scholarly journals Spherical Trigonometry in the Astronomy of the Medieval Kerala School

1998 ◽  
Vol 11 (2) ◽  
pp. 722-723
Author(s):  
Kim Plofker
Keyword(s):  
Exact Solution ◽  
Great Circle ◽  
Spherical Trigonometry ◽  
Mathematical Astronomy ◽  
Per Se ◽  
Spherical Triangles ◽  
Approximate Formulas

Although the methods of plane trigonometry became the cornerstone of classical Indian mathematical astronomy, the corresponding techniques for exact solution of triangles on the sphere’s surface seem never to have been independently developed within this tradition. Numerous rules nevertheless appear in Sanskrit texts for finding the great-circle arcs representing various astronomical quantities; these were presumably derived not by spherics per se but from plane triangles inside the sphere or from analemmatic projections, and were supplemented by approximate formulas assuming small spherical triangles to be plane.

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.

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.

A Study of a Species of Short-Method Table

1974 ◽  
Vol 27 (3) ◽  
pp. 395-401
Author(s):  
Charles H. Cotter
Keyword(s):  
Celestial Body ◽  
Great Circle ◽  
Spherical Triangles ◽  
Modified Forms ◽  
Better Than ◽  
Made In ◽  
Short Method ◽  
Comprehensive Study

This paper discusses navigation tables based on the decomposition of the astronomical triangle into two right-angled spherical triangles by a great circle arc extending from the zenith to the meridian of an observed celestial body. In a recent fairly comprehensive study of some thirty short-method tables in which the division of the PZX-triangle forms the principle of construction, nineteen are of the species to be discussed.Following the introduction in 1871 of the first short-method table by Thomson, some twenty years were to pass before any real advance was made in this field. Thomson's table was in fact re-issued by Kortazzi in 1880 and by Collet in 1891, in modified forms, but it was Professor F. Souillagouët of France who is to be credited for introducing something novel and decidedly better than Thomson's table. Unlike the earlier ones it was designed specifically for the Marcq Saint Hilaire method of sight reduction and, in contrast to Thomson's table which was based on the division of the PZX-triangle by a perpendicular from X, Souillagouët's was based on division by a perpendicular from Z.

scholarly journals Metode Azimuth Kiblat dan Rashdul Kiblat dalam Penentuan Arah Kiblat

2017 ◽  
Vol 15 (2) ◽  
pp. 191
Author(s):  
Ila Nurmila
Keyword(s):  
Great Circle ◽  
Spherical Trigonometry ◽  
The Earth

This article examines the methods of determining the Qibla direction, namely the Qibla azimuth and Rashdul Qibla methods. In this research, the writer tries to describe and interpret the concept of Qibla direction and the concept of Qibla azimuth and Rasdul Qibla in astronomical formulations. The Qiblah problem is nothing but talking about the direction of praying exactly to the Kaaba in Mecca from a point where it is located one line in the great circle of the earth and is the closest distance between the point of place and the Kaaba. Given that every point on the Earth’s surface is on the surface of the Earth’s sphere, then the calculation uses spherical trigonometry. To know the Qibla direction correctly, it is necessary to do calculations and measurements. In calculating and measuring the Qibla direction, there are several methods, and the results are quite varied.

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.

scholarly journals An Axiom of True Courses Calculation in Great Circle Navigation

2021 ◽  
Vol 9 (6) ◽  
pp. 603
Author(s):  
Mate Baric ◽  
David Brčić ◽  
Mate Kosor ◽  
Roko Jelic
Keyword(s):  
Trigonometric Function ◽  
Great Circle ◽  
Trigonometric Functions ◽  
Spherical Trigonometry ◽  
Software Packages ◽  
Vector Algebra ◽  
Computer Aided ◽  
Essential Knowledge

Based on traditional expressions and spherical trigonometry, at present, great circle navigation is undertaken using various navigational software packages. Recent research has mainly focused on vector algebra. These problems are calculated numerically and are thus suited to computer-aided great circle navigation. However, essential knowledge requires the navigator to be able to calculate navigation parameters without the use of aids. This requirement is met using spherical trigonometry functions and the Napier wheel. In addition, to facilitate calculation, certain axioms have been developed to determine a vessel’s true course. These axioms can lead to misleading results due to the limitations of the trigonometric functions, mathematical errors, and the type of great circle navigation. The aim of this paper is to determine a reliable trigonometric function for calculating a vessel’s course in regular and composite great circle navigation, which can be used with the proposed axioms. This was achieved using analysis of the trigonometric functions, and assessment of their impact on the vessel’s calculated course and established axioms.

Building the Latitude Equation of the Mid-longitude

2006 ◽  
Vol 60 (1) ◽  
pp. 164-170 ◽  
Author(s):  
Wei-Kuo Tseng ◽  
Hsuan-Shih Lee
Keyword(s):  
Great Circle ◽  
Spherical Trigonometry ◽  
The Cross ◽  
Cross Track ◽  
New Formula ◽  
Latitude Equation ◽  
Derivation Process

We were amused by the Cross Track Distance at Mid-Longitude problem posed by Paul Hickley in The Journal of Navigation 57, 320. Two professors, John Ponsonby and Peter Hoare, replied to the invitations immediately. Both their solutions to the original article give superb accuracy. The two solutions are certainly ingenious and creative and encouraged us to develop new formula for building the Mid-Longitude Equation on great circle. Regrettably, the original author doesn’t think that the two were the solution that ATPL examiners were looking for. I also think that the two solutions would be beyond the capacity of the average undergraduate. Our method gives a good understanding logically and easily to be mnemonic, and the derivation process is found without any need to appeal to any formula of spherical trigonometry.

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.

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