DOCSLIB.ORG

On the Geometrical Solution of the Navigational Triangle John A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Geometrical Solution of the Navigational Triangle John A Copyright © 2007, Institute of Navigation, www.ion.org Vol. 4, No. 6 Jacobs: Navigational Aspects of the let Stream 249 ice, APO 953, San Francisco, California, Pan-American World Airways, San Francisco Mnnunl 55-1, Tokyo-Honolulu C-97 Non- International Airport, November 1953. Stop Procedures, October 1954. lOStaff Members, Department of Meteorology, ‘Ruskin, R. E., et al, Development of the NRL University of Chicago, “On the General Cir- Ax&Flow Vortex Thevnzomete,; Report culation of the Atmosphere in Middle Lati- 4008, Naval Research Laboratory, Washing- tudes,” Bulletin of the American Meteoro- ton, D. C., 1952. logical Society, Vol. XXVIII, June 1947. “Serebreny, S. M., and Wiegman, E. J., The llBellamy, John C., “Four Dimensional Flight Characteristic Properties of the /et Stream Planning,” Navigation, Vol. 4, No. 3, Sep- OzJer the Pacific: Case History No. 1, Part 1, tember 1954. ON THE GEOMETRICAL SOLUTION OF THE NAVIGATIONAL TRIANGLE JOHN A. RUSSELL University of Southern C&f ornia, Los Angeles, California The Ukted States Naval Institute Proceed- line of position problems).” The triangular ings for March, 1952, contain a paper by pyramid that Mr. Breed discusses, although Joseph B. Breed III entitled “The Navigational sufficient for the determination of computed Triangle: How to Solve It by Drawing It.” In altitude or great circle distance, will not yield this paper Mr. Breed capably expresses the con- the azimuth or the great circle course. The fol- viction that although the supplanting of the lowing construction extends Mr. Breed’s solu- trigonometric solution of the navigational tri- tion to include the azimuth, without which, the angle by tabulated solutions represents a great solution of a line of position problem is not forward step in celestial navigation, instruction complete. in the newer methods often leaves the student In Figure 1, E is the center of the earth; 2 in ignorance of the basic relationships in the is the navigator’s position; P is nearer geo- navigational triangle. Mr. Breed does not pro- graphic pole; S is the geographic position of pose that spherical trigonometry be reinstated in the celestial object observed. ZP’ and ZS’ are the navigational curriculum. Instead, he sug- tangent to the earth at 2 and intersect the ex- gests, as an instructional tool, a geometrical tended radii EP and ES respectively. The de- solution of the navigational triangle that re- sired azimuth angle is the angle at 2 in the quires only a protractor, a straight edge, and a plane triangle P’ZS’. This triangle is the base pair of compasses. The paper describes with of a triangular pyramid whose vertex is at E. admirable clarity how to obtain the computed The three angles between the three radii to P, altitude of a celestial object by constructing the 2, and S are the complements of the latitude triangular pyramid whose vertex is the center of 2, of the altitude of the celestial object of the earth; whose three sides are the extended viewed from 2, and of the declination of the radii of the earth through the nearer geo- celestial object. graphic pole, the geographical position of the Figure 2 shows how a flattened version of celestial object, and the navigator’s position; the pyramid of Figure 1 can be constructed and and whose base is the intersection of the pyra- the azimuth angle obtained. First, draw EZ to mid with a plane tangent to the earth at the any convenient scale. Next, draw EP, so as to nearer pole. make an angle of 9o”-L, with EZ, and ES SO Mr. Breed suggests that his readers “Work as to make an angle of 90°-H, with EZ. (H, out several navigation problems with this will already have been obtained from Mr. graphic method (azimuth, great circle course Breed’s construction.) Extend EP, and ES until and distance, and star identification as well as they meet the perpendicular to EZ drawn Copyright © 2007, Institute of Navigation, www.ion.org 250 Navigation June 1955 Figure 1 Figure 2 through 2 at P: and S’. Next draw EP:, equal of spherical trigonometry, but they have not in length to EP:, so as to make an angle 90°- been as widely exploited in the manner sug- Dec. with ES’. Then strike an arc from 2 with gested by Mr. Breed as their instructiveness radius ZP{ and an arc from S’ with radius S’P:. merits. The author has found Mr. Breed’s ap- Angle PGZS’ is the desired azimuth angle. proach, as extended in this paper, a very useful Geometrical solids of the sort used in this manner of presenting the derivation of the tab- and Mr. Breed’s constructions are often em- ulated data in HO 214 and HO 249 without ployed in deriving the fundamental formulae introducing the concepts of trigonometry. PUTTING THE MOON IN THE POCKET ALMANAC ROBERT W. BYERLY The omission of all moon data from the gator his true position is a moon sight taken Pocket Almanac was recognized by the author at the time of a sun sight. In the night, the of the Almanac as a ground for criticism. TWO moon may provide a line of position which can reasons were given for omitting moon data.l be obtained in no other way. When the altitude One of the reasons was that “the moon is of the moon is not too high, the moon some- little used.” It is no doubt true that the moon times lights the horizon below it sufficiently is not much used in routine navigation, but in to permit the taking of a sight. It thus provides some situations the moon is of critical impor- a means, and the only celestial means, by which tance to the navigator. In the daytime, the moon a marine navigator may be able to obtain accu- enables the navigator to obtain a point fix. rate data as to his position during the some- When dead reckoning is uncertain because of times long interval between evening twilight an unknown current or wind, two or even more and morning twilight. Moon data thus has a position lines obtained from observations of very practical value for the navigator. the sun taken several hours apart may have Mr. Wylie’s other reason for omitting moon apparent crossing points several miles from the data was: “To maintain our standards of accu- true positi0n.l In this situation, the only racy, inclusion of the moon would involve, on astronomical means which can give the navi- account of that body’s rapid movement among .
Load more

Recommended publications
  • Understanding Celestial Navigation by Ron Davidson, SN Poverty Bay Sail & Power Squadron
    Understanding Celestial Navigation by Ron Davidson, SN Poverty Bay Sail & Power Squadron I grew up on the Jersey Shore very near the entrance to New York harbor and was fascinated by the comings and goings of the ships, passing the Ambrose and Scotland light ships that I would watch from my window at night. I wondered how these mariners could navigate these great ships from ports hundreds or thousands of miles distant and find the narrow entrance to New York harbor. Celestial navigation was always shrouded in mystery that so intrigued me that I eventually began a journey of discovery. One of the most difficult tasks for me, after delving into the arcane knowledge presented in most reference books on the subject, was trying to formulate the “big picture” of how celestial navigation worked. Most texts were full of detailed "cookbook" instructions and mathematical formulas teaching the mechanics of sight reduction and how to use the almanac or sight reduction tables, but frustratingly sparse on the overview of the critical scientific principles of WHY and HOW celestial works. My end result was that I could reduce a sight and obtain a Line of Position but I was unsatisfied not knowing “why” it worked. This article represents my efforts at learning and teaching myself 'celestial' and is by no means comprehensive. As a matter of fact, I have purposely ignored significant detail in order to present the big picture of how celestial principles work so as not to clutter the mind with arcane details and too many magical formulas. The USPS JN & N courses will provide all the details necessary to ensure your competency as a celestial navigator.
    [Show full text]
  • Mathematics for Celestial Navigation
    Mathematics for Celestial Navigation Richard LAO Port Angeles, Washington, U.S.A. Version: 2018 January 25 Abstract The equations of spherical trigonometry are derived via three dimensional rotation matrices. These include the spherical law of sines, the spherical law of cosines and the second spherical law of cosines. Versions of these with appropri- ate symbols and aliases are also provided for those typically used in the practice of celestial navigation. In these derivations, surface angles, e.g., azimuth and longitude difference, are unrestricted, and not limited to 180 degrees. Additional rotation matrices and derivations are considered which yield further equations of spherical trigonometry. Also addressed are derivations of "Ogura’sMethod " and "Ageton’sMethod", which methods are used to create short-method tables for celestial navigation. It is this author’sopinion that in any book or paper concerned with three- dimensional geometry, visualization is paramount; consequently, an abundance of figures, carefully drawn, is provided for the reader to better visualize the positions, orientations and angles of the various lines related to the three- dimensional object. 44 pages, 4MB. RicLAO. Orcid Identifier: https://orcid.org/0000-0003-2575-7803. 1 Celestial Navigation Consider a model of the earth with a Cartesian coordinate system and an embedded spherical coordinate system. The origin of coordinates is at the center of the earth and the x-axis points through the meridian of Greenwich (England). This spherical coordinate system is referred to as the celestial equator system of coordinates, also know as the equinoctial system. Initially, all angles are measured in standard math- ematical format; for example, the (longitude) angles have positive values measured toward the east from the x-axis.
    [Show full text]
  • Celestial Navigation Practical Theory and Application of Principles
    Celestial Navigation Practical Theory and Application of Principles By Ron Davidson 1 Contents Preface .................................................................................................................................................................................. 3 The Essence of Celestial Navigation ...................................................................................................................................... 4 Altitudes and Co-Altitudes .................................................................................................................................................... 6 The Concepts at Work ........................................................................................................................................................ 12 A Bit of History .................................................................................................................................................................... 12 The Mariner’s Angle ........................................................................................................................................................ 13 The Equal-Altitude Line of Position (Circle of Position) ................................................................................................... 14 Using the Nautical Almanac ............................................................................................................................................ 15 The Limitations of Mechanical Methods ........................................................................................................................
    [Show full text]
  • Bringing Core Seamanship and Navigational Competencies to Your Organization
    Bringing Core Seamanship And Navigational Competencies to Your Organization By Ronald Wisner Copyright Ronald Wisner 2019. All rights reserved. Not for reproduction, transmission, performance or use without express permission of the author. Seamanship How do we define it? Why do we espouse it? Seamanship We can answer these questions with the simple statement: Seamanship…. Is the set of skills and the character traits which makes someone want to go on a boat with you. Do I trust my life and limb with this person? Do I trust my boat with this person? What are Seamanship Skills? 1. Boat handling 2. Decision making 3. Technical skills 4. Navigation As I break down each of these categories, think about how your organization furthers these skills within your membership… 1. Boat handling 2. Decision making 3. Technical skills 4. Navigation 1. Boat handling • Sail setting • Sail trim • Steering (sailing to weather, steering by the compass) • Docking • Line handling 2. Decision making • Judgement • Situational awareness • Experience 3. Technical skills • Marlinspike seamanship: knot tying, splicing, line handling, rigging • Familiarity with boat hardware: shackles, snatch-blocks, cars, winches • Engines 4. Navigation • Chart literacy (reading scales, knowing symbols) • Understanding the compass • How to take bearings • How to plot a position • Navigational aids Can you improvise? The club or organization generally has, as one of their missions, the promotion of seamanship skills The Technology Trap It coddles us, it takes care of us, it gives us easy answers and solutions, it makes us feel smart. The Technology Trap But We Are Losing Basic Literacy • Driving a car (standard shift anyone?) • Reading an analog clock (ask your kid) • Chart and map reading • Using a library • Basic Hand tools Ultimately leading to a loss of self reliance Many young sailors don’t know how to read charts Or even the difference between longitude and latitude The lost art of map reading • Four out of five 18 to 30-year-olds can’t navigate without GPS.
    [Show full text]
  • Bowditch on Cel
    These are Chapters from Bowditch’s American Practical Navigator click a link to go to that chapter CELESTIAL NAVIGATION CHAPTER 15. NAVIGATIONAL ASTRONOMY 225 CHAPTER 16. INSTRUMENTS FOR CELESTIAL NAVIGATION 273 CHAPTER 17. AZIMUTHS AND AMPLITUDES 283 CHAPTER 18. TIME 287 CHAPTER 19. THE ALMANACS 299 CHAPTER 20. SIGHT REDUCTION 307 School of Navigation www.starpath.com Starpath Electronic Bowditch CHAPTER 15 NAVIGATIONAL ASTRONOMY PRELIMINARY CONSIDERATIONS 1500. Definition ing principally with celestial coordinates, time, and the apparent motions of celestial bodies, is the branch of as- Astronomy predicts the future positions and motions tronomy most important to the navigator. The symbols of celestial bodies and seeks to understand and explain commonly recognized in navigational astronomy are their physical properties. Navigational astronomy, deal- given in Table 1500. Table 1500. Astronomical symbols. 225 226 NAVIGATIONAL ASTRONOMY 1501. The Celestial Sphere server at some distant point in space. When discussing the rising or setting of a body on a local horizon, we must locate Looking at the sky on a dark night, imagine that celes- the observer at a particular point on the earth because the tial bodies are located on the inner surface of a vast, earth- setting sun for one observer may be the rising sun for centered sphere. This model is useful since we are only in- another. terested in the relative positions and motions of celestial Motion on the celestial sphere results from the motions bodies on this imaginary surface. Understanding the con- in space of both the celestial body and the earth. Without cept of the celestial sphere is most important when special instruments, motions toward and away from the discussing sight reduction in Chapter 20.
    [Show full text]
  • Celestial Navigation
    A Short Guide to Celestial Navigation Copyright © 1997-2006 Henning Umland All Rights Reserved Revised April 10 th , 2006 First Published May 20 th , 1997 Legal Notice 1. License and Ownership The publication “A Short Guide to Celestial Navigation“ is owned and copyrighted by Henning Umland, © Copyright 1997-2006, all rights reserved. I grant the user a free nonexclusive license to download said publication for personal or educational use as well as for any lawful non-commercial purposes provided the terms of this agreement are met. To deploy the publication in any commercial context, either in-house or externally, you must obtain a written permission from the author. 2. Distribution You may distribute copies of the publication to third parties provided you do not do so for profit and each copy is unaltered and complete. Extracting graphics and/or parts of the text is prohibited. 3. Warranty Disclaimer I provide the publication to you "as is" without any express or implied warranties of any kind including, but not limited to, any implied warranties of fitness for a particular purpose. I do not warrant that the publication will meet your requirements or that it is error-free. You assume full responsibility for the selection, possession, and use of the publication and for verifying the results obtained by application of formulas, procedures, diagrams, or other information given in said publication. January 1, 2006 Henning Umland Felix qui potuit boni fontem visere lucidum, felix qui potuit gravis terrae solvere vincula. Boethius Preface Why should anybody still practice celestial navigation in the era of electronics and GPS? One might as well ask why some photographers still develop black-and-white photos in their darkroom instead of using a high-color digital camera.
    [Show full text]
  • Chapter 15 Navigational Astronomy
    CHAPTER 15 NAVIGATIONAL ASTRONOMY PRELIMINARY CONSIDERATIONS 1500. Definition ing principally with celestial coordinates, time, and the apparent motions of celestial bodies, is the branch of as- Astronomy predicts the future positions and motions tronomy most important to the navigator. The symbols of celestial bodies and seeks to understand and explain commonly recognized in navigational astronomy are their physical properties. Navigational astronomy, deal- given in Table 1500. Table 1500. Astronomical symbols. 225 226 NAVIGATIONAL ASTRONOMY 1501. The Celestial Sphere server at some distant point in space. When discussing the rising or setting of a body on a local horizon, we must locate Looking at the sky on a dark night, imagine that celes- the observer at a particular point on the earth because the tial bodies are located on the inner surface of a vast, earth- setting sun for one observer may be the rising sun for centered sphere. This model is useful since we are only in- another. terested in the relative positions and motions of celestial Motion on the celestial sphere results from the motions bodies on this imaginary surface. Understanding the con- in space of both the celestial body and the earth. Without cept of the celestial sphere is most important when special instruments, motions toward and away from the discussing sight reduction in Chapter 20. earth cannot be discerned. 1502. Relative And Apparent Motion 1503. Astronomical Distances Celestial bodies are in constant motion. There is no Consider the celestial sphere as having an infinite radi- fixed position in space from which one can observe abso- us because distances between celestial bodies are lute motion.
    [Show full text]
  • Longitude by the Method of Lunar Distance
    Longitude by the Method of Lunar Distance By Wendel Brunner, PhD, MD, Mar 21, 2005 Finding longitude at sea without a chronometer by the method of lunar distance is the most difficult and complex aspect of celestial navigation. It is unlikely, in the 21st century, that you would ever have to rely on this method to find your way home. However, a celestial navigator who theroughly understands “lunars” will also have a sound grasp of all aspects of celestial navigation, and will be comfortable with any celestial navigation problem encountered. And for the celestial enthusiast, there is no greater satisfaction than dispensing with the chronometer and successfully finding a longitude at sea through careful lunar observation. This article provides a concise but complete description of the lunar distance method, suitable for anyone who has basic familiarity with the use of a sextant to find position by celestial navigation. The only tools required are a good quality metal sextant (the plastic ones are not adequate for lunars), the Nautical Almanac, and an inexpensive scientific calculator. The iterative approach to the calculations used here simplifies the observation requirements, and allows concentration on carefully determining the Lunar Distance. The article is divided into separate sections, which cover each aspect of the lunar distance problem: 1. A Short History of Lunars 2. Measuring the Observed Lunar Distance (OLD) 3. Averaging Multiple Observations to Reduce Error 4. A Brief Trigonometric Interlude 5. The Spherical Triangle and Calculator Methods 6. Calculating the OLD for a Known Position and Time 7. Determining GMT and Longitude at Sea from Observation of the Lunar Distance (OLD) 8.
    [Show full text]
  • Celestial Navigation Standard
    Effective Date: 22 February 2021 CELESTIAL NAVIGATION STANDARD Standard Description A classroom-based course leading to this standard teaches navigation by taking sights on heavenly bodies. Topics covered include positioning using the sun, the moon and selected stars. Methodology taught is H.O. 249. Objective To be able to demonstrate the celestial navigation theory required to safely navigate a sailing cruiser on an offshore passage. The Standard is applied practically in the Offshore Cruising Standard. Prerequisites Sail Canada Intermediate Coastal Navigation standard or Sail Canada Coastal Navigation Standard. Ashore Knowledge The candidate must be able to: 1. a) Convert longitude into time b) Convert standard time and zone time to GMT and vice versa c) Calculate the zone time for a given longitude, and d) Calculate the chronometer (or watch) error given a previous error and the daily rate 2. Apply the corrections for index error, dip of the horizon, and total correction to convert sextant altitudes of the sun, stars, planets, and moon to true altitude; 3. Calculate the time of meridian passage of the sun and calculate the boat’s latitude from the observed meridian altitude of the sun; 4. Determine the latitude at twilight by means of the Pole Star; 5. Solve the navigational triangle by means of navigation tables (electronic calculators may be used as a supplementary method only); 6. Plot celestial lines of position on a Mercator projection or on an appropriate plotting sheet; 7. Calculate the times (ship’s time and GMT) of sunrise, sunset and twilight; 8. Determine the approximate azimuths and altitudes of the navigational stars and planets at twilight; 9.
    [Show full text]
  • Chapter 16 Time
    CHAPTER 16 TIME TIME IN NAVIGATION 1600. Apparent and Solar Time makes a similar measurement of the transit of the Sun, the resulting time interval would be about four minutes longer The Earth's rotation on its axis presents the Sun and than the period determined by the Earth's rotation. This is other celestial bodies to appear to proceed across the sky due to the Earth's yearly orbital motion around the Sun, from east to west each day. If a navigator measures the time which continuously changes the apparent position of the interval between two successive transits across the local Sun against the background of stars, traditionally observed meridian of a very distant star by the passage of time against another physical time reference such as a chronom- as the cyclical procession of the zodiac. Thus, during the eter, he or she would be measuring the period of the Earth's course of a day, the Sun appears to lag a little to the east rotation. with respect to the stars, and the Earth's rotation must ex- In the most practical sense, the Earth’s rotation is the ceed a complete rotation (360°) in order to have the Sun navigator's standard of time. When the navigator then appear overhead at the local meridian. Figure 1600a. Apparent eastward movement of the Sun with respect to the stars. Apparent eastward lag of the Sun with diurnal ob- ground of stars. servations - when the Sun is on the observer's meridian at The apparent positions of individual stars against the point A in the Earth's orbit around the Sun (see Figure celestial background are commonly determined with refer- 1600a), it will not be on the observer's meridian after the ence to an imaginary point called the vernal equinox.
    [Show full text]
  • Chapter 4 Copyright © 1997-2004 Henning Umland All Rights Reserved
    Chapter 4 Copyright © 1997-2004 Henning Umland All Rights Reserved Finding One's Position (Sight Reduction) Lines of Position Any geometrical or physical line passing through the observer's (still unknown) position and accessible through measurement or observation is called a line of position or position line , LOP . Examples are circles of equal altitude, meridians, parallels of latitude, bearing lines (compass bearings) of terrestrial objects, coastlines, rivers, roads, railroad tracks, power lines, etc. A single position line indicates an infinite series of possible positions. The observer's actual position is marked by the point of intersection of at least two position lines, regardless of their nature. A position thus found is called fix in navigator's language. The concept of the position line is essential to modern navigation. Sight Reduction Finding a line of position by observation of a celestial object is called sight reduction . Although some background in mathematics is required to comprehend the process completely, knowing the basic concepts and a few equations is sufficient for most practical applications. The geometrical background (law of cosines, navigational triangle) is given in chapter 10 and 11. In the following, we will discuss the semi-graphic methods developed by Sumner and St. Hilaire . Both methods require relatively simple calculations only and enable the navigator to plot lines of position on a navigation chart or plotting sheet (see chapter 13). Knowing altitude and GP of a body, we also know the radius of the corresponding circle of equal altitude (our line of position) and the location of its center. As mentioned in chapter 1 already, plotting circles of equal altitude on a chart is usually impossible due to their large dimensions and the distortions caused by map projection.
    [Show full text]
  • Navigation of the James Caird on the Shackleton Expedition
    Records of the Canterbury Museum, 2018 Vol. 32: 23–66 Canterbury Museum 2018 23 Navigation of the James Caird on the Shackleton Expedition Lars Bergman1, George Huxtable2†, Bradley R Morris3, Robin G Stuart4 1Saltsjöbaden, Sweden 2Southmoor, Abingdon, Oxon, UK; † Deceased 3Manorville, New York, USA 4Valhalla, New York, USA Email: [email protected] In 1916, Frank A Worsley famously navigated the 22½ foot (6.9 m) James Caird from Elephant Island to South Georgia Island on a mission to seek rescue for the other 22 men of the Shackleton Expedition. The 800 nautical mile (1,500 km) journey remains one of history’s most remarkable feats of seamanship in a small boat on treacherous seas. The contents of the original log book of the voyage, housed at Canterbury Museum in Christchurch, New Zealand, have been interpreted. Photographic images of the navigational log book are provided along with a transcription that allows all characters to be read. The numbers appearing in the log have been independently recomputed and the navigation principles and procedures used to obtain them explained in detail. Keywords: celestial navigation, chronometer, dead reckoning, Elephant Island, Ernest Shackleton, Frank Worsley, Imperial Trans-Antarctic Expedition, James Caird, noon sight, time sight, South Georgia Introduction The Imperial Trans-Antarctic Expedition as a possible destination where help could be of 1914 under the command of Sir Ernest sought. With the Antarctic winter approaching, Shackleton consisted of 28 men and planned to Shackleton, Worsley and four others set off on cross the Antarctic continent from the Weddell 24 April in the 22½ foot (6.9 m) James Caird to the Ross Sea via the South Pole.
    [Show full text]