Direct Methods of Sight Reduction: An Historical Review

1982 ◽  
Vol 35 (2) ◽  
pp. 260-273 ◽  
Author(s):  
Charles H. Cotter
Keyword(s):  
Direct Method ◽  
Historical Review ◽  
Direct Methods ◽  
The Other ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
General Terms ◽  
Spherical Triangles ◽  
Two Sides ◽  
Fundamental Formula

In general terms the principal problem in astronomical navigation is the solving of a spherical triangle - the PZX-triangle. The fundamental formula of spherical trigonometry for finding an angle given the three sides of a spherical triangle is the cosine formula. By transposition this formula can be used for finding a side given the opposite angle and the other two sides. Because the cosine formula is not suitable for use with logarithms numerous formulae have been derived from it with the aim of simplifying logarithmic computation. The term ‘direct method’ applies to a method the basis of which is generally the cosine formula or any of its derivatives although some direct methods are based on Napier's Rules for right-angled spherical triangles.

Remembering spherical trigonometry

2016 ◽  
Vol 100 (547) ◽  
pp. 1-8 ◽  
Author(s):  
John Conway ◽  
Alex Ryba
Keyword(s):  
High School ◽  
20Th Century ◽  
Great Circle ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
The Subject ◽  
Image Position ◽  
Two Sides

Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.

New Formulae for Combined Spherical Triangles

2018 ◽  
Vol 72 (2) ◽  
pp. 503-512
Author(s):  
Tsung-Hsuan Hsieh ◽  
Shengzheng Wang ◽  
Wei Liu ◽  
Jiansen Zhao
Keyword(s):  
Research Field ◽  
Great Circle ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
Calculation Process ◽  
Different Types ◽  
Celestial Bodies ◽  
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.

Spherical trigonometry constrained kinematics for a dexterous robotic hand with an articulated palm

Robotica ◽  
2015 ◽  
Vol 34 (12) ◽  
pp. 2788-2805 ◽  
Author(s):  
Evangelos Emmanouil ◽  
Guowu Wei ◽  
Jian S. Dai
Keyword(s):  
Point Clouds ◽  
Joint Space ◽  
Singularity Analysis ◽  
Spherical Triangle ◽  
Design Criteria ◽  
Robotic Hand ◽  
Spherical Trigonometry ◽  
Joint Angles ◽  
Singularity Avoidance ◽  
Spherical Triangles

SUMMARYThis work presents a method based on spherical trigonometry for computing all joint angles of the spherical metamorphic palm. The spherical palm is segmented into spherical triangles which are then solved and combined to fully solve the palm configuration. Further, singularity analysis is investigated with the analysis of each spherical triangle the palm is decomposed. Singularity-avoidance-based design criteria are then presented. Finally, point clouds are generated that represent the joint space of the palm as well as the workspace of the hand with the advantage of an articulated palm is shown.

The Modern Approach: Right-Angled Triangles

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Pythagorean Theorem ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
Modern Approach ◽  
Locality Principle ◽  
Spherical Triangles

This chapter discusses the modern approach to solving right-angled triangles. After a brief background on John Napier's trigonometric work, in which he referred mostly to right-angled spherical triangles, the chapter describes the theorems for right triangles. It then considers an oblique triangle split into two right triangles and the ten fundamental identities of a right-angled spherical triangle, how the locality principle can be applied to derive the Pythagorean Theorem, and how to find a ship's direction of travel using the theorem. It also looks at Napier's work on logarithms which was devoted to trigonometry, along with Napier's Rules. The chapter concludes with an overview of “pentagramma mirificum,” a pentagram in spherical trigonometry that was discovered by Napier.

Vector Solutions for Great Circle Navigation

2005 ◽  
Vol 58 (3) ◽  
pp. 451-457 ◽  
Author(s):  
Michael A. Earle
Keyword(s):  
Great Circle ◽  
North Pole ◽  
Spherical Triangle ◽  
Vector Analysis ◽  
Small Computer ◽  
Spherical Trigonometry ◽  
The North ◽  
Document Vector ◽  
Spherical Triangles ◽  
Vector Approach

Traditionally, navigation has been taught with methods employing Napier's rules for spherical triangles while methods derived from vector analysis and calculus appear to have been avoided in the teaching environment. In this document, vector methods are described that allow distance and azimuth at any point on a great circle to be determined. These methods are direct and avoid reliance on the formulae of spherical trigonometry. The vector approach presented here allows waypoints to be established without the need to either ascertain the position of the vertex or select the nearest pole; the method discussed here requires only one spherical triangle having an apex at the North Pole and is also easy to implement on a small computer.

Eurasian "Theory of Economy Management" as an Organic Part of the Russian Economic School

2003 ◽  
pp. 95-110
Author(s):  
M. Voeykov
Keyword(s):  
Economic Development ◽  
Theoretical Model ◽  
Initial Step ◽  
State Regulation ◽  
Soviet Economy ◽  
The Other ◽  
Large Share ◽  
Organic Part ◽  
Two Sides ◽  
Enterprise Activity

The original version of "the theory of economy management", developed in the 1920s by Russian economists-emigrants who called themselves "Eurasians" (N. Trubetskoy, P. Savitskiy, etc.) is analyzed in the article. They considered this theory to be the basis of the original Russia's way of economic development. The Eurasian theory of economy management focuses on two sides of enterprise activity: managerial as well as social and moral. The Eurasians accepted the Soviet economy with the large share of state regulation as the initial step of development. On the other hand they paid much attention to the private sector activity. Eurasians developed a theoretical model of the mixed economy which can be attributed as the Russian economic school.

The Other Side Of The Coin: An Enquiry Into The Difficulties And Apprehensions Encountered By Consumers With Green Products

2019 ◽  
Vol 118 (8) ◽  
pp. 152-159
Author(s):  
Jijimon M J ◽  
Dr. S. Anthony Rahul Golden ◽  
Dr. S. Bulomine Regi
Keyword(s):  
The Other ◽  
Green Products ◽  
Two Sides

Every reality has its own positives and negatives. As the proverb goes coin has two sides. It is very much true in the case of green products too. There is no doubt that green products have many benefits and positives. Despite all the good things about green products, there exist a few glitches and shadows, thereby creating a few doubts and apprehensions in the minds of consumers. The present paper tries to understand these problems associated with green products from the perspectives of the consumers and analyses them with an intention of providing the green brands the means and ways to eliminate such anomalies. The study finds out that the unavailability of products is the most difficult thing the consumers have experienced while purchasing.

Size Selectivity of Diamond and Square Mesh Codends in Pelagic Herring Trawls: Only Small Herring Will Notice the Difference

1992 ◽  
Vol 49 (10) ◽  
pp. 2104-2117 ◽  
Author(s):  
Petri Suuronen ◽  
Russell B. Millar
Keyword(s):  
Relative Size ◽  
Mesh Size ◽  
The Other ◽  
Size Selectivity ◽  
The Difference ◽  
Catch Rates ◽  
Two Sides ◽  
High Catch

A twin codend trawl was fished in the northern Baltic to study the size selectivity of square mesh and diamond mesh codends of 36-mm nominal mesh size. For each codend, 15 hauls were completed with a small mesh (20 mm) codend deployed on the other side of the trawl. The relative size of the catches in the two sides of the trawl varied considerably from haul to haul (the separator section was not operating properly) and selection curves were estimated from each individual haul using a method that incorporated the differences in catching efficiency of the two sides. The length of 50% retention decreased with increased catch for both the diamond and square mesh codends, although in neither case was this relationship statistically significant. Selection curves fitted to the combined haul data were asymmetric. The square mesh codend retained significantly less small herring than the diamond mesh codend, and for larger herring the two codends had similar selectivity. In both codends, most escapes occurred at the front of the catch bulge, from the upper side of the codend. At high catch rates, mesh blockage was observed for several metres ahead of the catch bulge during the later part of the tow.

scholarly journals Exploring language attitudes in ELF research: Contrasting approaches in conversation

2016 ◽  
Vol 3 (4) ◽  
pp. 74-109 ◽  
Author(s):  
Tomokazu Ishikawa ◽  
Sonia Morán Panero
Keyword(s):  
Present Article ◽  
Social Practice ◽  
Language Attitudes ◽  
A Priori ◽  
The Other ◽  
Research Projects ◽  
Doctoral Research ◽  
Two Sides ◽  
Ontological Position ◽  
Linguistic Phenomenon

AbstractWith reference to two recent doctoral research projects on ELF, the present article examines the characterisation of language attitudes as either stable or variable evaluative phenomena, and provides a detailed account of methodological practices that may be favoured from each ontological position. The durability of language attitudes is more specifically conceptualised as a stable (but not enduring) construct directed to a linguistic phenomenon in one thesis, and as variable and emergent forms of evaluative social practice around a language-related issue in the other. With these two different approaches in conversation, the authors consider the extent to which stability and variability of language attitudes may be two sides of the same coin, and question whether it is safe to assume a priori the inferability of stable language attitudes from the observation of evaluative practice. This article evidences the need for ELF researchers working in this area to contemplate what and how it is being researched in the name of language attitudes while having awareness of possible alternatives in any given study.

scholarly journals PHOTIC STIMULATION AND LEG MOVEMENTS IN THE CRAYFISH

1933 ◽  
Vol 16 (6) ◽  
pp. 905-910 ◽  
Author(s):  
B. Kropp ◽  
E. V. Enzmann
Keyword(s):  
The Other ◽  
Photic Stimulation ◽  
Two Sides ◽  
Leg Movements

When Cambarus clarkii is exposed to a source of light so that both eyes are equally illuminated, leg movements of the two sides are equal in frequency and amplitude. On covering one eye and exposing the uncovered eye to light, leg movements on the side of the uncovered eye are more frequent and are of greater amplitude than on the side of the covered eye. On covering the exposed eye also the leg movements on the two sides again tend to become equal in frequency and amplitude. When one eye is lost and the other remains functional, the leg movements on the side of the lost eye will be similar to those on the side of a normal, covered eye.

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