SINS Based on Pseudo-Earth Frame in Polar Regions.

An accurate initial alignment must be required for inertial navigation system (INS). The performance of initial alignment directly affects the following navigation accuracy. However, the rapid convergence of meridians and the small horizontal component of rotation of Earth make the traditional alignment methods ineffective in polar regions. In this paper, from the perspective of global inertial navigation, a novel alignment algorithm based on pseudo-Earth frame and backward process is proposed to implement the initial alignment in polar regions. Considering that an accurate coarse alignment of azimuth is difficult to obtain in polar regions, the dynamic error modeling with large azimuth misalignment angle is designed. At the end of alignment phase, the strapdown attitude matrix relative to local geographic frame is obtained without influence of position errors and cumbersome computation. As a result, it would be more convenient to access the following polar navigation system. Then, it is also expected to unify the polar alignment algorithm as much as possible, thereby further unifying the form of external reference information. Finally, semi-physical static simulation and in-motion tests with large azimuth misalignment angle assisted by unscented Kalman filter (UKF) validate the effectiveness of the proposed method.

Computational Modes and Grid Imprinting on Five Quasi-Uniform Spherical C Grids

Abstract Currently, most operational forecasting models use latitude–longitude grids, whose convergence of meridians toward the poles limits parallel scaling. Quasi-uniform grids might avoid this limitation. Thuburn et al. and Ringler et al. have developed a method for arbitrarily structured, orthogonal C grids called TRiSK, which has many of the desirable properties of the C grid on latitude–longitude grids but which works on a variety of quasi-uniform grids. Here, five quasi-uniform, orthogonal grids of the sphere are investigated using TRiSK to solve the shallow-water equations. Some of the advantages and disadvantages of the hexagonal and triangular icosahedra, a “Voronoi-ized” cubed sphere, a Voronoi-ized skipped latitude–longitude grid, and a grid of kites in comparison to a full latitude–longitude grid are demonstrated. It is shown that the hexagonal icosahedron gives the most accurate results (for least computational cost). All of the grids suffer from spurious computational modes; this is especially true of the kite grid, despite it having exactly twice as many velocity degrees of freedom as height degrees of freedom. However, the computational modes are easiest to control on the hexagonal icosahedron since they consist of vorticity oscillations on the dual grid that can be controlled using a diffusive advection scheme for potential vorticity.

A General Filter for Stretched-Grid Models: Application in Two-Dimension Polar Geometry

Abstract Variable-resolution grids are used in global atmospheric models to improve the representation of regional scales over an area of interest: they have reduced computational cost compared to uniform high-resolution grids, and avoid the nesting issues of limited-area models. To address some concerns associated with the stretching and anisotropy of the variable-resolution computational grid, a general convolution filter operator was developed. The convolution filter that was initially applied in Cartesian geometry in a companion paper is here adapted to cylindrical polar coordinates as an intermediate step toward spherical polar latitude–longitude grids. Both polar grids face the so-called “pole problem” because of the convergence of meridians at the poles. In this work the authors will present some details related to the adaptation of the filter to cylindrical polar coordinates for both uniform as well as stretched grids. The results show that the developed operator is skillful in removing the extraneous fine scales around the pole, with a computational cost smaller than that of common polar filters. The results on a stretched grid for vector and scalar test functions are satisfactory and the filter’s response can be optimized for different types of test function and noise one wishes to remove.

Gauss Krüger projection for areas of wide longitudinal extent

This paper describes a new algorithm for calculating all equations (direct and inverse transformation, convergence of meridians, linear distortion, calculus of surfaces, and arc‐to‐chord correction) for the Gauss–Krüger projection. Instead of using different equations for each problem, all the calculi are based on the equations used to obtain the direct transformation. These equations are also more accurate than previous ones and can be extended to an arbitrary width. This paper also explains how the Gauss–Krüger projection may satisfy the needs and requirements of civil engineers as well as Geographical Information Systems (GIS) users, particularly if extended beyond its usual 4° longitudinal range, and therefore can be used as a global reference system for GIS.

A reduced grid for a parallel global ocean general circulation model

A limitation of many global climate models with explicit finite-difference numerics is the timestep restriction caused by the decrease in cell size associated with the convergence of meridians near the poles. To keep the longitudinal width of model cells as uniform as possible, we apply a “reduced” grid to a three-dimensional primitive equation ocean-climate model. With this grid the number of cells in the longitudinal direction is reduced at high latitudes. The grid consists of subgrids which interact at interfaces along their northern and southern boundaries, where the resolution changes by a factor of three. We extend the finite-difference techniques to these interfaces, focusing on the conservation required to perform long time integrations, while preserving the staggered spatial arrangement of variables and the numerics used on subgrids. The common alternative used to reduce the timestep restriction caused by the spherical grid is the filtering of high-frequency modes from the high-latitude solution. The reduced grid allows an increased timestep while eliminating the need for filtering and reduces execution time per model step by roughly 20%. We implement the reduced grid model for parallel computer architectures with two-dimensional domain decomposition and message passing, with speedup results similar to those of the original model. We present results of model runs showing small effects on the solution and sizable improvements to the execution time.

Jean Rotz and the Marine Chart, 1542

This paper was first published in the Journal in 1954 (Vol. 7, p. 138). It is followed by comments from Michael Richey.At the time (1542) when Jean Rotz presented Henry VIII with a variation compass, and an accompanying treatise on its making and use, the question of magnetic variation was an acute one. It formed part of a wider controversy concerning the general validity of the sea-chart which had been raised among seamen. As to this, Dr Pedro Nunez, writing a treatise in 1537 addressed to his friend and pupil the Infante Dom Luys of Portugal, said that there were skilled pilots who derided the chart and declared it to be ‘a mais falsa causa do mundo’ – the falsest thing in the world. Hastening to its defence, Nunez dealt at length with the navigating errors introduced by ignoring the convergence of the meridians, and showed besides that a rhumb, or line of constant bearing was a spiral curve on the globe.

Pedro Nunes' Discovery of the Loxodromic Curve (1537). How Portuguese Sailors in the Early Sixteenth Century, Navigating with Globes, had Failed to Solve the Difficulties Encountered with the Plane Chart

In the early sixteenth century, Portuguese navigators on voyages outside the tropics ran into an unexpected problem in navigating with the plane chart since the latter took no account of the convergence of the meridians. Voyages between Brazil and the Cape of Good Hope were accomplished much faster than the chart led them to forsee. To overcome this problem, were Portuguese pilots of the first half of the sixteenth century using globes to set their courses with on great circle courses? The general opinion is that they were not. Yet the mathematician Pedro Nunes (1502–78) describes in a short treatise entitled Tratado que ho Doutor Pedro Nunez fez sobre certas duvidas na navigação (published in 1537, but probably written in 1534) how the navigator Martim Afonso de Sousa, on his return from the east coast of Brazil in 1530–2, had asked him to solve several problems of navigation encountered on the voyage.

Experiments with a polar filter and a one‐dimensional semi‐implicit algorithm

Abstract The numerical integration of the dynamical equations on a latitude‐longitude grid requires excessively small time steps due to the convergence of meridians. Space filtering in the polar regions has generally been used to alleviate this restriction. Such filtering does not distinguish between low and high frequencies so that it can lead to errors in the meteorological modes. An alternative solution may lie in the use of a semi‐implicit technique in one dimension. These problems are explored with the aid of numerical models of the shallow‐water equations on a sphere.

Polar Navigation—A New Transverse Mercator Technique

When flying at high latitudes the rapid convergence of Earth meridians makes it difficult to establish true heading at any particular instant. Magnetic compass stabilizing forces are weak and steering must normally be carried out using a directional gyro. The navigation problem is normally solved on the chart by using a ‘grid’ technique.

THE CONVERGENCE OF MERIDIANS

Some publications that have dealt with the question of convergence of meridians seem, to the present writer, to be clouded with misconception, and these notes are intended to clarify some points of apparent obscurity. For instance, A. E. Young, in “Some Investigations in the Theory of Map Projections”, I920, devoted a short chapter to the subject, and appeared surprised to find that the convergence on the Transverse Mercator projection differs from the spheroidal convergence; the explanation which he advanced can be shown to be faulty. Captain G. T. McCaw, in E.S.R., v, 35, 285, derived an expression for the Transverse Mercator convergence which is equal to the spheroidal convergence, and described this as “a result which might be expected in an orthomorphic system”. Perhaps McCaw did not intend his remark to be so interpreted, but it seems to imply that the convergence on any orthomorphic projection should be equal to the spheroidal convergence, and it is easily demonstrated that this is not so. Also, in the second edition of “Survey Computations” there is given a formula for the convergence on the Cassini projection which is identical, as far as it goes, with that given for the Transverse Mercator, while the Cassini convergence as given by Young is actually the spheroidal convergence. Obviously, there is some confusion somewhere, and it is small wonder that Young prefaced his remarks with the admission that the subject had always presented some difficulty to him.