Mercator Chart Research Articles

Komparasi Perhitungan Kadet Secara Manual Dan Dengan Menggunakan Program – Program Pelayaran Dalam Pelaksanaan Passage Plan

To improve the academic ability of a cadet, it is necessary to apply the principles of shipping with the use of computer technology which is accommodated by existing shipping programs on board. In the implementation of a voyage, it always uses a passage plan to make it easier to navigate either flat or electronically. A cadet must really be equipped with the ability to calculate manually and the ability to understand the use of shipping tables and be able to apply it in a computer program. The research method is descriptive. Data collection was carried out by manual calculation of the concept of Flat Sailing using a Mercator map and comparison of the results of calculations with programs on the ship, observation, structured interviews, documentary studies and library research. The purpose of this study was to compare the results of manual calculations and by using shipping programs carried out by cadets in the implementation of the Passage Plan. The results showed that the cadets still did not understand the application of cruise calculation formulas on the projection of the mercator chart or mercator map. Cadets in entering data in a program are still wrong and less thorough so that it will affect the implementation of the passage plan.

Determine the effect of distortion of Mercator chart, Gnomonic chart on maritime safety and method of navigation when using Gnomonic chart

Nautical charts are an important tool to support safe navigation at sea. Several projections have been studied to develop charts for various maritime applications. Among them, the only Mercator projection satisfies the requirements of a nautical chart. However, the disadvantage of Mercator projection is the distortion that increases gradually to both poles. Therefore, Gnomonic charts are used to replace Mercator charts for navigation in some areas. However, the Gnomonic chart does not satisfy the basic requirements of a nautical chart, not built by an equiangular projection and the rhumb line is the curve on the chart. For building the nautical charts, the article will study the theoretical and practical basis of using Gnomonic projection instead of Mercator projection in determining the limit that the distortion of the Mercator projection affects the safe navigation, the conditions which the distortion of a Gnomonic projection has an influence on the safety of navigation in the allowable limits and finally determining the conditions and providing a method of navigation when using the Gnomonic chart instead of the Mercator chart.

Approximation of a Great Circle by using a Circular Arc on a Mercator Chart

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.

基于墨卡托海图的大圆航线设计

随着船舶自动化程度的不断提升与航海仪器设备的日益升级，船舶的航行变得更加安全、高效、稳定、经济。但随之带来的问题是驾驶员的理论知识不扎实，当仪器设备出现故障时，处理问题的能力不足，可能造成巨大损失。在航线设计中，如果脱离ECDIS，则在墨卡托海图中绘制大圆航线显得异常困难，为了避免损失，可根据球面三角形的数学特性，利用Excel来计算数据并设计航线，然后在电子海图中显示数据点，比较大圆航线与恒向线的差异。 With the continuous improvement of the degree of ship automation and the upgrading of marine equipment, the navigation becomes more safe, efficient, stable and economical. But the problem is that the driver’s theoretical knowledge is not solid; when the equipment breaks down, the ability to deal with the problem is insufficient, which may cause huge losses. If separated from ECDIS, the route design drawing of the great circle route based on Mercator chart is very difficult. In this paper, in order to avoid the loss, according to the mathematical properties of the spherical triangle, Excel can be used to calculate the data and design route, then data are displayed in the electronic chart, and the differences between the great circle route and rhumb line are compared.

Improved Adaptive Neural Network Autopilot for Track-keeping Control of Ships: Design and Simulation

This paper presents an improved adaptive neural network autopilot based on our previous study for track-keeping control of ships. The proposed optimal neural network controller can automatically adapt its learning rate and number of iterations. Firstly, the track-keeping control system of ships is described For the track-keeping control task, a way-point based guidance system is applied To improve the track-keeping ability, the off-track distance caused by external disturbances is considered in learning process of neural network controller. The simulations of track-keeping performance are presented under the influence of sea current and wind as well as measurement noise. The toolbox for track-keeping simulation on Mercator chart is also introduced.

Simple Approximation of Area Determination on Mercator Chart of Sphere

In this paper, simple approximate methods have been given to calculate the true value of an area on a Mercator chart. The introduction touches on the general characteristics of Mercator projection, and the area deformation on this projection is studied. As is known, in this conformal projection area deformations increase with the latitude and differentiate according to the location of the region. Therefore, the areas calculated or measured directly on these commonly used charts are not the true values. First, these two approximate methods—of which areas values very close to the true values can be obtained—are introduced. Then, these methods have been applied to the regions located at low, middle, and high latitudes. In the conclusion, calculated areas have been compared with the true areas, closeness of the results have been examined, and the validity of the methods has been proved.

Halley, cotes, and the nautical meridian

The “true division of the nautical meridian,” that is, the correct spacing of the parallels of latitude on the Mercator chart, presented theoretical problems to 17th-century mathematicians. Edmond Halley solved the problem by showing that under stereographic projection a rhumb line (line of constant bearing) projected into a logarithmic spiral. Hence he developed a rule, simple to apply, for calculating the meridional distance corresponding to any given latitude interval. Halley's younger contemporary, Roger Cotes, showed how his own logometric methods yielded Halley's rule more economically, and further, that the use of the logarithmic spiral was not necessary. La “vrai division du méridien maritime,” c'est-à-dire, le correct espacement des parallèles de latitude sur la carte de Mercator, présentait des problèmes théorique aux mathématiciens du dix-septième siècle. Edmond Halley résolut le problème en démontrant que, sous projection stéréographique, la ligne de rumb (ligne possédant un relèvement constant) se traduisait en spirale logarithmique. De là, il développa une règle, facile à appliquer, pour le calcul de la distance méridienne correspondant à un intervalle de latitude donné. Roger Cotes, contemporain de Halley, plus jeune de quelques années, montra que ses propres méthodes logarithmique arrivaient à la règle de Halley en façon plus économique, et que par surcroît il n'était pas nécessaire d'utiliser la spirale logarithmique. Die “genaue Division des nautischen Meridians,” d.h. die richtige Bestimmung des Abstandes der Breitengrad-Parallelen auf der Mercator Seekarte, war die Ursache theoretischer Probleme für Mathematiker des siebzehnten Jahrhunderts. Edmond Halley gelang es das Problem zu lösen indem er zeigte, daβ bei stereobildlicher Projektion der Kompassstrich (der Kurs der konstanten Ortung) in eine logarithmische Spirale vorstand. Deshalb entwickelte er eine einfach anzuwendende Regel für die Errechnung des Meridionalen Abstandes, korrespondierend mit jedem gegebenen Breitengrad-Abstand. Halleys jüngerer Zeitgenosse, Roger Cotes, zeigte wie seine einzelne logometrische Methode dazu führte Halleys Regel wirtschaftlicher anzuwenden and auβerdem, daβ der Gebrauch der logaritmischen Spirale nicht nötig war.

Measuring True Distance on a Mercator Chart

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The Analysis of Great-circle Tracks

It is the common practice to draw great-circle tracks on gnomonic charts, on which they appear as straight lines, to pick off the latitudes of intermediate points where the track is intersected by the charted meridians, and finally to transfer the pairs of coordinates thus measured to a Mercator chart. The purpose of this note is to present the derivation of some formulae by means of which the coordinates of points along the track may be calculated very easily.

The appearance of Mars from 1907 to 1971: Graphic synthesis of photographs from the I.A.U. center at Meudon

Using selected high quality plates collected at the I.A.U. Planetary Data Center in Meudon (Paris), the author has drawn 32 Mercator charts of the planet Mars for every opposition from 1907 to 1971. This graphic synthesis of albedo distribution may be used in the study of major surface changes over the years.

Lunar Gravity via Apollo 14 Doppler Radio Tracking

Gravity measurements at high resolution were obtained over a 100-kilometer band from + 70 degrees to -70 degrees of longitude during the orbits of low periapsis altitude (approximately 16 kilometers). The line-of-sight accelerations are plotted on Aeronautical Chart and Information Center mercator charts (scale 1 : 1,000,000) as contours at 10-milligal intervals. Direct correlations between gravity variations and surface features are easily determined. Theophilus, Hipparchus, and Ptolemaeus are negative features, whereas Mare Nectaris is a large positive region. The acceleration profiles over Mare Nectaris are suggestive of a broad disk near the surface rather than a deeply buried spherical body. These data are in good agreement with the short arc of Apollo 12 lunar module descent data.

太平洋海域における天測とロランCによる正午船位の確率密度勢力分布図について(日本航海学会第43回講演会)

First, the writer drew the contours of constant probability density of Sun-Run-Sun's Meridian Altitude on the Mercator chart, and, using the chart, made a serviceable diagram showing the most suitable time for a morning sun sight for navigators. Secondly, supposing that the navigator had made the morning observation at the most suitable time, the writer drew parallels of the constant probability density of Noon Position on the ditto chart, and by comparing these of Loran C fixes with the previously mentioned parallels, the writer tried to make a practical chart by which navigators could select the appropriate triplets of Loran C stations or Sun-Run-Sun's Meridian Altitude to obtain the most accurate Noon Fix in the Pacific Ocean.

天測誤差の考察(日本航海学会第41回講演会)

The line of position by heaven body on the Mercator chart has some kinds of error. For example, there are 1. error due to the curveture of equall altitude circle 2. bearing error due to the difference between great circle bearing and Mercator bearing 3. distance (intercept) error due to the scale of Mercator chart. Usually they are so small that they are used to be negligible. But we have to understand the nature of these errors and we must know which error is bigger than other's. This paper gives a calculation some kinds of error in LOP on the Mercator and chart also some error tables. Another purpose of this paper is to investigate the formula, error due to curveture=tan (altitude)/6876. This formula gives a assumption that the equall altitude curve on the Mercator chart is a circle. This is not a circle and not a simple curve, as many bibliographies say. So this is not exact formula but approximate one. This paper gives the limit of its adaptation, exact and simple formula, some tables and digrams to obtain the error and so on.

Harriot's calculation of the meridional parts as logarithmic tangents

Many approximate solutions of the problem of constructing a true sea-chart or Mercator's projection were worked out in the sixteenth and seventeenth centuries. It was not until after the middle of the latter century that it became widely known that the solution was very closely related to logarithmic tangents. However, unpublished manuscripts in private possession in England show that the problem had been completely dealt with much earlier by Thomas Harriot (1560–1621), the contemporary of Napier and Briggs, who is best known in the history of mathematics for his work on algebra. Harriot was successively scientific and navigational adviser to Sir Walter Ralegh and under the patronage of the earl of Northumberland. As a young man he produced an approximate solution of the Mercator problem, which is now lost. He then developed theories of the conformality of stereographic projection, the rectification and quadrature of the equiangular (logarithmic) spiral, used the exponential series, devised interpolation formulae, and applied these results to the calculation of the so-called meridional parts (latitudes croissantes) used in the construction of a Mercator chart, which was most probably completed in 1614. Transcriptions of many of the relevant manuscripts are included in the article.

Harmonic analysis of the topography along parallels of the Earth

The topography along latitude circles for each 5° between the poles has been subjected to harmonic analysis. The amplitude and phase of the first 15 harmonics are presented in tabular form. The first four harmonics are represented on Mercator charts.

Some Aspects of Doppler Track Computer Navigation

The doppler track computer is a navigation device which accepts inputs of ground-speed and track from a doppler radar, and computes and displays miles to go and miles transverse to a selected track. Several types are available on the market.The heading input to the doppler may be from a magnetically slaved gyro or from a free gyro corrected for Earth rate. Thus, either magnetic tracks or gyro tracks are fed to the computer. Also, navigation may be carried out with reference to airways, mercator, conformal conic, oblique mercator, or polar stereographic charts.

Lightweight Integrated Doppler Navigation System for Army Aircraft

This paper presents some of the problems associated with the Signal Corps search for a suitable Doppler navigator for both lightweight fixed wing and rotary wing aircraft. The unique approaches discussed in detail are: The choice of a lightweight 3-axis stabilized antenna, position computation and display in cartesian coordinates for compatibility with Universal Transverse Mercator charts, a moving bug map display, a tape display of absolute and barometric altitude, and an integrated instrument system displaying navigation and flight information for all weather operation. These various components are combined to furnish the Army with a truly automatic self-contained navigation and flight instrument system.

ランベルト投影図と天文航法(日本航海学会第21回講演会)

Characters of celestial line of position plotted on chart of Lambert conformal conic projection, e.g. ICAO WAC one millionth, are investigated. Error of length of the intercept caused by neglect of change in the magnification along meridian, error of the position of intercept terminal caused by the difference between straight line intercept and great circle intercept, and error of the direction of position line due to the convergency of meridians give final effect of third order infinitesimal or less to the position of ship or aircraft, and is practically neglegible even at high latitude or for high speed air navigation, at lower latitude in which cases the ordinary position line on Mercator chart gives serious error. The full text of the treatise shall be shown on the Contributions from the Marine Research Laboratory, Hydrographic Office of Japan, 1, 1, 20 (1959).

高緯度における位置の線(II)(日本航海学会第16回講演会)

In the last paper, we treated the errors of line of position that is plotted on the Mercator's chart and found that the error due to the convergency of meridian was most severe for the high latitudes. At this paper, in order to exclude this error, we get the stereographic projection from the conditions of conformality for polar representation. We investigate the characters of the errors of line of position on this projection arising from: (i) the length of intercept taken from the latitude graduation, (ii) the position of the terminal of intercept plotted as straight line, (iii) the azimuth of position line plotted perpendicular at the terminal of straight line intercept, and (iv) the line of position as approximation of the circle of position. The resultant effect to the fixed position is sufficiently small for the ordinary air or surface navigation at higher latitudes, for example, less than one mile for latitude 80°, and less than four miles even for latitude 60°. For the polar navigation, we adopt the pole itself as an assumed position and can easily plot the line of position without computation. A simple correction table for the curvature of position line is provided, which can be used for the whole polar region higher than latitude 85°.

I—Thomas Hariot's Manuscript

In a previous article in this Journal mention was made of an unexamined manuscript by Hariot belonging to the Petworth Collection entitled The Doctrine of Nauticall Triangles Compendious, which promised to be of interest to navigators. When Lord Leconfield, the owner of the Collection, heard of this he very kindly himself brought this manuscript to the British Museum for study there, and meanwhile Dr. Shirley of the United States, who is at work on other aspects of Hariot's life, sent over from America a transcript from a microfilm copy which was in his possession. The contents of the manuscript more than fulfilled expectation, for it appeared that the main theme was the computation of meridional parts for drawing a Mercator chart and, as Mr. Sadler, the Superintendent of H.M. Nautical Almanac Office, shows in the second part of this paper, Hariot was remarkably in advance of his time in his mathematical treatment of this problem, which continued to be a favourite one for discussion until the discovery of the calculus. As is well known, the first table of meridional parts to be published was that compiled (by simple addition of secants) by Edward Wright (1558–1615) of Caius College, Cambridge, a mathematician who had been nautical adviser to the Earl of Cumberland, and on occasion to the Lord High Admiral himself.