An Axiom of True Courses Calculation in Great Circle Navigation

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.

Download Full-text

The Vector Function for Distance Travelled in Great Circle Navigation

Journal of Navigation ◽

10.1017/s0373463307214122 ◽

2006 ◽

Vol 60 (1) ◽

pp. 158-164 ◽

Cited By ~ 8

Author(s):

Wei-Kuo Tseng ◽

Hsuan-Shih Lee

Keyword(s):

Linear Combination ◽

Vector Function ◽

Personal Computer ◽

Great Circle ◽

Departure Point ◽

Easy Method ◽

Vector Algebra ◽

Elementary Knowledge ◽

Vector Basis ◽

Great Circle Path

Traditionally, on a great circle, the latitude or longitude of a waypoint is found by inspection. In this paper, using an elementary knowledge of vector algebra including linear combination of vectors and vector basis, we provide an easy method for finding the equation of a great circle path as a parameterized curve. By use of this vector function of distance travelled, the latitude and longitude of waypoints can be found based on the distance from departure point along a great circle. The approach is intended to appeal to the navigator who is interested in the mathematics of navigation and who, nowadays, solves his navigation problems with a personal computer.

Download Full-text

Metode Azimuth Kiblat dan Rashdul Kiblat dalam Penentuan Arah Kiblat

Istinbath | Jurnal Penelitian Hukum Islam ◽

10.36667/istinbath.v15i2.26 ◽

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.

Download Full-text

Analyzing up Front

Mechanical Engineering ◽

10.1115/1.2000-oct-6 ◽

2000 ◽

Vol 122 (10) ◽

pp. 88-91 ◽

Cited By ~ 1

Author(s):

Jean Thilmany

Keyword(s):

Lead Time ◽

Stress Tests ◽

Element Analysis ◽

Design Engineers ◽

Software Packages ◽

Engineering Company ◽

Computer Aided ◽

Depth Study ◽

The Relationship ◽

Senior Engineer

This article highlights that up-front computer-aided engineering (CAE) dramatically decreases product lead time. Up-front CAE entails vesting responsibility for performing finite element analysis tests and other analysis tests with the design engineers. The designers use specific software packages to analyze their first-stage designs. This way, they can easily change designs that do not pass analysis tests-such as vibration or stress tests-before passing them on to an analyst for in-depth study. Not every engineering company, however, is turning to up-front CAE even as it faces the need to get products to market faster. Some engineers, like Zlatko Penzar, find that their present analysis hierarchy works just fine. He is a senior engineer for the fuel systems division of Mannesmann in Dusseldorf, Germany, another auto components supplier. Engineering departments have to find their own answer to the relationship between designer and analyst. The important thing is that once an answer is agreed upon, it happens the same way every day. A working atmosphere that functions reliably and smoothly is really the key to successful product design.

Download Full-text

Milling Cutter Data Structures for Use in Force Models

ASME 2009 International Manufacturing Science and Engineering Conference, Volume 2 ◽

10.1115/msec2009-84368 ◽

2009 ◽

Author(s):

Cuneyt Yalcin ◽

Robert B. Jerard ◽

Barry K. Fussell

Keyword(s):

Data Structure ◽

Academic Research ◽

Simulation Software ◽

Trigonometric Functions ◽

General Representation ◽

Milling Cutter ◽

Software Packages ◽

Milling Simulation ◽

Edge Segment ◽

Force Calculation

In this study we present a new general representation for describing a milling cutter and an internal data structure that systematically stores the cutting edge segment properties of the milling cutter. The intention of this effort is to enable commercial milling simulation software packages to communicate and store complicated cutter information, and thus enable them to include improved models developed in academic research. Examples with various milling cutters are given, and the versatility of the structures is demonstrated by using two different cutting force models with three different milling cutters. Force calculation time was decreased by a factor of four by using the internal data structure to store pre-calculated trigonometric functions.

Download Full-text

Remembering spherical trigonometry

The Mathematical Gazette ◽

10.1017/mag.2016.3 ◽

2016 ◽

Vol 100 (547) ◽

pp. 1-8 ◽

Cited By ~ 2

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.

Download Full-text

New Formulae for Combined Spherical Triangles

Journal of Navigation ◽

10.1017/s0373463318000772 ◽

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.

Download Full-text

Advanced 3D Modeling and Simulation Techniques for Decommissioning of A-1 NPP

9th ASME International Conference on Radioactive Waste Management and Environmental Remediation: Volumes 1, 2, and 3 ◽

10.1115/icem2003-4762 ◽

2003 ◽

Author(s):

Pavol Voza´r ◽

Vladimi´r Sleza´k ◽

Kamil Krava´rik

Keyword(s):

Laser Scanner ◽

3D Models ◽

3D Modelling ◽

Modelling And Simulation ◽

High Dose ◽

Dose Rates ◽

Simulation Techniques ◽

Software Packages ◽

Computer Aided ◽

Selection Of

This paper deals with advanced 3D computer-aided technologies used for modelling and simulation for decommissioning purposes. Within the A-1 NPP decommissioning process a set of activities is needed to perform successful dismantling and decontamination of rooms and equipment. Optimal process of performance of D&D of underground storage tanks and auxiliary rooms were used on the base of simulation outputs. The mockup tests were performed before using remotely controlled manipulators. The human presence during decontamination and dismantling is case by case excluded due to the radiation safety and ALARA approach. Within Bohunice A-1 Decommissioning Project an advanced computer-aided technologies were/are developed and used. Modelling software packages EUCLID and 3Dipsos together with 3D-laser scanner SOISIC are used for creating of 3D models and also for the verification of as-built state of selected systems and facilities. Software IGRIP is used for computer simulations of all D&D tasks. The 3D modelling and simulation of selected rooms and technological equipment of the A-1 NPP are used consequently in the process of decommissioning preparation and implementation. 3D modelling for the verification and simulation of operating steps is presented in the paper and its contribution to avoiding of collisions and non-optimal interventions into the building and technological parts during performing particular works is evaluated. The application of 3D models for the verification and simulation of operating steps significantly contribute to the optimal planning of D&D procedures. Minimisation of occupation doses of realisation personnel is main reason why the 3D modelling and simulations are used. The paper also presented 3D models of rooms chosen to simulate specific operations (decontamination, handling of radioactive wastes and/or dismantling by remote controlled manipulators) without risk accident, high dose rates of personnel etc. Process of selection of optimal operating procedure for decontamination and dismantling is presented as well as achieved experiences and recommendations for further work.

Download Full-text

A Simple Approach to Great Circle Sailing: The COFI Method

Journal of Navigation ◽

10.1017/s0373463313000751 ◽

2013 ◽

Vol 67 (3) ◽

pp. 403-418 ◽

Cited By ~ 4

Author(s):

Chih-Li Chen ◽

Pin-Fang Liu ◽

Wei-Ting Gong

Keyword(s):

Simple Approach ◽

Great Circle ◽

Spherical Triangle ◽

Straightforward Method ◽

Vector Algebra ◽

Triangle Method ◽

Conventional Methods ◽

Crossing Point

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.

Download Full-text

Analogues between 2D Linear Equations and Great Circle Sailing

Journal of Navigation ◽

10.1017/s0373463313000532 ◽

2013 ◽

Vol 67 (1) ◽

pp. 101-112 ◽

Cited By ~ 5

Author(s):

Wei-Kuo Tseng ◽

Wei-Jie Chang

Keyword(s):

Linear Equations ◽

Great Circle ◽

Basic Algebra ◽

The Other ◽

Spherical Triangle ◽

Other Hand ◽

Vector Algebra

This paper presents the similarities between equations used for great circle sailing and 2D linear equations. Great circle sailing adopts spherical triangle equations and vector algebra to solve problems of distance, azimuth and waypoints on the great circle; these equations are sophisticated and deemed hard for those unfamiliar with them, whereas on the other hand, 2D linear equations can be solved easily with basic algebra and trigonometry definitions. By pointing out the similarities, readers can quickly comprehend great circle equations and grasp just how similar they are to the corresponding 2D linear equations.

Download Full-text