Great Circle Track Research Articles

New Formulae for Combined Spherical Triangles

Spherical trigonometry formulae are widely adopted to solve various navigation problems. However, these formulae only express the relationships between the sides and angles of a single spherical triangle. In fact, many problems may involve different types of spherical shapes. If we can develop the different formulae for specific spherical shapes, it will help us solve these problems directly. Thus, we propose two types of formulae for combined spherical triangles. The first set are the formulae of the divided spherical triangle, and the second set are the formulae of the spherical quadrilateral. By applying the formulae of the divided spherical triangle, waypoints on a great circle track can be obtained directly without finding the initial great circle course angle in advance. By applying the formulae of the spherical quadrilateral, the astronomical vessel position can be yielded directly from two celestial bodies, and the calculation process concept is easier to comprehend. The formulae we propose can not only be directly used to solve corresponding problems, but also expand the spherical trigonometry research field.

Research on SINS/GPS integrated navigation system based on grid reference frame in the polar region

For the rapid response of the aircraft, it is required that the aircraft is not only quick but also chooses the shortest flight routes. The great circle track is the shortest flight route on the earth between any two points, but the great circle track needs to cross the polar region when the aircraft flies over the hemispheres of East and West. Because the polar region has some particularity, such as the landform characteristics, the climatic conditions, and the vast territory, the transpolar aircraft is the key technology of global navigation. This article researched on the algorithm of SINS/GPS integrated navigation system and analyzed the result of simulation when the SINS/GPS integrated navigation system works in the middle- and low-latitude region and the polar region. The result of simulation shows that the working performance of SINS/GPS integrated navigation system is improved and makes use of their respective advantages to overcome the shortcomings. By comparing the result of simulation, the navi...

A Systematic Approach for Solving the Great Circle Track Problems based on Vector Algebra

Abstract A systematic approach, based on multiple products of the vector algebra (S-VA), is proposed to derive the spherical triangle formulae for solving the great circle track (GCT) problems. Because the mathematical properties of the geometry and algebra are both embedded in the S-VA approach, derivations of the spherical triangle formulae become more understandable and more straightforward as compared with those approaches which use the complex linear combination of a vector basis. In addition, the S-VA approach can handle all given initial conditions for solving the GCT problems simpler, clearer and avoid redundant formulae existing in the conventional approaches. With the technique of transforming the Earth coordinates system of latitudes and longitudes into the Cartesian one and adopting the relative longitude concept, the concise governing equations of the S-VA approach can be easily and directly derived. Owing to the advantage of the S-VA approach, it makes the practical navigator quickly adjust to solve the GCT problems. Based on the S-VA approach, a program namely GCTPro_VA is developed for friendly use of the navigator. Several validation examples are provided to show the S-VA approach is simple and versatile to solve the GCT problems.

A Novel Approach to Solve the Great Circle Sailings Based on Rotation Transformation

The great circle track (GCT) is composed of several rhumb lines in practice. Given the initial conditions, the waypoints along the GCT can be determined by the navigator. To reduce the complex calculation in a space system, transforming the system into a planar coordinate one is considered. Thus, an idea of the plane of the celestial meridian in celestial navigation is proposed to determine the waypoints, which are presented in equator coordinate system and horizontal one. For locating the waypoints on the GCT, the rotation transformation is introduced to derive the governing equations for the computation procedures with respect to different given initial conditions. Due to the rotation transformation, all initial conditions to obtain the waypoints can be dealt with simultaneously. A program, GCTPro_RT, is developed to solve the GCT problems for any smart devices. Its effectiveness and real applications are presented through demonstrated examples.

New Approach for Modeling of Air Traffic Flow

<strong><strong></strong></strong> Firstly, from the strategic level flow prediction and tactical traffic flow prediction analysis of two methods of flight flow forecast, put forward by the combination method of strategic level forecast, the dead reckoning for the tactical traffic flow prediction. According to the characteristics of dynamic air traffic flow prediction, inference algorithm for aircraft trajectory conjecture by great circle track and isometric track, so as to realize the dynamic traffic prediction. Basic factors that error in in-depth study on track, according to the actual situation of Guangzhou area control center control operation, analyzes and summarizes the influence of dead reckoning uncertainty factors, proposes an equiangular track improved predictive models, and flight data is validated using actual controllers in Guangzhou, the results show that the prediction accuracy of the improved model significant improvement.

A Simple Approach to Great Circle Sailing: The COFI Method

An approach formulated by vector algebra is proposed to deal with great circle sailing problems. Using the technique of the fixed coordinates system and relative longitude concept, derivations of formulae for this approach are simpler than those of the conventional methods. Due to fixing the initial great circle course, the great circle track (GCT) is determined. Since the course is fixed (known as “COFI” in this paper), the proposed approach, which we have named the “COFI method”, can directly calculate the waypoints along the GCT. It is considered that the COFI method is a more understandable and straightforward method to solve waypoint problems than older approaches in the literature. Based on the COFI method, a program has been developed for the navigator. In addition, the spherical triangle method with respect to the equator crossing point (STM-E) is developed by supplemental theorem. Several examples are demonstrated to validate the proposed COFI method and STM-E.

A new SINR equation based on the polarization ellipse parameters

For optimal polarization reception in the presence of interference and noise, a new signal-to-interference-and-noise-ratio (SINR) equation is derived from two triangles in polarization sphere based on the polarization ellipse parameters. The local optimal solutions of SINR on the great circle tracks and the small circle tracks are obtained. Some analytic solutions in special polarization schemes are presented. Several optimal strategies are proposed. An optimal polarization scheme called Three Steps' Searching and Comparing (TSSC) is suggested. The comparison among these optimal schemes is given. Simulation results show that TSSC scheme closely approaches the global optimum quickly and efficiently.

Is the Martian crust also the Martian elastic lithosphere?

Correlations between topography, gravity, and areoid on Mars are used to constrain the crustal and lithospheric thicknesses on the planet. Assuming that the Hellas basin is in isostatic equilibrium with Airy compensation, point correlations between areoid anomalies and topography are used to obtain the mean crustal density and the crustal thickness. We find that the crustal thickness at the reference zero elevation is 90 ± 10 km. We also find that the mean crustal density is 2960 ± 50 kg m−3. We have also used several approaches to constrain the thickness of the elastic lithosphere. Using the spherical harmonic coefficients of the gravity potential and topography as a function of degrees, a relatively weak constraint on the globally averaged thickness of the elastic lithosphere is obtained. An improved constraint is obtained using wavelet transform analyses of great circle tracks of gravity and topography. The gravity‐topography admittance as a function of wavelet wavelength constrains the globally averaged thickness of the elastic lithosphere to be in the range 90 ± 10 km. The observation that the mean thicknesses of the crust and elastic lithosphere are likely to be equal suggests that a rheologically tougher crust is the elastic lithosphere.

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.

Spatial power spectra of the crustal geomagnetic field and core geomagnetic field

Lowes (1966, 1974) has introduced the function R n defined by R n =(n + 1) ∑ m=0 ∞ [(g m n) 2 + (h m n) 2] where g n m and h n m are the coefficients of a spherical harmonic expansion of the scalar potential of the geomagnetic field at the Earth's surface. The mean squared value of the magnetic field B = −∇ V on a sphere of radius r > α is given by 〈 B ·〉 = ∑ n=1 ∞ R n( a/ r ) 2n=4 where a is the Earth's radius. We refer to R n as the spherical harmonic spatial power spectrum of the geomagnetic field. In this paper it is shown that R n = R M n = R C n where the components R n M due to the main (or core) field and R n C due to the crustal field are given approximately by R M n = [ (n =1)/ (n + 2) ](1.142 × 10 9)(0.288 n Λ 2 R C n = [ (n =1){[1 — exp(-n/290)]/ (n/290)} 0.52 Λ 2 where Iγ = 1 nT . The two components are approximately equal for n = 15. Lowes has given equations for the core and crustal field spectra. His equation for the crustal field spectrum is significantly different from the one given here. The equation given in this paper is in better agreement with data obtained on the POGO spacecraft and with data for the crustal field given by Alldredge et al. (1963). The equations for the main and crustal geomagnetic field spectra are consistent with data for the core field given by Peddie and Fabiano (1976) and data for the crustal field given by Alldredge et al. The equations are based on a statistical model that makes use of the principle of equipartition of energy and predicts the shape of both the crustal and core spectra. The model also predicts the core radius accurately. The numerical values given by the equations are not strongly dependent on the model. Equations relating average great circle power spectra of the geomagnetic field components to R n are derived. The three field components are in the radial direction, along the great circle track, and perpendicular to the first two. These equations can, in principle, be inverted to compute the R n for celestial bodies from average great circle power spectra of the magnetic field components.

Cartography in the Round—The Orthographic Projection

Throughout its history the principal function of cartography has been the supply of maps for navigational, engineering or cadastral purposes to a clientele interested tn the map primarily as a basis of mensuration. The interest of the map maker tended therefore to be restricted to projections whose properties facilitated accurate representation of direction, distance and area, to the neglect of other systems less well adapted to such purposes. The characteristics of the orthographic projection, formerly discounted, should now commend it for a greater role in cartographic communication especially in the space age. However, construction of the most frequently required oblique version by methods dependent on draughtsmanship, the camera or the computer may overtax either the patience or resources of the average cartographer. As an alternative, general mapping equations are derived and it is suggested that the pocket calculator can be used to evaluate these to provtde sufficient co-ordinates for plotting useful graticules. Construction of some examples incorporating enlarged insets is discussed together with the plotting of great circle tracks and satellite orbits.

Cartography in the Round—The Orthographic Projection

Throughout its history the principal function of cartography has been the supply of maps for navigational, engineering or cadastral purposes to a clientele interested tn the map primarily as a basis of mensuration. The interest of the map maker tended therefore to be restricted to projections whose properties facilitated accurate representation of direction, distance and area, to the neglect of other systems less well adapted to such purposes. The characteristics of the orthographic projection, formerly discounted, should now commend it for a greater role in cartographic communication especially in the space age. However, construction of the most frequently required oblique version by methods dependent on draughtsmanship, the camera or the computer may overtax either the patience or resources of the average cartographer. As an alternative, general mapping equations are derived and it is suggested that the pocket calculator can be used to evaluate these to provtde sufficient co-ordinates for plotting useful graticules. Construction of some examples incorporating enlarged insets is discussed together with the plotting of great circle tracks and satellite orbits.

Acoustic sidebands from cw sources towed at long ranges in the deep ocean

Three cw sources (9.8, 110, and 262 Hz) were towed at approximately 7 knots along a great circle track from Antigua, W.I., to the Grand Banks of Newfoundland. Acoustic data from each were recorded on a bottom-mounted hydrophone near Antigua, providing source-to-receiver distances of from 100 to 2800 km. High-resolution spectral analysis centered at each frequency showed that while no acoustic sidebands were discernible at 9.8 Hz, they were observable over the entire track for both the higher frequencies. Except for expected statistical fluctuations, the upper and lower sidebands were of equal amplitude and symmetrically placed about the carrier frequency. In regions of strong convergence zone activity the sideband level displayed similar peak-to-null variation to those of the carrier, although the peak-to-null ratio was smaller than for the carrier. Detailed environmental data were available only near the sources. A high degree of correlation was observed between the sideband levels and the wave and swell heights when the sources were towed through a storm which occurred some 2000 km from the receiver. Sideband levels computed from the single bounce sinusoidal surface theory of Roderick and Cron [J. Acoust. Soc. Am. 48, 759–766 (1970)] using swell amplitude data at the sources agreed well with the measured levels for both frequencies during the stormy period. The separation of the sidebands in frequency from the carrier was the same for both source frequencies, was not affected by the convergence zone behavior of the sideband levels, and decreased as the sources approached and transited the stormy area.

Long-range low-frequency CW propagation in the deep ocean: Antigua-Newfoundland

The acoustic fields of shallow sources operating at 13.89 and 111.1 Hz were measured on a deep sound-channel hydrophone located near Antigua, W. I. The sources were towed away from the receiver along a great circle track to a maximum range of 2800 km. Convergence zones were observed at long range, and their positions were found to be frequency dependent. The average rate of increase of transmission loss with increasing range was less than that for cylindrical spreading. The propagation characteristics are explained in terms of the measured oceanographic features.

Search for Atmospheric Gravity Waves induced by the Eclipse of June 30, 1973

It was suggested that atmospheric gravity waves could be generated by the supersonic passage of the moon's shadow through the earth's atmosphere during a solar eclipse, and it was thought that the eclipse of June 30, 1973 might provide an opportunity of investigating this idea. Gravity waves were predicted to have a period of 4 hr and amplitude 10 mu bar at a position 10,000 km from the eclipse path. For the experimental observations a microbarograph was employed, having a variable pass band; this instrument is described in detail. The trace for the period during which the gravity waves might have been expected was very noisy'' due to weather conditions. There were two principal features on the trace; first, a sine wave with period 15 min and amplitude 60 mu bar starting at 1257 UT; secondly, a sharp decrease in pressure at 1314 UT which sert the instrument off scale for about 2 min, after which the trace returned to its original noisy pattern. These two features could possibly have been noise'', but no other such isolated feature was recorded during 50 days of almost continuous operation. There were no meteorological fronts in the vicinity at the time of observation,more » and the duration was far too long to be the product of a sonic boom. Assuming great circle tracks and a speed of propagation of 320 m/sec the gravity wave should have reached the instrument after 1353 UT. It therefore seems that the features recorded could not have been due to the eclipse. (UK)« less

Project KIWI ONE: Very Low Frequency Acoustic Propagation Measurements in the South Pacific Ocean

A very low frequency acoustic propagation experiment was conducted along a 10 000-km great circle track between New Zealand and the Galapagos Islands. Sound signals set to detonate at the sound channel axis were dropped at 30-km intervals along the entire route by a jet aircraft flying nonstop from Panama to New Zealand. Values of propagation and attenuation loss for the frequency range 10–1000 Hz have been computed from data obtained from listening stations located at both ends of the track.

Icosahedral-gnomonic projection and grid of the world ocean for wave studies

The world ocean has been mapped in a special projection, icosahedral-gnomonic, which is optimized for wind-wave studies. The map allows for relatively simple propagation of the waves. The graticule of the map has been computed in detail. The inverse function describing geographical coordinates in terms of map coordinates has been derived and used to tabulate grid point locations for a world network; the function describing great circle track adjustments on any icosahedral-gnomonic projection has been derived and tabulated. This projection features segmented straightline representation of great circles and small simple distortions of direction and scale. Large parts of the globe can be represented without interruption, and a regular, nearly equidistant grid can be fitted over the globe. This projection should be useful in other problems, especially problems involving great circles.

The Nauticator

The Nauticator is a vectorial computer for the solution of spherical trigonometrical problems, designed by Captain Leiv Jernæs of the Royal Norwegian Navy, who is at present captain of the training ship M.S. Sjöveien, Oslo. The instrument has been designed particularly for the rapid reduction of astronomical sights, and the calculation of great-circle tracks, distances and vertices. It has been sent to the Institute to assess its value, and this note will discuss its value to the air navigator.

Saving of Distance in Sailing from one to another on the Great Circle

In the following sections, we will consider the diffrence between Rhumb line distance and Great Circle distance which we call Saving of Distance D. § 1. Saving of Distance on Parallel Sailing. § 2. Saving of Distance on Composite Great Circle Sailing. § 3. Great Circle Distance on Middle Latitude for the same D. Long. § 4. Saving of Distance for the short D. Long. § 5. Saving of Distance for the sfort D. Lat. § 6. Saving of Distance for the short distance Principal Results. (1) Saving of Distance on Parallel Sailing for the same D. Long. Take the maximum value at Lat, 50°〜55°. (2) Within about 10' is Saving of Distance on the case of Composite Great Circle Track taking Lat, about 1°20' lower then vertex Latitude as Limiting Latitude.

A Graphical Aid to the Selection of Best-Time Tracks

It the “mean effective on-track wind components” along the Great Circle track and any other track of known length are known, it is possible to compare flight-times along the two tracks. A graph is devised whereby it may be determined whether or not the longer track offers a shorter flight-time. By comparing several likely-looking tracks, that one which most closely, approximates the best-time track may be found.