DOCSLIB.ORG

Meridian Distance.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Meridian Distance.Pdf MERIDIAN DISTANCE R.E.Deakin School of Mathematical & Geospatial Sciences, RMIT University, GPO Box 2476V, MELBOURNE, Australia email: [email protected] March 2006 ABSTRACT These notes provide a detailed derivation of the equations for computing meridian distance on an ellipsoid of revolution given ellipsoid parameters and latitude. Computation of meridian distance is a requirement for geodetic calculations on the ellipsoid, in particular, computations to do with conversion of geodetic coordinates φλ, (latitude, longitude) to Universal Transverse Mercator (UTM) projection coordinates E,N (East,North) that are used in Australia for survey coordination. The "opposite" transformation, E,N to φλ, requires latitude given meridian distance and these note show how series equations for meridian distances are "reversed" to give series equations for latitude. The derivation of equations follow methods set out in Lehrbuch Der Geodäsie (Baeschlin, 1948), Handbuch der Vermessungskunde (Jordan/Eggert/Kneissl, 1958), Geometric Geodesy (Rapp, 1982) and Geodesy and Map Projections (Lauf, 1983) and some of the formula derived are used in the Geocentric Datum of Australia Technical Manual (ICSM, 2002) – an on-line reference manual available from Geoscience Australia. An understanding of the methods introduced in the following pages, in particular the solution of elliptic integrals by series expansion and reversion of a series, will give the student an insight into other geodetic calculations. MATLAB functions for computing (i) meridian distance given latitude (function mdist.m) and (ii) latitude given meridian distance (function latitude.m) are also given. Meridian Distance.doc 1 INTRODUCTION For an ellipsoid defined by the parameter pairs ()af, , semi-major axis and flattening or (ae, 2 ) , semi-major axis and first-eccentricity squared or (ab, ), semi-major and semi-minor axes, let m be the length of an arc of the meridian from the equator to a point in latitude φ then dm= ρφ d (1) where ρ is the radius of curvature in the meridian plane and ae(11−−22) ae( ) ρ ==3 3 (2) (1sin−e22φ)2 W where the first-eccentricity squared e2 and W are ab22− eff2 ==−()2 a2 ab− f = (3) a 1 We=−(1sin22φ)2 Alternatively, the radius of curvature in the meridian plane is also given by ac2 ρ ==3 3 (4) be(1cos+ ′22φ)2 V where the polar radius c, the second-eccentricity squared e′2 and V are aa2 c == bf1 − ab22− ff()2 − e′2 == (5) b2 ()1 − f 2 1 Ve=+(1cos′22φ)2 Substituting the first group of equations [equations (2) and (3)] into equation (1) lead to series formula for the meridian distance m as a function of latitude φ and powers of e2 . Substituting the second group, [(4) and (5)] lead to series formula for m as a function of φ ab− and powers of another ellipsoid constant n = . Series formula for m involving powers ab+ of e2 are more commonly found in the geodetic literature but as will be shown in the following sections, series formula for m involving powers of n are more compact and give identical results at the level of the 5th decimal place of a metre. A further advantage of Meridian Distance.doc 2 the series formula involving powers of n, is that they are easier to "reverse", i.e., given m as a function of latitude φ and powers of n develop a series formula (by reversion of a series) that gives φ as a function m. This is very useful in the conversion of UTM projection coordinates E,N to geodetic coordinates φλ, . MERIDIAN DISTANCE AS A SERIES FORMULA IN POWERS OF e2 Using equations (1), (2) and (3) the meridian distance is given by the integral φφφ2 ae(1 − ) 1 − 3 mdaedaeed==−=−−φφ()111sin2222()( )2 φ (6) ∫∫∫WW33 000 This is an elliptic integral of the second kind that cannot be evaluated directly; instead, 1 − 3 the integrand =−(1sine22φ) 2 is expanded in a series and then evaluated by term- W 3 by-term integration. 1 − 3 The integrand =−(1sine22φ) 2 can be expanded by use of the binomial series W 3 ∞ ⎛⎞β ()1 +=xxβ ⎜ ⎟ n (7) ∑ ⎜n⎟ n =0 ⎝⎠⎜ ⎟ An infinite series where n is a positive integer, β is any real number, −<11x < and the ⎛⎞β β ⎜ ⎟ binomial coefficients Bn = ⎜ ⎟ are given by ⎝⎠⎜n⎟ ⎛⎞β Γ+(β 1) β ⎜ ⎟ Bn ==⎜ ⎟ (8) ⎝⎠⎜n⎟ Γ+Γ−+()nn11()β 1 In equation (8) the gamma function Γ satisfies Γ=( 2 ) π with Γ+=Γ(βββ1) ( ) for all β ≠−−0, 1, 2, and for integer values of n, Γ+=(nn1!) with 0 != 1 . In the case where β is a positive integer, say k, and −<1x < 1, the binomial series (7) can be expressed as the finite sum k ⎛⎞k ()1 +=xxk ⎜ ⎟ n (9) ∑ ⎜n⎟ n =0 ⎝⎠⎜ ⎟ ⎛⎞k k ⎜ ⎟ where the binomial coefficients Bn = ⎜ ⎟ in series (9) are given by ⎝⎠⎜n⎟ Meridian Distance.doc 3 ⎛⎞k k ! k ⎜ ⎟ Bn ==⎜ ⎟ (10) ⎝⎠⎜n⎟ nk!!()− n ⎛⎞− 3 − 3 2 2 ⎜ ⎟ The binomial coefficients Bn = ⎜ ⎟ for the series (7) are given by equation (8) as ⎝⎠⎜ n ⎟ 1 − 3 Γ−( ) B 2 = 2 with the following results for n = 0,1,2 and 3 n 1 nn! Γ−( 2 − ) 1 − 3 Γ−( ) n = 0 B 2 ==2 1 0 1 0!Γ−( 2 ) 133 − 3 Γ−( ) − Γ−( ) 3 n = 1 B 2 ==222 =− 1 33 1!Γ−( 22) 1! Γ−( ) 2 133355 3 Γ−( ) − Γ−( ) ( −)( −) Γ−( ) − 2 22222215 n = 2 B == = = 2 55 5 2!Γ−( 22) 2! Γ−( ) 2! Γ−( 2) 8 1 3577 − 3 Γ−( ) (−−−Γ−222)( )( ) ( 2) 105 n = 3 B 2 ==2 =− 3 77 3!Γ−( 22) 3! Γ−( ) 48 ⎛⎞− 3 − 3 2 2 ⎜ ⎟ Inspecting the results above, we can see that the binomial coefficients Bn = ⎜ ⎟ are a ⎝⎠⎜ n ⎟ sequence of the form 3 35⋅ 357 ⋅⋅ 3579 ⋅⋅⋅ 357911 ⋅⋅⋅⋅ 1,−− , , , , − , 2 24⋅ 246 ⋅⋅ 2468 ⋅⋅⋅ 246810 ⋅⋅⋅⋅ Using these coefficients gives (Baeschlin 1948, p.48; Jordan/Eggert/Kneissl 1958, p.75; Rapp 1982, p.26) 1−3 3 35⋅⋅⋅ 357 =−()1eeee22 sinφφφφ2 =+ 1 22 sin + 44 sin + 66 sin W 3 2 24⋅⋅⋅ 246 3579⋅⋅⋅ 357911 ⋅⋅⋅⋅ ++ee8sin 8φφ 10 sin 10 + (11) 2468⋅⋅⋅ 246810 ⋅⋅⋅⋅ To simplify this expression, and make the eventual integration easier, the powers of sin φ can be expressed in terms of multiple angles using the standard form 11⎛⎞2n ()− n ⎪⎧⎛⎞222nnn ⎛⎞ ⎛⎞ 2n ⎜⎪⎟ ⎪⎜⎜⎜⎟⎟⎟ sinφφφφ=+221nn⎜ ⎟ − ⎨ ⎟⎟⎟ cos() 2nnn −−−+− 0 cos () 2 2 cos() 2 4 22⎜ n ⎟ ⎪⎜⎜⎜012⎟⎟⎟ ⎝⎠ ⎩⎪⎝⎠ ⎝⎠ ⎝⎠ ⎛⎞22nn ⎛ ⎞ ⎪⎫ ⎜⎜⎟⎟n ⎪ −−+−⎜⎜⎟⎟cos() 2n 6φφ () 1 cos 2 ⎬ (12) ⎜⎜31⎟⎟n − ⎪ ⎝⎠ ⎝ ⎠ ⎭⎪ ⎛⎞2n 2n ⎜ ⎟ Using equation (12) and the binomial coefficients Bn = ⎜ ⎟ computed using equation ⎝⎠⎜ n ⎟ (10) gives Meridian Distance.doc 4 11 sin2 φφ=− cos 2 22 31 1 sin4 φφφ=+ cos 4 − cos 2 88 2 51 3 15 sin6 φφφφ=−+− cos 6 cos 4 cos 2 16 32 16 32 35 1 1 7 7 sin8 φφφφφ=+ cos 8 − cos 6 + cos 4 − cos 2 128 128 16 32 16 63 1 5 45 15 105 sin10 φ=− cos10 φφφφφ + cos 8 − cos 6 + cos 4 − cos 2 (13) 256 512 256 512 64 256 Substituting equations (13) into equation (11) and arranging according to cos 2φ , cos 4φ , etc, we obtain (Baeschlin 1948, p.48; Jordan/Eggert/Kneissl 1958, p.75; Rapp 1982, p.27) 1 − 3 =−(1eABCDEF22 sinφφφφφφ) 2 =− cos 2 + cos 4 − cos 6 + cos 8 − cos10 +(14) W 3 where the coefficients A, B, C, etc., are 3 45 175 11025 43659 Aeeee=+1 24 + + 6 + 8 + e 10 + 4 64 256 16384 65536 3 15 525 2205 72765 Beeee=++++24 6 8 e 10 + 4 16 512 2048 65536 15 105 2205 10395 Ceeee=++++46 8 10 64 256 4096 16384 (15) 35 315 31185 Deee=+++68 10 512 2048 131072 315 3465 Ee=+8 e10 + 16384 65536 693 Fe=+10 131072 Substituting equation (14) into equation (6) gives the meridian distance as φ ma=−()1 e2 ∫ {} AB − cos 2φφφφ + C cos 4 − D cos 6 + E cos 8 − F cos10 φφ + d 0 x sin ax Integrating term-by-term using the standard integral result cosax dx = gives the ∫ a 0 meridian distance m from the equator to a point in latitude φ as BC DE F ma=−()1 eA2 φφφφφ − sin 2 + sin 4 − sin 6 + sin 8 − sin 10 φ + (16) { 246810 } Meridian Distance.doc 5 where φ is in radians and the coefficients A, B, C, etc., are given by equations (15) For the Geodetic Reference System 1980 (GRS80) ellipsoid (af==6378137, 1 298.257222101) the coefficients A, B, C, etc., have the numeric values A = 1.005 052 501 813 087 B = 0.005 063 108 622 224 C = 0.000 010 627 590 263 (17) D = 0.000 000 020 820 379 E = 0.000 000 000 039 324 F = 0.000 000 000 000 071 Multiplying each of the coefficients in equation (16) by ae(1 − 2 ) and for latitude φ in degrees, meridian distance on the GRS80 ellipsoid can be expressed as m =−111 132.952 546 998φφ 16 038.508 741 268 sin 2 + 16.832 613 327 sin 4φ − 0.021 984 374 sin 6φ + 0.000 031 142 sin 8φ − 0.000 000 045 sin 10φ (18) 5π and the meridian distance at latitude φ ==50 radians is 18 m50 = 5 540 847.041 560 963 metres (19) From equation (16), the quadrant distance Q, the meridian distance from the equator to the pole, for the GRS80 ellipsoid is 2 1 Qa=−(1 eA) ( 2 π) = 10 001 965.729 229 864 metres (20) Equation (16) may be simplified by multiplying the coefficients by (1 −e2 ) and expressing the meridian distance as maAA=−{ 02φφsin 2 + A 4 sin 4 φφφ − A 6 sin 6 + A 8 sin 8 − A 10 sin 10 φ +} (21) Meridian Distance.doc 6 B C where AeA=−(1 2 ) , Ae=−(1 2 ) , Ae=−()1 2 , etc., and 0 2 2 4 4 1 3 5 175 441 Aeeee=−1 24 − − 6 − 8 − e 10 + 0 4 64 256 16384 65536 3⎛⎞24 1 15 6 35 8 735 10⎟ Aeee2 =++++⎜ e e +⎟ 8⎝⎠⎜ 4 128 512 16384 ⎟ 15⎛⎞46 3 35 8 105 10⎟ Aeeee4 =++++⎜ ⎟ 256⎝⎠⎜ 4 64 256 ⎟ (22) 35⎛⎞68 5 315 10⎟ Aeee6 =+++⎜ ⎟ 3072⎝⎠⎜ 4 256 ⎟ 315⎛⎞810 7 ⎟ Aee8 =++⎜ ⎟ 131072⎝⎠⎜ 4 ⎟ 693 Ae=+()10 10 131072 For the GRS80 ellipsoid, the coefficients AAA024,,,etc., have the numeric values A0 = 0.998 324 298 423 043 A2 = 0.002 514 607 124 555 A4 = 0.000 002 639 111 298 (23) A6 = 0.000 000 003 446 837 A8 = 0.000 000 000 004 883 A10 = 0.000 000 000 000 071 Multiplying each of the coefficients in equation (21) by a and for latitude φ in degrees, meridian distance m on the GRS80 ellipsoid can be expressed as m =−111 132.952 547 005φφ 16 038.508 741 587 sin 2 + 16.832 613 418 sin 4φ − 0.021 984 397 sin 6φ + 0.000 031 145 sin 8φ − 0.000 000 453 sin 10φ (24) 5π and the meridian distance at latitude φ ==50 radians is 18 m50 = 5 540 847.041 560 711 metres From equation (21), the quadrant distance Q on the GRS80 ellipsoid is 1 QaA==0 ( 2 π) 10 001 965.729 230 469 metres Meridian Distance.doc 7 Inspection of equations (18) and (24) for the meridian distance m and the values for the distances m50 and the quadrant distances Q computed from equations (16) and (21) show that for all practical purposes these equations [(16) and (21)] give identical results at the 5th decimal place of a metre.
Load more

Recommended publications
  • So, What Is Actually the Distance from the Equator to the Pole? – Overview of the Meridian Distance Approximations
    the International Journal Volume 7 on Marine Navigation Number 2 http://www.transnav.eu and Safety of Sea Transportation June 2013 DOI: 10.12716/1001.07.02.14 So, What is Actually the Distance from the Equator to the Pole? – Overview of the Meridian Distance Approximations A. Weintrit Gdynia Maritime University, Gdynia, Poland ABSTRACT: In the paper the author presents overview of the meridian distance approximations. He would like to find the answer for the question what is actually the distance from the equator to the pole ‐ the polar distance. In spite of appearances this is not such a simple question. The problem of determining the polar distance is a great opportunity to demonstrate the multitude of possible solutions in common use. At the beginning of the paper the author discusses some approximations and a few exact expressions (infinite sums) to calculate perimeter and quadrant of an ellipse, he presents convenient measurement units of the distance on the surface of the Earth, existing methods for the solution of the great circle and great elliptic sailing, and in the end he analyses and compares geodetic formulas for the meridian arc length. 1 INTRODUCTION navigational receivers and navigational systems (ECDIS and ECS [Weintrit, 2009]) suggest the Unfortunately, from the early days of the necessity of a thorough examination, modification, development of the basic navigational software built verification and unification of the issue of sailing into satellite navigational receivers and later into calculations for navigational systems and receivers. electronic chart systems, it has been noted that for the The problem of determining the distance from the sake of simplicity and a number of other, often equator to the pole is a great opportunity to incomprehensible reasons, this navigational software demonstrate the multitude of possible solutions in is often based on the simple methods of limited common use.
    [Show full text]
  • From Lacaille to Gill and the Start of the Arc of the 30 Meridian
    From LaCaille to Gill and the start of the Arc of the 30th Meridian Jim R. SMITH, United Kingdom Key words: LaCaille. Maclear. Gill. Meridian Arcs. 30th Arc. SUMMARY If one looks at a map of European arcs of meridian and parallel at the beginning of the 20th century there will be seen to be a plethora them. Turning to Africa there was the complete opposite. Other than the arcs of Eratosthenes (c 300 BC) and the short arcs by LaCaille (1752) and Maclear (1841-48) the continent was empty. It was in 1879 that David Gill had the idea for a Cape to Cairo meridian arc but 1954 before it was completed. This presentation summaries the work of LaCaille and Maclear and then concentrates on a description of the 30th Meridian Arc in East Africa. The only other comparable arcs at the time were that through the centre of India by Lambton and Everest observed between 1800 and 1843; the Struve arc of 1816 to 1852 and the various arcs in France during the 18th century. The usefulness of such arcs was highlighted in 2005 with the inscription by UNESCO of the Struve Geodetic Arc on the World Heritage Monument list. A practical extension to the Struve Arc Monument is the 30th Meridian Arc since there is a connection between the two. At an ICA Symposium in Cape Town, 2003, Lindsay Braun gave a presentation that detailed the political machinations of the history of the 30th Arc; here it is hoped to fill in other aspects of this work including some facts and figures since that is what surveyors thrive upon.
    [Show full text]
  • Geodesy Methods Over Time
    Geodesy methods over time GISC3325 - Lecture 4 Astronomic Latitude/Longitude • Given knowledge of the positions of an astronomic body e.g. Sun or Polaris and our time we can determine our location in terms of astronomic latitude and longitude. US Meridian Triangulation This map shows the first project undertaken by the founding Superintendent of the Survey of the Coast Ferdinand Hassler. Triangulation • Method of indirect measurement. • Angles measured at all nodes. • Scaled provided by one or more precisely measured base lines. • First attributed to Gemma Frisius in the 16th century in the Netherlands. Early surveying instruments Left is a Quadrant for angle measurements, below is how baseline lengths were measured. A non-spherical Earth • Willebrod Snell van Royen (Snellius) did the first triangulation project for the purpose of determining the radius of the earth from measurement of a meridian arc. • Snellius was also credited with the law of refraction and incidence in optics. • He also devised the solution of the resection problem. At point P observations are made to known points A, B and C. We solve for P. Jean Picard’s Meridian Arc • Measured meridian arc through Paris between Malvoisine and Amiens using triangulation network. • First to use a telescope with cross hairs as part of the quadrant. • Value obtained used by Newton to verify his law of gravitation. Ellipsoid Earth Model • On an expedition J.D. Cassini discovered that a one-second pendulum regulated at Paris needed to be shortened to regain a one-second oscillation. • Pendulum measurements are effected by gravity! Newton • Newton used measurements of Picard and Halley and the laws of gravitation to postulate a rotational ellipsoid as the equilibrium figure for a homogeneous, fluid, rotating Earth.
    [Show full text]
  • Great Elliptic Arc Distance1
    GREAT ELLIPTIC ARC DISTANCE1 R. E. Deakin School of Mathematical & Geospatial Sciences, RMIT University, GPO Box 2476V, MELBOURNE VIC 3001, AUSTRALIA email: [email protected] January 2012 ABSTRACT These notes provide a detailed derivation of series formula for (i) Meridian distance M as a function of latitude φ and the ellipsoid constant eccentricity-squared e2, and φ and the ellipsoid constant third flattening n; (ii) Rectifying latitude µ as functions of φ,e2 and φ,n; and (iii) latitude φ as functions of µ,e2 and µ,n These series can then be used in solving the direct and inverse problems on the ellipsoid (using great elliptic arcs) and are easily obtained using the Computer Algebra System Maxima. In addition, a detailed derivation of the equation for the great elliptic arc on an ellipsoid is provided as well as defining the azimuth and the vertex of a great elliptic. And to assist in the solution of the direct and inverse problems the auxiliary sphere is introduced and equations developed. INTRODUCTION In geodesy, the great elliptic arc between P1 and P2 on the ellipsoid is the curve created by intersecting the ellipsoid with the plane containing P1 , P2 and O (the centre of the ellipsoid) and these planes are great elliptic planes or sections. Figure 1 shows P on the great elliptic arc between P1 and P2 . P is the geocentric latitude of P and P is the longitude of P. There are an infinite number of planes that cut the surface of the ellipsoid and contain the chord PP12 but only one of these will contain the centre O.
    [Show full text]
  • The Meridian Arc Measurement in Peru 1735 – 1745
    The Meridian Arc Measurement in Peru 1735 – 1745 Jim R. SMITH, United Kingdom Key words: Peru. Meridian. Arc. Triangulation. ABSTRACT: In the early 18th century the earth was recognised as having some ellipsoidal shape rather than a true sphere. Experts differed as to whether the ellipsoid was flattened at the Poles or the Equator. The French Academy of Sciences decided to settle the argument once and for all by sending one expedition to Lapland- as near to the Pole as possible; and another to Peru- as near to the Equator as possible. The result supported the view held by Newton in England rather than that of the Cassinis in Paris. CONTACT Jim R. Smith, Secretary to International Institution for History of Surveying & Measurement 24 Woodbury Ave, Petersfield Hants GU32 2EE UNITED KINGDOM Tel. & fax + 44 1730 262 619 E-mail: [email protected] Website: http://www.ddl.org/figtree/hsm/index.htm HS4 Surveying and Mapping the Americas – In the Andes of South America 1/12 Jim R. Smith The Meridian Arc Measurement in Peru 1735-1745 FIG XXII International Congress Washington, D.C. USA, April 19-26 2002 THE MERIDIAN ARC MEASUREMENT IN PERU 1735 – 1745 Jim R SMITH, United Kingdom 1. BACKGROUND The story might be said to begin just after the mid 17th century when Jean Richer was sent to Cayenne, S. America, to carry out a range of scientific experiments that included the determination of the length of a seconds pendulum. He returned to Paris convinced that in Cayenne the pendulum needed to be 11 lines (2.8 mm) shorter there than in Paris to keep the same time.
    [Show full text]
  • The History of Geodesy Told Through Maps
    The History of Geodesy Told through Maps Prof. Dr. Rahmi Nurhan Çelik & Prof. Dr. Erol KÖKTÜRK 16 th May 2015 Sofia Missionaries in 5000 years With all due respect... 3rd FIG Young Surveyors European Meeting 1 SUMMARIZED CHRONOLOGY 3000 BC : While settling, people were needed who understand geometries for building villages and dividing lands into parts. It is known that Egyptian, Assyrian, Babylonian were realized such surveying techniques. 1700 BC : After floating of Nile river, land surveying were realized to set back to lost fields’ boundaries. (32 cm wide and 5.36 m long first text book “Papyrus Rhind” explain the geometric shapes like circle, triangle, trapezoids, etc. 550+ BC : Thereafter Greeks took important role in surveying. Names in that period are well known by almost everybody in the world. Pythagoras (570–495 BC), Plato (428– 348 BC), Aristotle (384-322 BC), Eratosthenes (275–194 BC), Ptolemy (83–161 BC) 500 BC : Pythagoras thought and proposed that earth is not like a disk, it is round as a sphere 450 BC : Herodotus (484-425 BC), make a World map 350 BC : Aristotle prove Pythagoras’s thesis. 230 BC : Eratosthenes, made a survey in Egypt using sun’s angle of elevation in Alexandria and Syene (now Aswan) in order to calculate Earth circumferences. As a result of that survey he calculated the Earth circumferences about 46.000 km Moreover he also make the map of known World, c. 194 BC. 3rd FIG Young Surveyors European Meeting 2 150 : Ptolemy (AD 90-168) argued that the earth was the center of the universe.
    [Show full text]
  • B10.Pdf (774.2Kb)
    Page 1 Coordinates Series B, No. 10 The History of Cartography in a Nutshell Persistent URL for citation: http://purl.oclc.org/ coordinates/b10.pdf Vladimiro Valerio Vladimiro Valerio (e-mail: [email protected]) is full Professor of Projective Geometry at the Date of Publication: 06/03/08 Department of History of Architecture at the University IUAV of Venice. His publications include "Cartography in the Kingdom of Naples During the Early Modern Period" in vol. 3 of The History of Cartography (University of Chicago Press). Abstract This is a very short history of cartography. Notes and links to images are included at the end. Keywords: maps, cartography, history, portolan charts, map projections, Claudius Ptolemy, Gerardus Mercator, Fra Mauro, Peutinger Table, C.F. Cassini de Thury Editor’s Note: About five years ago Professor Valerio was asked to prepare a short article on the history of cartography for a multimedia presentation by the Istituto e Museo di Storia della Scienza of Florence. He was astounded to learn that his article could only be thirteen lines long, but he nonetheless complied. The following is a translation of that article. Most of us are familiar with much longer accounts of this subject, such as the multi-volume History of Cartography being published by the University of Chicago Press, or the single volume edited by James R. Akerman and Robert W. Karrow, Jr., Maps: Finding Our Place in the World (Chicago: University of Page 2 Chicago Press, 2007). But, as the Reader’s Digest has taught us, there are few limits to condensation. Prof.
    [Show full text]
  • On the Ancient Determination of the Meridian Arc Length by Eratosthenes of Kyrene
    On the Ancient Determination of the Meridian Arc Length by Eratosthenes of Kyrene Dieter LELGEMANN, Germany Key words: SUMMARY For topographic/geographic purposes the size of the Earth is of utmost importance in modern time as well as for Eratosthenes facing the task to draw a map of the "oikumene". His determination of the radius of the Earth resulted in 252000 stadia. If he has used a stadion definition of 1 stadion = 158,7m = 300 Egyptian royal cubits = 600 Gudea units (length of the yardstick at the statue of Gudea (2300 b c) in the Louvre/Paris), he has already observed the meridian arc length to 252000 ⋅ 0,1587 = 40000km. How could Eratosthenes obtain in ancient times already such an accurate result? Ptolemaios describes in his "Geographike hyphegesis" a method used by the "elder" to determine the size of the Earth; this ancient method to measure the meridian arc length between the latitude circles of two cities (e.g. Alexandria/Syene, Syene/Meroe) is based on a traversing technique, as will be shown. Geographical latitudes could be measured using a "Skiotheron" (shadow seizer). An according to ancient information reconstructed instrument will be shown and explained; the accuracy of sun observations with such a kind of instrument is comparable with those of a modern sextant. A recovery of the two systems of ancient geographical stadia (Alexandrian and Greek) is presented. It is presently used for a rectification of the digitalised maps given in Ptolemy's "Geographike hyphegesis". The stadion definition Eratosthenes has used (1 meridian degree = 700 stadia) was applied also in northern and western Europe and in Asia east of the Tigris river; using it as a scale factor we got very good results for the rectification.
    [Show full text]
  • The International Association of Geodesy 1862 to 1922: from a Regional Project to an International Organization W
    Journal of Geodesy (2005) 78: 558–568 DOI 10.1007/s00190-004-0423-0 The International Association of Geodesy 1862 to 1922: from a regional project to an international organization W. Torge Institut fu¨ r Erdmessung, Universita¨ t Hannover, Schneiderberg 50, 30167 Hannover, Germany; e-mail: [email protected]; Tel: +49-511-762-2794; Fax: +49-511-762-4006 Received: 1 December 2003 / Accepted: 6 October 2004 / Published Online: 25 March 2005 Abstract. Geodesy, by definition, requires international mitment of two scientists from neutral countries, the collaboration on a global scale. An organized coopera- International Latitude Service continued to observe tion started in 1862, and has become today’s Interna- polar motion and to deliver the data to the Berlin tional Association of Geodesy (IAG). The roots of Central Bureau for evaluation. After the First World modern geodesy in the 18th century, with arc measure- War, geodesy became one of the founding members of ments in several parts of the world, and national geodetic the International Union for Geodesy and Geophysics surveys in France and Great Britain, are explained. The (IUGG), and formed one of its Sections (respectively manifold local enterprises in central Europe, which Associations). It has been officially named the Interna- happened in the first half of the 19th century, are tional Association of Geodesy (IAG) since 1932. described in some detail as they prepare the foundation for the following regional project. Simultaneously, Gauss, Bessel and others developed a more sophisticated Key words: Arc measurements – Baeyer – Helmert – definition of the Earth’s figure, which includes the effect Figure of the Earth – History of geodesy – of the gravity field.
    [Show full text]
  • A General Formula for Calculating Meridian Arc Length and Its Application to Coordinate Conversion in the Gauss-Krüger Projection 1
    A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection 1 A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection Kazushige KAWASE Abstract The meridian arc length from the equator to arbitrary latitude is of utmost importance in map projection, particularly in the Gauss-Krüger projection. In Japan, the previously used formula for the meridian arc length was a power series with respect to the first eccentricity squared of the earth ellipsoid, despite the fact that a more concise expansion using the third flattening of the earth ellipsoid has been derived. One of the reasons that this more concise formula has rarely been recognized in Japan is that its derivation is difficult to understand. This paper shows an explicit derivation of a general formula in the form of a power series with respect to the third flattening of the earth ellipsoid. Since the derived formula is suitable for implementation as a computer program, it has been applied to the calculation of coordinate conversion in the Gauss-Krüger projection for trial. 1. Introduction cannot be expressed explicitly using a combination of As is well known in geodesy, the meridian arc elementary functions. As a rule, we evaluate this elliptic length S on the earth ellipsoid from the equator to integral as an approximate expression by first expanding the geographic latitude is given by the integrand in a binomial series with respect to e2 , regarding e as a small quantity, then readjusting with a a1 e2 trigonometric function, and finally carrying out termwise S d , (1) 2 2 3 2 0 1 e sin integration.
    [Show full text]
  • Article Is Part of the Spe- Senschaft, Kunst Und Technik, 7, 397–424, 1913
    IUGG: from different spheres to a common globe Hist. Geo Space Sci., 10, 151–161, 2019 https://doi.org/10.5194/hgss-10-151-2019 © Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License. The International Association of Geodesy: from an ideal sphere to an irregular body subjected to global change Hermann Drewes1 and József Ádám2 1Technische Universität München, Munich 80333, Germany 2Budapest University of Technology and Economics, Budapest, P.O. Box 91, 1521, Hungary Correspondence: Hermann Drewes ([email protected]) Received: 11 November 2018 – Revised: 21 December 2018 – Accepted: 4 January 2019 – Published: 16 April 2019 Abstract. The history of geodesy can be traced back to Thales of Miletus ( ∼ 600 BC), who developed the concept of geometry, i.e. the measurement of the Earth. Eratosthenes (276–195 BC) recognized the Earth as a sphere and determined its radius. In the 18th century, Isaac Newton postulated an ellipsoidal figure due to the Earth’s rotation, and the French Academy of Sciences organized two expeditions to Lapland and the Viceroyalty of Peru to determine the different curvatures of the Earth at the pole and the Equator. The Prussian General Johann Jacob Baeyer (1794–1885) initiated the international arc measurement to observe the irregular figure of the Earth given by an equipotential surface of the gravity field. This led to the foundation of the International Geodetic Association, which was transferred in 1919 to the Section of Geodesy of the International Union of Geodesy and Geophysics. This paper presents the activities from 1919 to 2019, characterized by a continuous broadening from geometric to gravimetric observations, from exclusive solid Earth parameters to atmospheric and hydrospheric effects, and from static to dynamic models.
    [Show full text]
  • Why Does the Meter Beat the Second?
    ROMA 1 - 1394/04 December 2004 Why does the meter beat the second? P. Agnoli1 and G. D’Agostini2 1 ELECOM s.c.r.l, R&D Department, Rome, Italy ([email protected]) 2 Universit`a“La Sapienza” and INFN, Rome, Italy ([email protected], www.roma1.infn.it/˜dagos) Abstract The French Academy units of time and length — the second and the meter — are traditionally considered independent from each other. However, it is a matter of fact that a meter simple pendulum beats virtually the second in each oscillation, a ‘surprising’ coincidence that has no physical reason. We shortly review the steps that led to the choice of the meter as the ten millionth part of the quadrant of the meridian, and rise the suspicion that, indeed, the length of the seconds pendulum was in fact the starting point in establishing the actual length of the meter. 1 Introduction Presently, the unit of length in the International System (SI) [1] is a derived unit, related to the unit of time via the speed of light, assumed to be constant and of exact value c = 299 792 458 m/s.1 The unit of time is related to the period of a well defined electromagnetic wave.2 The proportionality factors that relate conventional meter and second to more fundamental and reproducible quantities were chosen to make these units back compatible to the original units of the Metric System. In that system the second was defined as 1/86400 of the average solar day and the meter was related to the Earth meridian (namely 1/10 000 000 of the distance between the pole and the equator on the Earth’s surface).
    [Show full text]