Great Ellipse Research Articles

Vector-Algebra Algorithms to Draw the Curve of Alignment, the Great Ellipse, the Normal Section, and the Loxodrome

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.

NUMERICAL SIMULATION OF SHIP NAVIGATION IN ROUGH SEAS BASED ON ECMWF DATA

Recently, several changes have been observed in the Earth’s environment. This is also applicable to the ocean environment. The concept of weather routing has been applied for ship navigation for a long time. Many service providers offer weather routing service with the availability of high-quality satellite data. Unfortunately, not much information is available in the public domain as to how much the recent change in the weather pattern has affected ship navigation. The purpose of this paper is to fill this information gap. We investigate the influence of recent changes in the ocean environment on ship navigation. Weather data from ECMWF, namely ERA-Interim, is used for this purpose. The ECMWF data for the last 27 years is analysed. We compute the statistical characteristics of this data for the first 10 years, last 10 years, and 27 years. The statistical characteristics of the data are determined based on “summer” and “winter” zones as defined by international maritime regulations. Six different worldwide commercial ship routes are selected covering all the ocean regions. Navigation on great ellipse with waypoint is considered. MMG type ship manoeuvring model for 3 different ship types (DTMB 5415, PCC, VLCC) is used. The added resistance due to wave, wind and the effort of keeping the ship on the desired course using autopilot in the rough ocean environment is included in the MMG model. The fuel consumption and the duration of each one of the voyage are computed. Based on the analysis and simulation results it is shown that: (i) The mean wave height, wave period, and wind speed has increased in some ocean zones and decreased in other ocean zones. If any change has occurred, it is uniform for both seasons (summer and winter). (ii) In which ocean regions there is a perceptible change in fuel consumption, average ship speed and voyage time due to the changes in the weather pattern. (iii) The changing weather pattern in different ocean zones affects each ship type differently.

Evaluation and Execution of Great Elliptic Sailing

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.

Solutions to the direct and inverse navigation problems on the great ellipse

Abstract The Great Ellipse (GE) is the curve of intersection between the surface and a plane through the center of an ellipsoid. For arcs within a few thousands of kilometres it agrees within a few metres with the geodesic. As the direct and indirect navigation problems for the GE can be solved almost entirely by closed formulas (in contrast to the corresponding geodetic problems of the geodesic), navigation on the GE is mostly preferred. Here we take advantage of the Clairaut constant on the GE in solving the navigation problems.

The vector solutions for the great ellipse on the spheroid

Compact vector formulas are given for the great ellipse on the spheroid providing vector solutions to both the forward and the inverse problems and conversions of longitude and latitude. The solution incorporates a closed form for the azimuth and the derivation of this equation is presented and illustrated. This paper also shows that a computer algebra system and linear algebra are powerful tools to solve some mathematical derivations that may be useful in navigation, geodesy, and cartography.

Accurate Harmonic Series for Inverse and Direct Solutions for the Great Ellipse

In this paper, formulae drawn from the discipline of map projections are applied to provide simple and accurate solutions for the Inverse and Direct problems on the Great Ellipse. Distance along the meridional arc of the spheroid as a function of geodetic latitude is defined in terms of an elliptic integral which will be replaced here with a compact harmonic series approximation possessing simplicity and high accuracy. Latitude as a function of distance along the meridional arc will also be obtained via another equally simple inversion series that also possesses high accuracy. When these two series are applied with their constants modified to suit the section ellipse, they will be shown to provide accurate solutions to the inverse and direct navigation problems pertaining to the Great Ellipse, and thereby provide a complete solution that is also simple to implement.

NAVIGATION ON A GREAT ELLIPSE

This paper presents contact formulae for the computation of the position, the distance, and the azimuth along a great ellipse. The proposed alternative formulae are to be primarily used for accurate sailing calculations on the ellipsoid in a GIS environment as in ECDIS and other ECS. Among the ECDIS Requirements is the need for a continuous system with a degree of accuracy consistent with the requirements of safe navigation. At present, this requirement is best fulfilled by the Global Positioning System (GPS). The GPS system is referenced to World Geodetic System 1984 (WGS 84) Datum. Using the ellipsoid model for the spherical model attains more accurate calculation of sailing on the Earth. Therefore, we construct a computational procedure for solving the length of the arc of a great ellipse, the waypoints and azimuths along a great ellipse. In this paper, we provide the straightforward formulae involving the great elliptic sailing based on two scenarios. The first scenario is that the departure point and the destination point are known. The second scenario is that the departure point and the initial azimuth are given.

Sphere to Spheroid Comparisons

The notion of the earth as a perfect sphere has served navigators quite well over the centuries and it continues to provide a basis for instruction and practical navigation. Aside from the fact that modern navigators employ refined models of the earth such as the WGS84 ellipsoid, which together with satellite positioning gives unprecedented positioning accuracy, the question arises as to just how good are the spherical results for distance when compared to the results for distance obtained from the spheroidal model. In this document a comparison is made of great circle distance (GCD) on the sphere with great ellipse distance (GED) on the spheroid. This comparison is then repeated for rhumbline distances. In each case it is concluded that the difference in using the sphere when compared to the spheroid is near 0·5%.

‘A Vector Solution for Navigation on a Great Ellipse’

Distance and course to steer along part of a great ellipse can be readily determined with the use of vector analysis and differential geometry. Although the resulting integral for distance is not amenable to direct solution, it can nevertheless be handled efficiently and quickly by available commercial mathematical software. In this paper, a method of computing distance along with course to steer is presented and is one that has been prepared with the syntax of a particular commercial mathematics package in mind. This approach was intended to appeal to the navigator who is interested in the mathematics of navigation, and who nowadays solves his problems with a portable computer.

A Vector Solution for Navigation on a Great Ellipse

Distance and course to steer along part of a great ellipse can be readily determined with the use of vector analysis and differential geometry. Although the resulting integral for distance is not amenable to direct solution, it can nevertheless be handled efficiently and quickly by available commercial mathematical software. In this paper, a method of computing distance along with course to steer is presented and is one that has been prepared with the syntax of a particular commercial mathematics package in mind. This approach was intended to appeal to the navigator who is interested in the mathematics of navigation, and who nowadays solves his problems with a portable computer.

The Great Ellipse Solution for Distances and Headings to Steer between Waypoints

The normal Great Circle method of computing the shortest distance between two positions on the Earth – e.g. from an aircraft's present position (PP) to a waypoint (WP) – is not accurate enough to meet present-day requirements for aircraft Nav–Attack systems.On the surface of an Ellipsoid (or Spheroid), the true ‘shortest distance’ is along a geodesic curve between the two points, but the computation of this curve is complex, and as shown by R. Williams at Reference, the difference between the geodesic and Great Ellipse distances between two points is negligible (<0·01 nm).The Great Ellipse through two points on a spheroid is defined as the ellipse that passes through the two points and the centre of the spheroid; it therefore has a major axis equal to the Earth's, and a minor axis that is between the Earth's major axis (for two points on the Equator) and minor axis (for two points on the same, or diametrically opposite, longitudes). Thus the problem of deciding on which Great Ellipse the two points lie is equivalent to determining the magnitude of the minor axis β of the ellipse on which they both lie.

The Great Ellipse on the Surface of the Spheroid

On any surface which fulfils the required continuity conditions, the shortest path between two points on the surface is along the are of a geodesic curve. On the surface of a sphere the geodesic curves are the great circles and the shortest path between any two points on this surface is along the arc of a great circle, but on the surface of an ellipsoid of revolution, the geodesic curves are not so easily defined except that the equator of this ellipsoid is a circle and its meridians are ellipses.

Elimination of near-source ellipticity corrections to body-wave travel times by use of equidistant latitudes

AbstractThe use of equidistant latitudes has been proposed by the author to eliminate discrepancies between angular and kilometric epicentral distances. This is done in combination with a path-length correction which depends on the inclination of the great ellipse containing the epicenter-receiver path. If there were a one-to-one correspondence between source-receiver surface arc length (in kilometers) and, say, P-wave travel time (for constant focal depth) for a standard spheroidal Earth, the ellipticity (time) correction could then be replaced by the distance correction described. However, one would only expect this to be approximately valid for small epicentral distances Δ.In this paper, the travel-time corrections made by using equidistant latitudes (and the great-ellipse correction) are compared with the “true” ellipticity corrections due to Dziewonski and Gilbert. It is seen that the present equidistant-latitude method gives P-wave correction values that, for example, are always within 0.05 sec of the “true” values for Δ ≦ 14° and normal focal depth (h ≦ 40 km). For large Δ(⪞ 45°) and/or great focal depth (h⪞ 475 km), these values may differ by more than 0.2 sec. This equidistant-latitude method of correcting body-wave travel times is thus not recommended for routine use, but it could be used to advantage in special studies involving smaller Δ and h.

The Saghalien Trade: A Contribution to Ainu Studies

FEW PEOPLE have so aroused the curiosity of anthropologists as have the Ainu of northern Japan and Saghalien. This is largely due to the fact that they are a tiny island of alleged Caucasian people within a great Mongoloid sea. More important, their origin and history is so obscure that they constitute a problem. That is, Who are the Ainu and from where did they come? To some extent they are a problem within a problem, for the question of who are the Japanese and from where did they come is still unanswered. Indeed we may go further and say we have a threefold problem, for both the Ainu and the Japanese are but part of the whole complex of peoples living in the great sea and land area of the maritime northeast of Asia, and the study of this area and its inhabitants has barely begun. That great ellipse of mankind running from Kamchatka through the Kuriles to the islands of the Japanese archipelago, thence to Korea, and then turning north through the maritime province of eastern Siberia to the Chuckchee peninsula and back to Kamchatka, is one of the most fertile and untouched fields for ethnological study that we have. In light of this, the Ainu problem must be answered in relation to the questions asked and solutions derived about all the peoples of this entire area; for some sort of satisfactory relationship or non-relationship must be worked out before one can venture to say who any of these people are or from whence any of them probably came. A basic step should be the most accurate historical descriptions we can get of the and development of these people. However, in what little work Western scholars are doing on the area, little attention is being paid to what I call the pure description of the round of life based on recorded observations.1 Much work has gone into erecting superstructures of theory based on linguistic, sociological, and even psychological methods without attention to the dull work of reconstructing the of these people from the observations of men who lived with and knew the Ainu before they were Japanized, Sinicised, or Russianized. Western language literature is not of much value, although it comprises a great deal of publication from sixteenth century missionary reports to contemporary theoretical studies. One thing immediately stands out after a survey