scholarly journals Orthodrome-Loxodrome Correlation by the Middle Latitude Rule

2013 ◽  
Vol 67 (3) ◽  
pp. 539-543 ◽  
Author(s):  
Miljenko Petrović

In this note the Middle Latitude Rule is derived. Namely, to reach Great Circle vertex in two steps such that the total distance is a minimum, an initial rhumb line course equal to the orthodromic course at Middle latitude is to be used. The shortest distance is achieved if the rhumb line course is altered towards the vertex at the orthodrome-loxodrome intersection point.

2017 ◽  
Vol 24 (2) ◽  
pp. 13-16
Author(s):  
Miljenko Petrović

Abstract This work tackles the problem of misconception when using sophisticated mathematical tools, nonlinear optimization in this particular case, to solve a navigational problem. Namely, to reach the Great Circle vertex with two rhumb line legs ensuing the optimized distance, an initial rhumb line course equal to the orthodromic course at middle latitude may be used. The initial course is thereupon optimized by the incremental value steps. The optimized distance is achieved if the rhumb line course is altered towards the vertex at the orthodrome-loxodrome intersection point. As determination of this point cannot be formulated in a closed form, an iterative solution is to be applied. The derived transcendental equation forms a basis for an iterative solution of intersection using the Newton-Raphson method. To the contrary, finding solutions to a system of nonlinear equations can mislead a researcher unable to comprehend and grasp the mathematical meanings of the algorithm. The gist of this essay is a novel concept showing an intrinsic property i.e. orthodrome-loxodrome correlation using a well-known formula.


Kinesiology ◽  
2021 ◽  
Vol 53 (2) ◽  
pp. 280-287
Author(s):  
Fernando González-Mohíno ◽  
Arturo Casado ◽  
Andrew Renfree ◽  
José María González-Ravé ◽  
Brian Hanley

The aim of this study was to determine the influence of running wide on bends regarding intermediate changes of position, dispersion of athletes, and speed relative to season best (SB) between medalists and non-medalists in men’s and women’s 800 m championship finals. Extra distance covered on bends was measured using official videos. The total distance run was calculated and described as the minimum calculable distance. Theoretical mean speeds, theoretical finishing times and theoretical finishing positions were calculated. In addition, intermediate and total changes of position (on bends and straights) and dispersion between athletes in each 100-m segment were calculated. Finishing times and segment times were calculated relative to SB. Theoretical finishing times were faster than official finishing times (p<.001). Finishing positions were influenced by extra distance covered. Medalists displayed a higher number of positive changes of intermediate positions than non-medalists (p<.001), occurring mainly on the straights (p=.003) and were greatest between 500 and 600 m (p=.003), without differences between groups in total extra distance and in extra distance covered on each bend. Medalists displayed higher total speeds relative to SB (p=.042) and over the 500-600 m segment onwards (p<.05), increasing the dispersion from this segment onwards (p<.001). These findings indicate that running the shortest distance on the bends and being able to accelerate during the last 300 m of the race positively influence finishing position in major championships 800 m finals.


Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1492
Author(s):  
Sajina Pradhan ◽  
Suk-seung Hwang ◽  
Dongbin Lee

The time of arrival (TOA) trilateration is one of the representative location detection technologies (LDT) that determines the true location of a mobile station (MS) using a unique intersection point of three circles based on three radii corresponding to distances between MS and base stations (BSs) and center coordinates of BSs. Since the distance between MS and BS is estimated by using the number of time delays, three circles based on the estimated radii are generally increased and they may not meet at a single point, resulting in the location estimation error. In order to compensate this estimation error and to improve estimation performance, we present two advanced TOA trilateration localization algorithms with detail mathematical expressions. The considered algorithms are the shortest distance algorithm, which calculates an average of three interior intersection points among an entire six intersection points from three intersecting circles, and the line intersection algorithm, which calculates an intersection point of three lines connecting two intersection points of two circles among the three circles, as the estimated location of the MS. In this paper, we present both algorithms with detailed mathematical expressions. The computer simulation results are provided to compare the location estimation performance of both algorithms. In addition, in this paper, mathematical analysis is provided to indicate the relation between the line intersection algorithm and the shortest distance algorithm. In this analysis, we verify that line equations based on the intersection points obtained from the shortest distance algorithm are identical to those obtained from the line intersection algorithm.


2017 ◽  
Vol 14 (03) ◽  
pp. 1750019 ◽  
Author(s):  
Suk-Seung Hwang ◽  
Sajina Pradhan

The time of arrival trilateration method is one of the representative algorithms for the location detection technology, which estimates the location of mobile station (MS) at a unique intersection point of three circles with radiuses corresponding to distances between MS and base stations (BSs) and centers corresponding to coordinates of BSs. However, there may be serious estimation errors, when they do not meet at a point because the estimated radiuses of them are increased. The solutions for reducing the estimation position error in the main case of meeting three circles with the extended radius have been recently provided as the shortest distance algorithm and the line intersection algorithm. In general, they have good performance for the location estimation, but they may have serious errors in some cases. In this paper, we propose the efficient location estimation algorithm for the specific case of two large circles and one relatively small circle, which is located in the area of two large circles. In this case, there are six intersections in total based on the three extended circles and a small circle has four intersections with two large circles. The proposed approach compares four distances based on four neighboring intersections and selects the shortest one. Finally, it determines the averaged coordinate of two intersections corresponding to the shortest distance, as the location of MS. The location-estimating performance of the proposed algorithm is illustrated by the computer simulation example.


Author(s):  
Jim Bennett

What was the state of the mathematical science of navigation at the end of the 17th century? ‘Dead reckoning, longitude, and time’ describes the different techniques of sailing—plane sailing, Mercator sailing, middle-latitude sailing, and great-circle sailing—employed to set and reckon a course in the 17th‐18th centuries. Although longitude could be found by dead reckoning, a more reliable method was required to improve navigation. The Royal Observatory, founded in 1675, and the Board of Longitude, which followed in 1714, were crucial to the emergence of longitude methods by chronometers and lunar distances (or ‘lunars’). The work of key characters such as John Hadley and John Harrison is described.


1999 ◽  
Vol 52 (3) ◽  
pp. 421-424 ◽  
Author(s):  
P. R. Walwyn

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.


Paleobiology ◽  
10.1666/13015 ◽  
2014 ◽  
Vol 40 (3) ◽  
pp. 428-439 ◽  
Author(s):  
Shuang-Ye Wu ◽  
Arnold I. Miller

In the assessment of Phanerozoic marine global biodiversity, there has been longstanding interest in quantifying compositional similarities among sampling points as a function of their distances from one another (geodisparity). Previous research has demonstrated that faunal similarity between any two locations tends to decrease significantly as the great circle distance (GCD) between the locations increases, but the rate of decrease begins to stabilize at transoceanic distances. The accuracy of these assessments, and comparisons among different temporal intervals, may suffer, however, because of intervening landmasses that are not accounted for when distance is calibrated simply as GCD. Here, we present a new method for determining the shortest overwater distance (WD) between two marine locations, and we use the method to recalibrate for several Phanerozoic intervals previous measures of global geodisparity in the taxonomic compositions of marine biotas. WD was determined by using a cost-distance approach in ArcGIS, modified to work on a spherical, as opposed to a planar, surface. Results demonstrate two notable effects of using WD. First, mean compositional similarity between locations tends to decrease more continuously as a function of distance with WD than with GCD. Second, pairs of locations with WDs that are at least 50% greater than their GCDs tend to have lower compositional similarity to one another than those with more closely matching WDs and GCDs. These differences are expected as WD better represents the “true” distance between locations; they diminish at GCDs of 5000 km or more when clear, transoceanic paths between locations become more common. Despite these effects, using WD does not alter fundamental temporal trends in global geodisparity through the Phanerozoic observed in previous research, but it is likely to have more significant ramifications for more confined paleobiogeographic investigations.


2013 ◽  
Vol 280 (1750) ◽  
pp. 20122114 ◽  
Author(s):  
L. A. Hawkes ◽  
S. Balachandran ◽  
N. Batbayar ◽  
P. J. Butler ◽  
B. Chua ◽  
...  

Bar-headed geese are renowned for migratory flights at extremely high altitudes over the world's tallest mountains, the Himalayas, where partial pressure of oxygen is dramatically reduced while flight costs, in terms of rate of oxygen consumption, are greatly increased. Such a mismatch is paradoxical, and it is not clear why geese might fly higher than is absolutely necessary. In addition, direct empirical measurements of high-altitude flight are lacking. We test whether migrating bar-headed geese actually minimize flight altitude and make use of favourable winds to reduce flight costs. By tracking 91 geese, we show that these birds typically travel through the valleys of the Himalayas and not over the summits. We report maximum flight altitudes of 7290 m and 6540 m for southbound and northbound geese, respectively, but with 95 per cent of locations received from less than 5489 m. Geese travelled along a route that was 112 km longer than the great circle (shortest distance) route, with transit ground speeds suggesting that they rarely profited from tailwinds. Bar-headed geese from these eastern populations generally travel only as high as the terrain beneath them dictates and rarely in profitable winds. Nevertheless, their migration represents an enormous challenge in conditions where humans and other mammals are only able to operate at levels well below their sea-level maxima.


Author(s):  
B.C. Muddle ◽  
G.R. Hugo

Electron microdiffraction has been used to determine the crystallography of precipitation in Al-Cu-Mg-Ag and Al-Ge alloys for individual precipitates with dimensions down to 10 nm. The crystallography has been related to the morphology of the precipitates using an analysis based on the intersection point symmetry. This analysis requires that the precipitate form be consistent with the intersection point group, defined as those point symmetry elements common to precipitate and matrix crystals when the precipitate crystal is in its observed orientation relationship with the matrix.In Al-Cu-Mg-Ag alloys with high Cu:Mg ratios and containing trace amounts of silver, a phase designated Ω readily precipitates as thin, hexagonal-shaped plates on matrix {111}α planes. Examples of these precipitates are shown in Fig. 1. The structure of this phase has been the subject of some controversy. An SAED pattern, Fig. 2, recorded from matrix and precipitates parallel to a <11l>α axis is suggestive of hexagonal symmetry and a hexagonal lattice has been proposed on the basis of such patterns.


1885 ◽  
Vol 20 (501supp) ◽  
pp. 7991-7991
Author(s):  
Richard A. Proctor
Keyword(s):  

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