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.

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.

7. Spheres and more

2020 ◽  
pp. 128-152
Author(s):  
Glen Van Brummelen
Keyword(s):  
Hyperbolic Geometry ◽  
Spherical Geometry ◽  
Pythagorean Theorem ◽  
Spherical Trigonometry ◽  
Spherical Triangles ◽  
The Times ◽  
The Impact

‘Spheres and more’ considers the ten formulas for right-angled spherical triangles (and how they can be generated), the spherical Pythagorean theorem, and Napier’s rules. Spherical trigonometry was intended originally for astronomers, but medieval Islamic scholars used it to predict the beginning of the sacred month of Ramadan and the times of the five daily prayers. The impact of spherical geometry on Euclid’s axioms resulting in two types of non-Euclidean geometries—elliptical geometry and hyperbolic geometry—is also considered.

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.

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.

Areas, Angles, and Polyhedra

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Eighteenth Century ◽  
Unit Sphere ◽  
Regular Polyhedron ◽  
Spherical Triangle ◽  
French Mathematician ◽  
Regular Polyhedra ◽  
Leonhard Euler ◽  
Spherical Triangles

This chapter explains how to find the area of an angle or polyhedron. It begins with a discussion of how to determine the area of a spherical triangle or polygon. The formula for the area of a spherical triangle is named after Albert Girard, a French mathematician who developed a theorem on the areas of spherical triangles, found in his Invention nouvelle. The chapter goes on to consider Euler's polyhedral formula, named after the eighteenth-century mathematician Leonhard Euler, and the geometry of a regular polyhedron. Finally, it describes an approach to finding the proportion of the volume of the unit sphere that the various regular polyhedra occupy.

scholarly journals On the special spherical triangles for physical and cosmological applications

2021 ◽  
pp. 112-114
Author(s):  
Kalimuthu S
Keyword(s):  
Spherical Triangle ◽  
Spherical Triangles ◽  
The Way

It is well known that a spherical triangle of 270 degree triangle is constructible on the surface of a sphere; a globe is a good example. Take a point (A) on the equator, draw a line 1/4 the way around (90 degrees of longitude) on the equator to a new point (B).

scholarly journals Dihedral f-tilings of the sphere by rhombi and triangles

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Ana Breda ◽  
Altino F. Santos
Keyword(s):  
Spherical Triangle ◽  
International Audience ◽  
Spherical Triangles

International audience We classify, up to an isomorphism, the class of all dihedral f-tilings of S^2, whose prototiles are a spherical triangle and a spherical rhombus. The equiangular case was considered and classified in Ana M. Breda and Altino F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles. Here we complete the classification considering the case of non-equiangular rhombi.

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.

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