259
1 INTRODUCTION
Unfortunately, from the early days of the
developmentofthebasic navigationalsoftwarebuilt
into satellite navigational receivers and later into
electronicchartsystems,ithasbeennotedthatforthe
sake of simplicity and a number of other, often
incomprehensiblereasons,this navigationalsoftware
is often based
on the simple methods of limited
accuracy.Itis surprisingthat evennowadays, atthe
beginning of the twenty‐first century, the use of
navigationalsoftwareisstillusedinaloosemanner,
sometimes ignoring basic computational principles
andadoptingoversimplifiedassumptions anderrors
such as the wrong combination of
spherical and
ellipsoidal calculations (while in car navigation
systems – even primitive simple calculations on flat
surfaces) in different steps of the solution of a
particular sailing problem. The lack of official
standardizationonboththe“accuracyrequired”and
theequivalent“methodsemployed”,inconjunctionto
the “black box solutions” provided
by GNSS
navigational receivers and navigational systems
(ECDIS and ECS [Weintrit, 2009]) suggest the
necessity of a thorough examination, modification,
verification and unification of the issue of sailing
calculations for navigational systems and receivers.
The problem of determining the distance from the
equator to the pole is a great opportunity
to
demonstrate the multitude of possible solutions in
commonuse.
2 THEMAINQUESTIONANDFIVETHEBESTAD
HOCANSWERS
Well, let’s put the title question‐what is actually
distance from the Equator to the Pole? And let us
consider what actually answer would we expect?
There will answers simple,
crude, naive, almost
primitive,butalsoverysophisticatedandrefined,full
of mathematics. As it might seem at first glance,
surelytheproblemisnottrivial.
So, What is Actually the Distance from the Equator to
the Pole? – Overview of the Meridian Distance
Approximations
A.Weintrit
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Inthepapertheauthorpresentsoverviewofthemeridiandistanceapproximations.Hewouldlike
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
beginningofthepapertheauthordiscussessomeapproximationsandafewexactexpressions(infinitesums)to
calculateperimeterandquadrantofanellipse,he
presentsconvenientmeasurementunitsofthedistance onthe
surfaceoftheEarth,existingmethodsforthesolutionofthegreatcircleandgreatellipticsailing,andintheend
heanalysesandcomparesgeodeticformulasforthemeridianarclength.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 7
Number 2
June 2013
DOI:10.12716/1001.07.02.14
260
2.1 AnswerNo.1
Itʹsexactly10,000km.Thisisbecausethedefinitionof
a meter is 1 10,000,000th of the distance from the
North Pole to the equator. So itʹs exactly 10,000,000
metersfrom the NorthPole to the equator,which is
exactly10,000km.
2.2 AnswerNo.2
10,002 kilometres. The original definition of a
kilometre was 1/10,000 of the distance from the
equator to the North Pole, but measurements have
improved.
2.3 AnswerNo.3
Easy, there are 90 degrees of distance from the
equator to the North Pole. Each degree has 60
minutes,eachminute=1
nauticalmile,therefore60x
90=5,400nauticalmiles.
2.4 AnswerNo.4
Angle between the equator and North Pole is 90°.
1nautical mile = 1852 meters = 1’; 1° = 60’; just
multiply60x90x1852.Theansweris10,000,800m.
2.5 AnswerNo.5
Ifthequestion
is:whatisthedistancefromtheNorth
Poletotheequatorindegrees?‐theanswerismuch
easier.
Themeasureofacircleindegreesis360degrees.
SothedistancefromPoletoequatorisonequarterof
this;namely,90degrees.
2.6 WhatisImportantinThat
Calculation?
Frankly speaking, all five answers are correct, and
also ... completely wrong. First of all we should
decide what length unit we will use for the
measurement,what model of theEarth will be used
forourcalculations,andtheaccuracyoftheresultwe
expect.
We know already that
the Earth is not a sphere;
therefore our calculations should be a bit more
difficult.Wewillusetheellipsoidofrevolution.Early
literatureusesthetermoblatespheroidtodescribea
sphereʺsquashedatthepolesʺ.Modernliteratureuses
the termʺellipsoid of revolutionʺ although the
qualifyingwordsʺ
ofrevolutionʺareusuallydropped.
Anellipsoidwhichisnotanellipsoidofrevolutionis
calledatri‐axialellipsoid. Spheroidandellipsoidare
usedinterchangeablyinthispaper.Currentlyweuse
to navigate the ellipsoid WGS‐84 (World Geodetic
System 1984). The WGS‐84 meridionalellipse has an
ellipticity
=0.081819191.
Figure1.ParametersoftheellipsoidWGS‐84[Dana,1994]
3 MEASUREMENTOFTHEDISTANEON
SURFACEOFTHEEARTH
We have to decide what unit of measurement we
wouldliketouseformeasuringthedistance:milesor
metres. While the measure of one meter has been
strictly defined, miles seem to be made of chewing
gum.Therearea
lotofdifferentmiles,someofthem
aremeasuresofa fixedlength,suchas:geographical
mile,International NauticalMile(INM), statue mile,
otherofvariablelengthdependentonthelatitudeof
locationofmeasurement,suchas:nauticalmileorsea
mile.
3.1 GeographicalMile
Distancesonthesurfaceof
asphereoranellipsoidof
revolutionareexpressedinanaturalwayinunitsof
thelengthofoneminuteofarc,mea suredalongthe
equator.ThisunitisknownastheGeographicalMile.
Its value is determined by the dimensions of the
spheroid in use. We will use it
throughout in our
treatment of navigational methods. Its length varies
accordingtotheellipsoidwhichisbeingusedasthe
model but, in these units, the radius of the Earth is
fixed at a value of 108,000/π. The length of one
minute of arc of the equator on the
surface of the
WGS‐84ellipsoidisapproximately1,855.3284metres.
3.2 TheInternationalNauticalMile
The international nautical mile was defined by the
First International Extraordinary Hydrographic
Conference, Monaco (1929) as exactly 1852 metres.
Thisistheonlydefinitioninwidespreadcurrentuse,
and is the one accepted by the International
Hydrographic Organization (IHO) and by the
International Bureau of Weights and Measures
(BIPM).Before1929,differentcountrieshaddifferent
definitions, and the United Kingdom, the United
States,theSovietUnionandsomeothercountriesdid
notimmediatelyaccepttheinternationalvalue.
Both the Imperial and U.S. definitions of the
nautical
mile were based on the Clarke (1866)
spheroid: they were different approximations to the
lengthof one minuteof arcalong a great circle of a
sphere having the same surface area as the Clarke
spheroid.TheUnitedStatesnauticalmilewasdefined
261
as 1,853.248 metres (6,080.20 U.S. feet, based on the
definition of the foot in the Mendenhall Order of
1893):itwasabandonedinfavouroftheinternational
nauticalmilein1954.TheImperial(UK)nauticalmile,
also known as the Admiralty mile, was defined in
terms of the knot, such
that one nautical mile was
exactly 6,080 international feet (1,853.184 m): it was
abandoned in 1970 and, for legal purposes, old
references to the obsolete unit are now converted to
1,853metresexactly[Weintrit,2010].
3.3 NauticalMile
A nautical mile is a unit of measurement used on
water by
sailors and/or navigators in shipping and
aviation.Itistheaveragelengthofoneminuteofone
degreealongagreatcircleoftheEarth.Onenautical
mile corresponds to one minute of latitude. Thus,
degrees of latitude are approximately 60 nautical
milesapart.Bycontrast,thedistanceofnauticalmiles
betweendegreesoflongitudeisnotconstantbecause
lines of longitude become closer together as they
convergeatthepoles.
Each country can keep different, arbitrarily
selectedvalueofthenauticalmile,butmostofthem
use the International Nautical Mile, although in the
pastitwasdifferent.
Theunit
usedby the United Kingdomuntil1970
was the British Standard nautical mile of 6,080 ft or
1,853.18m.
Today, one nautical mile still equals exactly the
internationallyagreeduponmeasureof1,852meters
(6,076 feet). One of the most important concepts in
understandingthenauticalmilethoughisitsrelation
tolatitude.
3.4 TheSeaMile
The sea mile is the length of 1 minute of arc,
measured along the meridian, in the latitude of the
position;itslength variesbothwith the latitude and
withthedimensionsofthespheroidinuse.
The sea mile is an ambiguous unit,
with the
followingpossiblemeanings:
InEnglishusage,aseamileis,foranylatitude,the
length of one minute of latitude at that latitude. It
varies from about 1,842.9 metres (6,046 ft) at the
equatortoabout1,861.7metres(6,108ft)atthepoles,
with a mean value of 1,852.3
metres (6,077ft). The
internationalnauticalmilewaschosenastheinteger
numberofmetresclosesttothemeanseamile.
Americanusehaschanged recently.Theglossary
inthe1966editionofBowditchdefinesaʺseamileʺas
aʺnauticalmileʺ.Inthe2002edition[Bowditch,2002],
the
glossarysays:ʺAnapproximatemeanvalueofthe
nautical mile equal to 6,080 feet; the length of a
minuteofarcalongthemeridianatlatitude48°.ʺ
Thesea milehasalsobeendefinedas6,000feetor
1,000 fathoms, for example in Dresnerʹs Units of
Measurement [Dresner, 1971]. Dresner
includes a
remark to the effect that this must not be confused
with the nautical mile. Richard Norwood in The
Seaman’s Practice (1637) determined that 1/60th of a
degreeofanygreatcircleonEarthʹssurfacewas6,120
feet(vs.themodernvalueof6,080feet).However,he
stated:ʺ
ifanymanthinkitmoresafeandconvenient
inSea‐reckoningsʺhemayassign6,000feettoamile,
relyingoncontexttodeterminethetypeofmile.
3.5 TheStatueMile
Thestatuemileistheunitofdistanceof1,760yards
or 5,280 ft) 1609.3 m. The difference
between a mile
andastatutemileishistorical,ratherthanpractical.
Hundredsofyearsamilemeantdifferentthingsto
differentpeople.Itbecamenecessary,eventually,for
a mile to be the same distance for all concerned.
DuringthereignofQueenElizabethI,astatutewas
passed by
the English Parliament that standardized
the measurement of a mile, thus giving rise to the
termʹstatuteʹ mile. The measurement of a mile at
5,280feetisnowaccepted almosteverywhere in the
world.
3.6 HistoryoftheMile
Thenauticalmilewashistoricallydefinedasaminute
of arc along
a meridian of the Earth (North‐South),
makingameridianexactly180×60=10,800historical
nautical miles. It can therefore be used for
approximate measures on a meridian as change of
latitude on a nautical chart. The originally intended
definitionof themetreas 10
−7
of a half‐meridian arc
makes the mean historical nautical mile exactly
(2×10
7
)/10,800 = 1,851.851851… historical metres.
Based on the current IUGG meridian of
20,003,931.4585(standard)metresthemeanhistorical
nauticalmileis1,852.216m.
The historical definition differs from the length‐
based standard inthata minute ofarc, and hence a
nauticalmile,isnotaconstantlengthatthesurface
of
theEarthbutgraduallylengthensinthenorth‐south
direction with increasing distance from the equator,
asacorollaryoftheEarthʹsoblateness,hencetheneed
forʺmeanʺ in the last sentence of the previous
paragraph. This length equals about 1,861 metres at
thepolesand1,843metres
attheEquator.
Other nations had different definitions of the
nautical mile. This variety, in combination with the
complexity of angular measure described aboveand
theintrinsicuncertaintyofgeodeticallyderivedunits,
mitigatedagainsttheextantdefinitionsinfavourofa
simple unit of pure length. International agreement
was achieved in
1929 when the IHB adopted a
definitionofoneinternationalnauticalmileasbeing
equalto1,852metresexactly,inexcellent agreement
(foraninteger)withboththeabove‐mentionedvalues
of1,851.851 historical metres and 1,852.216 standard
metres.
The use of an angle‐based length was first
suggested by
Edmund Gunter (of Gunterʹs chain
fame).Duringthe18thcentury,therelationofamile
of, 6000 (geometric) feet, or a minute of arc on the
earth surface, had been advanced as a universal
measurefor land and sea.The metric kilometrewas
selected to represent a centesimal minute of
arc, on
262
the same basis, with the circle divided into 400
degreesof100minutes.
3.7 HistoryoftheMetricSystem
Thehistoryofmetricsystemisstrictlyconnectedwith
polar distance calculation. The metre (meter in
AmericanEnglish),symbolm,isthefundamentalunit
of length in the International System of
Units (SI).
Originally intended to be one ten‐millionth of the
distancefromtheEarthʹsequatortotheNorthPole(at
sealevel),itsdefinitionhasbeenperiodicallyrefined
to reflect growing knowledge of metrology. Since
1983, it has been defined asʺthe length of the path
travelledby
lightinvacuumduringatimeintervalof
1/299,792,458ofasecondʺ.
TheoriginalʺSacredCubitʺwasaunitofmeasure
equalto25British inches, and also equal to one 10‐
millionthpartofthedistancebetweentheNorthPole
and the center of the Earth. In 1790 Charles
TalleyrandwassenttotheParisAcademyofSciences
inordertohelpestablishanewworldwidesystemof
weights and measures meant to replace the English
system of weights and measures that was in use all
over the world at the time. This new French
measuringsystemwouldbe
baseduponanewunitof
measure known astheʺmeter.ʺ Themeter(from the
Greek wordʺmetronʺ) was designed to be a
counterfeitcubit,equaltoone10‐millionthpartofthe
distancebetweentheNorthPoleandtheEquator:
Cubit = 1/10,000,000th part of distance from N.
Pole
toEarthʹsCenter;
Meter = 1/10,000,000th part of distance from N.
PoletoEarthʹsEquator.
TheoriginalSacredCubitwasalengthequalto25
English inches, or 7ʺhands.ʺ Theʺhandʺ measure is
still used today by people who raise horses, it is a
lengthofjust
under4inches(3.58inchestobeexact),
and is equal to the width of a manʹs hand, not
includingthethumb.
A decimal‐based unit of length, the universal
measureorstandardwasproposedinanessayof1668
bytheEnglishclericandphilosopherJohnWilkins.In
1675,
the Italian scientist Tito Livio Burattini, in his
workMisuraUniversale,usedthephrasemetrocattolico
(lit.ʺcatholic[i.e. universal]measureʺ),derived from
the Greek métron katholikón, to denote the standard
unitoflengthderivedfromapendulum.Inthewake
oftheFrenchRevolution,acommissionorganisedby
the French Academy of Sciences and charged with
determining a single scale for all measures, advised
the adoptionof a decimal system(27 October,1790)
andsuggestedabasicunitoflengthequaltooneten‐
millionthofthedistancebetweentheNorthPoleand
theEquator,tobecalledʹ
measureʹ(mètre)(19thMarch
1791).TheNationalConventionadoptedtheproposal
in1793.Thefirstoccurrenceofmetreinthissensein
Englishdatesto1797.
4 THEFORMULAFORTHEPERIMETEROFAN
ELLIPSE
The problem of calculating the distance from the
equatortothepolebasicallycomesdown
tocalculate
the perimeter of an ellipse and its quadrant. But
rather strangely, the perimeter of an ellipse is very
difficulttocalculate!
Figure2.Ellipseparameter:a‐majoraxis;b–minoraxis
ForanellipseofCartesianequationx
2
/a
2
+y
2
/b
2
=1
witha>b:
aiscalledthemajorradiusorsemimajoraxis,
bistheminorradiusorsemiminoraxis,
the quantity istheeccentricity
oftheellipse,
theunnamedquantityh=(a‐b)
2
/(a+b)
2
oftenpops
up.
There is no simple exact formula to calculate
perimeterofanellipse.Therearesimpleformulasbut
they are not exact, and there are exact formulas but
they are not simple. Here, weʹll discuss many
approximations, and two exact expressions (infinite
sums). There are many formulas,
here are a few
interestingonesonly,butnotall[Michon,2012]:
Approximation1
Thisapproximationwillbewithinabout5%ofthe
truevalue,solongasaisnotmorethan3timeslonger
than b (in other words, the ellipse is not too
ʺsquashedʺ):
22
2
2
ab
p
(1)
Approximation2
It is found in dictionaries and other practical
referencesasasimpleapproximationtotheperimeter
poftheellipse:
2
22
()
2
2
ab
pab
(2)
Approximation3
Anapproximateexpression,forenottoocloseto1,
is:
3
2
p
ab ab
(3)
263
Approximation4
The famous Indian mathematician S. Ramanujan
in1914cameupwiththisbetterapproximation:
333
p
ab aba b
(4)
Approximation5
TheaboveRamanujanformulaisonlyabouttwice
aspreciseasa formulaproposedbyLindnerbetween
1904and1920,whichisobtainedsimplybyretaining
only the first three terms in an exact expansionin
termsofh(thesethreetermshappentoformaperfect
square).
Firstlywemustcalculateʺhʺ:
2
2
()
()
ab
h
ab
(5)
2
()[1/8] pabh
(6)
Approximation6
A better 1914 formula, also due to Ramanujan,
calledRamanujanII,givestheperimeterp:
3
()1
10 4 3
h
pab
h
(7)
Approximation7
R.G.Hudsonistraditionallycreditedforaformula
without square roots which he did not invent and
which is intermediate in precision between the two
Ramanujanformulas.
2
64 3
()
64 16
h
pab
h
(8)
Approximation8
Amore precise Padéapproximantconsists of the
optimizedratiooftwoquadraticpolynomialsofhand
leadstothefollowingformula:
2
2
256 48 21
()
256 1 12 3
hh
pab
hh
(9)
Approximation9
One more popular approximation, Peano’s
formula:
3 (1 )
()
2
h
pab
(10)
InfiniteSeries1
Anexactexpressionoftheperimeterpofanellipse
was first published in 1742 by the Scottish
mathematicianColinMaclaurin.
Thisisanexactformula,butitrequiresanʺinfinite
seriesʺ of calculations to be exact, so in practice we
stillonlygetanapproximation.
Firstlywe mustcalculatee (theʺeccentricityʺ,not
Euler’snumber“e”):
22
ab
e
a
(11)
Thenusethisʺinfinitesumʺformula:
2
2
4
1
2!
21 ·
(2 · !) 2 1
i
i
i
i
e
pa
ii
(12)
whichmaylookcomplicated,butexpandslikethis:
22 2
46
2
11·3 1·3·5
21
2 2·4 3 2·4·6 5
ee
pa e
(13)
The terms continue on infinitely, and
unfortunatelywemustcalculatealotoftermstogeta
reasonablycloseanswer.
InfiniteSeries2
Author’s favourite exact “infinite sum” formula
(becauseitgivesaverycloseanswerafteronlyafew
terms)isasfollows:
2
0
0.5
()
n
n
p
ab h
n
(14)
Note:the
0.5
n
isthebinomialcoefficientwithhalf‐
integerfactorials.
Itmaylookabitscary,butitexpandstothisseries
ofcalculations,nowcalledtheGauss‐Kummerseries
ofh:
23
11 1
()1
464 256
pab h h h
(15)
Themoretermswecalculate,themoreaccurateit
becomes(thenexttermis25h
4
/16384,whichisgetting
quite small, and the next is 49h
5
/65536, then
441h
6
/1048576).
Comparison of the results of calculations done
according to all the methods described above is
showninTable1.
264
Table1.Comparisonofresultsofformulasforperimeterofanellipseanditsquadrant,forparametersaandbofellipsoid
WGS‐84,wherea =6,378,137m,b=6,356,752.3142452m
__________________________________________________________________________________________________
MethodFormulaPerimeterQuadrant
__________________________________________________________________________________________________
Approximation1(1)40,007,891.1205403010,001,972.78013510
Approximation2(2)40,007,862.9172360010,001,965.72930900
Approximation3(3)40,007,862.9172659010,001,965.72931650
Approximation4RamanujanI(4)40,007,862.9172509010,001,965.72931270
Approximation5Lindner(6)40,007,862.9172510010,001,965.72931270
Approximation6RamanujanII(7)40,007,862.9172510010,001,965.72931270
Approximation7Hudson(8)40,007,862.9172609010,001,965.72931520
Approximation8Pade(9)40,007,862.9172509010,001,965.72931270
Approximation9Peano(10)
40,007,862.9172659010,001,965.72931650
InfiniteSeries1Maclaurin(13)40,007,862.9181143010,001,965.72952860
InfiniteSeries2Gauss‐Kummer(15)40,007,862.9172510010,001,965.72931270
__________________________________________________________________________________________________
5 MERIDIANARC
Onany surface which fulfilstherequired continuity
conditions, the shortest path between two points on
thesurfaceisalongthearcofageodesiccurve.Onthe
surfaceofaspherethegeodesiccurvesarethegreat
circlesandtheshortestpathbetweenanytwo
points
onthissurfaceisalongthearcofagreatcircle,buton
thesurfaceofanellipsoidofrevolution,thegeodesic
curves are not so easily defined except that the
equatorof thisellipsoidisa circle andits meridians
areellipses[Williams,1996].
In geodesy, a meridian arc
measurement is a
highlyaccuratedeterminationofthedistancebetween
two points with the same longitude. Two or more
suchdeterminationsatdifferentlocationsthenspecify
the shape of the reference ellipsoid which best
approximatestheshape ofthegeoid.Thisprocess is
called the determination of the figure of
the Earth.
Theearliestdeterminations ofthe sizeof aspherical
Earthrequiredasinglearc.Thelatestdeterminations
useastrogeodetic measurements andthe methods of
satellitegeodesytodeterminethereferenceellipsoids.
5.1 TheEarthasanEllipsoid
High precision land surveys can be used determine
thedistancebetweentwo
placesatʺalmostʺthesame
longitude by measuring a base line and a chain of
triangles (suitable stations for the end points are
rarely at the same longitude). The distanceΔalong
the meridian from one end point to a point at the
same latitude as the second end point is
then
calculatedbytrigonometry.ThesurfacedistanceΔis
reducedtoΔʹ,thecorresponding distanceatmeansea
level. The intermediate distances to points on the
meridianatthesamelatitudesasotherstationsofthe
surveymayalsobecalculated.
The geographic latitudes of both end points,φ
s
(standpoint) andφ
f(forepoint)andpossibly atother
pointsaredeterminedbyastrogeodesy,observingthe
zenith distances of sufficient numbers of stars. If
latitudesaremeasuredatendpointsonly,theradius
ofcurvatureatthemid‐pointofthemeridianarccan
be calculated from R =Δʹ/(|φ
s ‐ φf|). A second
meridian arc will allow the derivation of two
parameters required to specify a reference ellipsoid.
Longerarcswithintermediatelatitudedeterminations
can completely determine the ellipsoid. In practice
multiplearcmeasurementsareusedtodeterminethe
ellipsoidparameters by the methodof least squares.
The parameters determined are
usually the semi‐
majoraxis,a,andeitherthesemi‐minoraxis,b,orthe
inverse flattening
1/
f
, (where the flattening is
/
f
ab a
).
Figure3.Anoblatespheroid(ellipsoid)
5.2 MeridianDistanceontheEllipsoid
Thedeterminationofthemeridiandistancethatisthe
distancefromtheequatortoapointatlatitude
on
theellipsoidisanimportantprobleminthetheoryof
mapprojections,particularlytheTransverseMercator
projection.Ellipsoidsarenormallyspecifiedinterms
of the parameters defined above, a, b,
1/
f
, but in
theoretical work it is useful to define extra
parameters, particularly the eccentricity, e, and the
third flattening n. Only two of these parameters are
independent and there are many relations between
them[Banachowicz,2006]:
2
1/ 2
22
2
, 2 ,
2
4
1 1 , .
1
ab ab f
feffn
aabf
n
ba f a e e
n
(16)
Theradiusofcurvatureisdefinedas
2
3/2
22
1
,
1sin
ae
M
e
(17)
so that the arc length of an infinitesimal element of
the meridian is
M
d
(with
in radians).
Thereforethe meridian distance from the equator to
latitude
is
265
3/2
222
00
11sin .mMdae e d
(18)
The distance from the equator to the pole, the
polardistance,is
/2 .
p
mm
(19)
Theaboveintegralisrelatedtoaspecialcaseofan
incompleteellipticintegralofthethirdkind.
2
1 Πφ,e,e .mae
(20)
Many methods have been used for the
computationoftheintegralofformula(18).Allthese
methodsandformulacanbeusedforthecalculation
ofthedistancealongthegreatellipticarcbyformula
(21).
2
0
3
2
0
2
(1 )
(1 sin )
ae
Md
e
(21)
Equation (21) can be transformed to an elliptic
integral of the second type, which cannot be
evaluated in a closed form. The calculation can be
performed either by numerical integration methods,
suchasSimpson’srule,orbythebinomialexpansion
of the denominator to rapidly converging series,
retention of
a few terms of these series and further
integration by parts. This process yields results like
formula(22).
22 24
0
3315
11 sin2
4832
Mae e e e
(22)
Equation(22)isthestandardgeodeticformulafor
the accurate calculation of the meridian arc length,
whichisproposedinanumberoftextbookssuchasin
Torge’sGeodesyusingupto
sin 2
terms.
According to Snyder [Snyder, 1987] and Torge
[Torge, 2001], Simpson’s numerical integration of
formula(21)doesnotprovidesatisfactoryresultsand
consequently the standard computation methods for
thelengthofthemeridianarcarebasedontheuseof
series expansion formulas, such as formula (22) and
moredetailed
formulaspresentedbelow.
Delambre
The above integral may be approximated by a
truncated series in the square of the eccentricity
(approximately1/150)byexpandingtheintegrandin
abinomialseries.Setting
sins
,
223/2 22 44 66 88
2468
(1 sin φ)1 ,ebesbesbesbes
(23)
where
24 6 8
3 15 35 315
, , .
2 8 16 128
bb b b
Using simple trigonometric identities the powers
of
sin
may be reduced to combinations of factors
of
cos 2
p
. Collecting terms with the same cosine
factorsandintegratinggivesthefollowing series,first
givenbyDelambrein1799.
02 4
68
sin2 sin4
sin 6 sin 8 ,
mAA A
AA
(24)
where:
2246 8
0
3 45 175 11025
11
4 64 256 16384
A
ae e e e e
2
24 6 8
2
1
3 15 525 2205
2 4 16 512 2048
ae
A
ee e e
2
46 8
4
1
15 105 2205
4 64 256 4096
ae
A
ee e
2
68
6
1
35 315
6 512 2048
ae
A
ee
2
8
8
1
315
8 16384
ae
A
e
Thenumericalvaluesforthesemi‐majoraxisand
eccentricityoftheWGS‐84ellipsoidgive,inmetres,
6367449.146 16038.509 sin 2
16.833sin 4 0.022sin 6 0.00003sin 8
m
(25)
The first four terms have been rounded to the
nearestmillimetrewhilsttheeighthordertermgives
risetosub‐millimetre corrections. Tenthorder series
are employed in modernʺwide zoneʺ
implementations of the Transverse Mercator
projection.
For the WGS‐84 ellipsoid the distance from
equatortopoleis
given(inmetres)by
0
1
10 001 965.729 .
2
p
mA m
Thethirdflatteningnisrelatedtotheeccentricity
by
223
2
4
412 3 4 .
1
n
ennnn
n
(26)
Withthissubstitutiontheintegralforthemeridian
distancebecomes
2
3/2
2
0
11
.
12cos2
an n
md
nn
(27)
Thisintegralhas beenexpanded inseveral ways,
allofwhichcanberelatedtotheDelambreseries.
Bessel’sformula
In1837Besselexpandedthisintegralinaseriesof
theform:
266
2
02 4 6
1 1 sin2 sin4 sin6 ,mannDD D D
(28)
where
24 24
04
9 225 15 105
1 , ,
464 1664
Dnn Dnn
35 3 5
26
3 45 525 35 315
, ,
2 16 128 48 256
Dnn n D n n
Sincenisapproximatelyonequarterofthevalue
of the squared eccentricity, the above series for the
coefficientsconverge16timesasfastastheDelambre
series.
Helmert’sformula
In 1880 Helmert extended and simplified the
aboveseriesbyrewriting
2
2
2
1
11 1
1
nn n
n
(29)
andexpandingthenumeratorterms.
02 4 6 8
sin2 sin4 sin6 sin8
1
a
mHHHHH
n
(30)
with
24
3
06
35
1
464 48
nn
HHn
3
4
28
3 315
2 8 512
n
Hn H n
4
2
4
15
16 4
n
Hn
UTM
Despite the simplicity and fast convergence of
HelmertʹsexpansiontheU.S.DMAadoptedthefully
expandedformoftheBesselseriesreportedbyHinks
in 1927. This expansion is important, despite the
poorerconvergenceofseriesinn,becauseitisusedin
thedefinitionofUTM[Bowring,
1983].
02 4 6 8
sin 2 sin 4 sin 6 sin8 ,mBBBBB
(31)
wherethecoefficientsaregiventoordern
5
by
23 4 5
0
5 5 81 81
1,
4 4 64 64
Ba n n n n n
234 5
2
37755
,
28864
Bannnnn
23 4 5
4
15 3 3
,
16 4 4
Bannnn
34 5
6
35 11
,
48 16
Bannn
45
8
315
,
512
Bann
Generalizedseries:
Theaboveseries,toeighthorderineccentricityor
fourthorderinthirdflattening,areadequateformost
practical applications. Each can be written quite
generally. For example, Kazushige Kawase (2009)
derivedfollowinggeneralformula[Kawase,2011]:
2
2
1
1/2
01 1 1
1
1
4sin2
m
m
jj
k
jm
jk m
a
m
n
(32)
where
3
.
2
i
n
n
i
Truncatingthesummationatj=2givesHelmertʹs
approximation.
The polar distance may be approximated by the
ThomasMuirʹsformula:
2/3
/2
3/2 3/2
0
.
22
p
ab
mMd
(33)
6 EXISTINGMETHODSFORTHESOLUTIONOF
THEGREATELLIPTICSAILING
6.1 BowringMethodfortheDirectandInverseSolutions
fortheGreatEllipticLine
Bowring [Bowring, 1984] provides formulas for the
solutionofthedirectandinversegreatellipticsailing
problem. Bowring’s formulas can be used for the
calculations of the great elliptic arc length and the
forwardandbackwardazimuths.
ThemethodofBowringforthecalculationofgreat
elliptic arc length employs the use of an auxiliary
geodetic sphere and various types of coordinates,
such as, geodetic, geocentric, Cartesian and polar.
These formulas for the great elliptic
distance have
beentestedanditwasprovedthattheyprovidevery
satisfactory results in terms of obtained accuracy.
Neverthelessothersimplercomputationsmethodsof
thelengthofthegreatellipticarccanbeusedbythe
employment of standard geodetic formulas for the
length of the arc of the
meridian, after the proper
modificationoftheparametersofthemeridianellipse
withthose of the greatellipse, such asformula (21).
TheformulasusedbyBowringforthecalculationof
theforwardandbackwardazimuths,unlikethosefor
the distance, are very much simpler than other
methods of the same
accuracy [Pallikaris &
Latsas,2009].
267
6.2 William’sMethodfortheComputationofthe
DistanceAlongtheGreatEllipticArc
Williams [Williams, 1996] provides formulas for the
computation of the sailing distance along the arc of
the great ellipse. These formulas have the general
form of the integral of formula (21). For the
computation of the
eccentricity egeand the geodetic
great elliptic angle φ
ge of formula (21), Williams
provides simple and compact formulas. For the
evaluationofthisintegralWilliamsemploysthecubic
spline integration method of Phythian and Williams
[Phythian&Williams,1985].
6.3 Earle’sMethodforVectorSolutions
Earle [Earle, 2000] has proposed a method of
computing distancealongagreat ellipse
that allows
theintegralfordistancetobecomputeddirectlyusing
the built‐in capabilities of commercial mathematical
software. This obviates the need to write code in
arcane computer languages. According to Earle, his
method has been prepared with the syntax of a
particularcommercialmathematicspackageinmind.
6.4 Walwyn’s
GreatEllipseAlgorithm
Walwyn [Walwyn, 1999] presented an algorithm for
the computation of the arc length along the great
ellipseandtheinitialheadingtosteer. Thealgorithm
usesvariousformulas forthe calculation of distance
and azimuths (courses). In some cases, probably for
thesakeofsimplicity,theseformulas
arenottheright
onesusedinstandard geodeticcomputations, asthe
formulas for the transformation of the geodetic
latitudestogeocentric.
6.5 ThePallikarisandLatsas’sNewAlgorithmforthe
GreatEllipticSailing
Algorithm proposed by Pallikaris and Latsas
[Pallikaris&Latsas,2009]wasinitiallydevelopedasa
supporting tool
in another research work of the
Pallikaris on the implementation of sailing
calculations in GIS‐based navigational systems
(ECDISandECS).Thecompletegreat ellipticsailing
problemissolvedincluding,inadditiontothegreat
elliptic arc distance, the geodetic coordinates of an
unlimited number of intermediate points along the
great elliptic arc. The algorithm has been developed
having a mind to avoid the use of advanced
numerical methods, in order to allow for the
convenient implementation even in programmable
pocketcalculators.
The algorithm starts with the calculation of the
eccentricityofthegreatellipseandthegeocentricand
geodetic great
elliptic angles of the points of
departure and destination. For this part of the
algorithm we used the formulas proposed by
Williams [Williams, 1996] because they are simple,
straightforwardandprovideaccurateresults.Forthe
calculation of the length of the great elliptic arc we
usedthestandardgeodeticseries
expansionformulas
forthemeridianarclengththatarepresentedinbasic
geodesytextbookslike[Torge,2001]aftertheirproper
modificationforthegreatellipse.
CalculationsoftheGreatEllipticDistance:
Lengthofthegreatellipticarc:
2
1
2
1
2
12
3
22
22 24
1
1sin
3315
11 sin2
4832
ge
ge
ge
ge
ge
ge ge
ge ge
ae
Sd
e
ae e e e
(34)
upto
sin 8φ terms.
6.6 TheSnyder’sSeriesApproximationsfortheMeridian
Ellipse
Equation21iseasilyevaluatednumericallyandeven
elementarymethodssuchasSimpsonʹsrulewillwork
but may not have sufficient precision, although an
algorithm described in [Williams, 1998] is known to
work well. It is preferable however, to
use an
adaptive algorithm that adjusts the intervals of the
integrandaccordingtotheslopeofthefunction.
Thefunction
1
f
belowisacompactharmonic
series approximation to equation 21 for meridional
distance[Snyder,1987].
3
10
1
sin 2
n
n
faaan
(35)
Thecoefficientsare:
24 6
0
13 5
1
4 64 256
a
24 6
1
33 45
8 32 1024
a
46
2
15 45
256 1024
a
6
3
35
3072
a
Distance
12
M
between two latitudes on the
meridional arc in the same hemisphere can be
determinedusingequation20i.e.
12 1 2 1 1
Mf f
(36)
Loss of significant digits is reduced for small
angular separations if differencing is applied to
equation20resultingin:
3
12 0 2 1 2 1 2 1
1
2cos sin
n
n
Maa a n n
(37)
which will be adapted later to give distance on the
great ellipse. There is also a companion harmonic
inversionseries to equation 35, described bySnyder
andattributedtoanearlierwork[Adams,1921]that
268
usedtheLagrangeInversionTheoremtoconstructthe
inversion series. It provides geodetic latitude as a
function of normalized meridional distance. The
condensedformofthisharmonicinversionseriesis:
4
20
1
sin 2
n
n
f
ubu b nu
(38)
theconstantsforwhichare:
0
1b
3
11 1
327
232
b
24
21 1
21 55
16 332
b
3
31
151
96
b
4
41
1097
512
b
and
1
1/1
For each value of the normalized distance
0
,
2
M
u
M
the function
2
f
u
returns a value of
geodetic latitude
corresponding to the given
meridional distance M. The constant
0
M
is the
meridionaldistance fromthe equatorto thepole i.e.
01
2
Mf
or, equivalently,
00
/2 .Maa
Bothoftheseseriesareperiodicandcanbeusedover
arcs spanning anyintervalin the range
02
[Earle,2011].
6.7 TheDeakin’sMeridianDistanceM
Meridian distance M is defined as the arc of the
meridian ellipse from the equator to the point of
latitude
.
Thisisanellipticintegralthatcannotbeexpressed
in terms of elementary functions; instead, the
integrandisexpandedbyintoaseriesusingTaylor’s
theoremthenevaluatedbyterm‐by‐termintegration.
The usual form of the series formula for M is a
function of
and powers of
2
e
obtained from
[Deakin&Hunter,2010],[Deakin,2012]
3/2
222
0
11sin
M
ae e d
(39)
ButtheGermangeodesistF.R.Helmert(1880)gave
a formula for meridian distance as a function of
and powers of n that required fewer terms for the
same accuracy. Helmertʹs method of developmentis
given in [Deakin & Hunter, 2010] and with some
algebrawemaywrite
23/2
22
0
112cos2
1
a
M
nnn d
n
(40)
Itcanbeshown,usingMaxima,that(39)and(40)
caneasilybeevaluatedandMwrittenas
02 4
2
6810
24
1
68 10
b b sin b sin
Ma e
b sin b sin b sin
(41)
where the coefficients
n
b
are to order
10
e as
follows:
24 6 8 10
0
3 45 175 11025 43659
1
4 64 256 16384 65536
beee e e
24 6 8 10
2
3 15 525 2205 72765
8 32 1024 4096 131072
bee e e e
46 8 10
4
15 105 2205 10395
256 1024 16384 65536
be e e e
68 10
6
35 105 10395
3072 4096 262144
bee e
810
8
315 3465
131072 524288
bee
10
10
693
1310720
be
or
02 4
6810
24
68 10
1
c c sin c sin
a
M
c sin c sin c sin
n
(42)
where the coefficients
n
c are to order
5
n as
follows
24
0
11
1
464
cnn
35
2
33 3
2 16 128
cnnn
24
4
15 15
16 64
cnn
35
6
35 175
48 768
cnn
4
8
315
512
cn
5
10
693
1280
cn
Note here that for WGS‐84 ellipsoid, where a =
6,378,137 m andf= 1/298.257223563 the ellipsoid
constants
n 1 .679220386383705e 003n
and
2
6.694379990141317e 003e
, and
10 12
5
1007 6.7
ee
n
[Williams,2002],[Deakin,2012].
269
This demonstrates that the series (42) with fewer
termsinthecoefficients
n
c
isatleastas‘accurate’
as the series (41). To test this consider the meridian
distance expressed as a sum of terms
024
MM M M,whereforseries(41)
2
002
22
24 4
1,
12, 14
Ma ebM
aebsinMaebsin
,etc.
andforseries(42)
0022
44
, sin 2 ,
11
sin 4 ,
1
aa
McMc
nn
a
Mc
n
,etc.
Maximum values for
024
, , MMM
occur at
latitudes
90 , 45 , 22.5 ,
when
max or
sin 1k
andtestingthedifferencesbetweenterms
at these maximums revealed no differences greater
than 0.5 micrometres. So series (42) should be the
preferable method of computation. Indeed, further
truncationofthecoefficients
n
c toorder
4
n and
truncating series (42) at
8
sin8c
revealed no
differencesgreaterthan1micrometre[Deakin,2012].
QuadrantLengthQ
ThequadrantlengthoftheellipsoidQisthelength
ofthemeridianarcfromtheequatortothepoleandis
obtainedfromequation(41)bysetting
1
2
, and
noting that
2, 4, 6sin sin sin
all equal zero,
giving
2
0
2
2246 8 10
1
3 45 175 11025 43659
11
4 64 256 16384 65536 2
Qa eb
ae e e e e e
(43)
Similarly,usingequation(42)
24
0
2
11
1
114642
aa
Qc nn
nn
(44)
7 GEODETICFORMULASFORTHEMERIDIAN
ARCLENGTH
7.1 TheSnyder’sSeriesApproximationsfortheMeridian
Ellipse
The methods and formulas used to calculate the
length of the arc of the meridian for precise sailing
calculations on the ellipsoid, such as “rhumb‐line
sailing”,“greatellipticsailing”and“geodesic
sailing”
are simplified forms of general geodetic formulas
used in geodetic applications [Pallikaris, Tsoulos,
Paradissis, 2009]. In this section an overview of the
mostimportantgeodeticformulasalongwithgeneral
commentsandremarksontheiruseiscarriedout.For
consistencypurposesandinordertoavoidconfusion
in certain formulas
the symbolization has been
changedfromthatoftheoriginalsources.
Equation (21) can be transformed to an elliptic
integral of the second type, which cannot be
evaluatedina“closed”form.The calculationcanbe
performed either by numerical integration methods,
suchasSimpson’srule,orbythebinomial
expansion
of the denominator to rapidly converging series,
retention of a few terms of these series and further
integration by parts. According to Snyder [Snyder,
1987] and Torge [Torge, 2001], Simpson’s numerical
integrationdoes not provide satisfactory results and
consequentlythe standardcomputationmethods are
based on the use of series
expansion formulas.
Expanding the denominator of (21) by the binomial
theoremyields:
2224466
0
0
31535
11
2816
M
a e e sin e sin e sin dx
(45)
Since the values of powers of e are very small,
equation (45) is a rapidly converging series.
Integrating(45)bypartsweobtain:
224
2
0
46
3315
1sin2
4832
1
15 105
sin 4
256 1024
eee
Mae
ee
(46)
Equation(46)isthestandardgeodeticformulafor
the accurate calculation of the meridian arc length,
whichisproposedinanumberoftextbookssuchasin
Torge’s“Geodesy”usinguptosin(2φ)terms,[Torge,
2001] and in Veis’ “Higher Geodesy” using up to
sin(8φ) terms [Veis,
1992]. A rigorous derivation of
(46)fortermsuptosin(6φ),ispresentedin[Pearson,
1990].
Equation (46) can be written in the form of
equation(47)providedbyVeis[Veis,1992]
2
002468
12468MaeM M M M M
(47)
24 6 8
0
3 45 175 11025
1
4 64 256 16384
Meee e
24 6 8
2
3 15 525 2205
8 32 1024 4096
M
ee e e
+…
46 8
4
15 105 2205
256 1024 16384
Me e e
68
6
35 315
3072 12288
Me e
8
8
315
131072
Me
270
Equation (48) is derived directly from equation
(47) for the direct calculation of the length of the
meridian arc between two points (A and B) with
latitudes φ
A and φB. In the numerical tests for the
assessment of the relevant errors of selected
alternative formulas, we will refer to equations (47)
and(48)asthe“Veis‐Torge”formulas.
2
0
24
68
1 [
(sin 2 sin 2 ) (sin 4 sin 4 )
sin 6 sin 6 ) (sin8 sin 8 )
B
A
AB
BA BA
BA BA
MaeM
MM
MM
(48)
Equations (47) and (48) are the basic series
expansion formulas used for the calculation of the
meridian arc. They are rapidly converging since the
value of the powers of e is very small. In most
applications, very accurate results are obtained by
formula(47)andtheretentionoftermsuptosin(6φ)
orsin(4φ)and8
th
or10
th
powersofe.
For sailing calculations on the ellipsoid it is
adequatetoretainonlyuptosin(2φ)terms,whereas
forothergeodeticapplicationsitisadequatetoretain
up to sin (4φ) or sin (6φ) terms. The basic formulas
(47) and (48) can be
further manipulated and
transformed to other forms. The most common of
theseformsisformula(49).Simplifiedversionsof(49)
(retaininguptoA
6ande
6
termsonly)areproposedin
textbookssuchasinBomford’s“Geodesy”[Bomford,
1985].
002468
2468 MaA A A A A
(49)
24 6 8
0
13 5 175
1
4 64 256 16384
A
ee e e
24 6 8
2
31135
8 4 15 512
Aeee e
46 8
4
15 3 35
256 4 64
Aeee
68
6
35 175
3072 12228
A
ee
8
8
315
131072
Ae
Inthe“AdmiraltyManualofNavigation”[AMN,
1987]forthesameformula(49)therearementioneda
littledifferentcoefficients(A
2inparticular):
24 6
0
13 5
1
464256
Aeee
24 6
2
3115
84128
Aee e
46
4
15 3
256 4
Aee
6
6
35
3072
Ae
Another formula for the meridian arc length is
equation (50), which is used by Bowring [Bowring,
1983] as the reference for the derivation of other
formulas, employing polar coordinates and complex
numbers. The basic difference of formula (50) from
(47), (48) and (49) is that (50) uses the ellipsoid
parameters
(a, b), instead of the parameters (a, e)
whichareusedinformulas(47),(48)and(49).
23 4
01 1
315 35 315
sin 2 sin 4 sin 6 sin8
216 48 512
MA Bn n n n
(50)
22
1
1
(1 )
8
1
an
A
n
2
1
3
1
8
B
n
ab
n
ab
.
Bowring [Bowring, 1985] proposed also formula
(51) for precise rhumb‐line (loxodrome) sailing
calculations.Thisformulacalculatesthemeridianarc
asafunctionofthemeanlatitudeφ
mandthelatitude
difference∆φofthetwopointsdefiningthearconthe
meridian.
B
A
φ
φ 02 m
4m 6m
8m
Ma(AΔφ Acos2φ sin Δφ
Acos4φ sin 2Δφ Acos6φ sin 3Δφ
Acos8 sin4Δφ
(51)
In(51), the coefficients A
0,A2, A4, A6, andA8 are
thesameasin(49).Equation(49)hasthegeneral form
ofequation(52).
02
46
8
ΔΔφcos 2 sin Δφ
cos 4 sin 2Δφ cos 6 sin 3Δφ
cos 8 sin 4Δφ
m
mm
m
Mk k
kk
k
(52)
In(52),thecoefficientsk
0,k2,k4,k6,k8are:k0=aA 0,
k
2=aA2,k4=aA4,k6=aA6,k8=aA8
7.2 TheProposedNewFormulasbyPallikaris,Tsoulos
andParadissis
Theproposednewformulasforthecalculationofthe
length of the meridian in sailing calculations on the
WGS‐84ellipsoidinmetersandinternationalnautical
milesare(53)and (54),respectively[Pallikaris,etal,
2009].
111132.95251 Δ 16038.50861
sin sin
90 90
B
A
BA
M
(53)
60.006994 Δ 8.660102
sin sin
90 90
B
A
BA
M
(54)
271
In both formulas (53) and (54) the values of
geodeticlatitudesφ
AandφBareindegreesandthe
calculated meridian arc length in meters and
internationalnauticalmilesrespectively.Formulas
(53)and(54)havebeen derivedfrom(48)for the
WGS‐84, since the geodetic datum employed in
ElectronicChartDisplayandInformationSystems
is WGS‐84. The derivation of the proposed
formulas
isbasedonthecalculationoftheM0and
M
2 terms of (48) using up to the 8th power of e.
Thisisequivalenttotheaccuracyprovidedby(49)
using A
0 and A2 terms with subsequent e terms
extended up to the 10th power since in formula
(48)thetermsM
0,M2,M4…aremultipliedby(1‐
e
2
). According to the numerical tests carried out,
which are presented in the next section, the
proposedformulashavethefollowingadvantages:
theyare muchsimplerthan andmore than twice
asfastastraditionalgeodeticmethodsofthesame
accuracy.
they provide extremely high accuracies for the
requirements of sailing calculations on the
ellipsoid.
7.3 TheAuthor’sProposal
TakingintoaccountthatthepolardistanceforWGS‐
84 is 10001965,7293127 m (see: Table 1) the author
proposes some modification to the formula (53)
proposedbyPallikaris,TsoulosandParadissis:
6367449.1458234 Δ
16038.50862 sin 2 sin 2
B
A
BA
M
(55)
with
inradians,andresultinmeters).
Thisformulawillbealittlebitmoreaccuratethan
formula(53).
7.4 NumericalTestsandComparisons
The different formulas and methods for the
calculation of meridian arc distances, which have
beeninitiallyevaluatedandcompared,are:
theproposednewformulasbyPallikaris,Tsoulos
and Paradissis (53) and (54), with author’s
modification(55);
“Veis–Torge”formulas(formulas(47)and(48))in
various versions, according to the number of
retained terms (1st version with up to M8 terms,
2ndversionuptoM6terms,3rdversionuptoM4
terms,4thversionuptoM2terms);
TheBowring[Bowring,1983]formula (50);
TheBowring[Bowring,1985]formula (51);
Thesenumericaltestsandcomparisonshavebeen
basedontheanalysisofthecalculationsofthelength
of the polar distance. The results of the evaluated
formulasareshowninTable2.
Itisnotsurprisethattheycorrespondtotheresults
presentedin
Table1.
Table2. Comparison of results of the calculations of polar
distance for ellipsoid WGS‐84 on the base of meridian
distanceformulas
_______________________________________________
MethodFormulaQuadrant
_______________________________________________
Deakin,2010(44) 10,001,965.72931270
Veis‐Torge(48) 10,001,965.72922300
Bomford,1985(49) 10,001,965.72931360
AMN,1987(49) 10,001,965.72952860
Bowring,1983(50) 10,001,965.72931270
Pallikaris,etal,2009 (53) 10,001,965.72590000
Weintrit,2013(55) 10,001,965.72931270
_______________________________________________
TheproposednewformulasbyPallikaris,Tsoulos
and Paradissis [Pallikaris, et al, 2009] for the
calculationofthemeridianarcaresufficientlyprecise
for sailing calculations on the ellipsoid. Higher sub
metreaccuracies canbeobtainedbytheuseofmore
complete equations with additional higher order
terms. Seeking
this higher accuracy for sailing
calculations does not have any practical value for
marinenavigationandsimplyaddsmorecomplexity
to the calculations only. In other than navigation
applications, where higher sub metre accuracy is
required, the Bowring formulas showed to be
approximately two times faster than alternative
geodeticformulasof
similaraccuracy.
8 CONCLUSIONS
NowwecansurelystatethatfortheWGS‐84ellipsoid
of revolution the distance from equator to pole is
10,001,965.729m,whichwasconfirmedbyanumber
of geometric and geodesic calculations presented in
thepaper.
The proposed formulas can be immediately used
not only
for the development of new algorithms for
sailingcalculations,butalsoforthe simplificationof
existingalgorithmswithoutdegradingtheaccuracies
requiredforprecisenavigation.Thesimplicityofthe
proposedmethodallowsforitseasyimplementation
even on pocket calculators for the execution of
accuratesailingcalculationsontheellipsoid.
Original
contribution affects and verifies
established views based on approxima ted
computational procedures used in the software of
marine navigational systems and devices. Current
stage of knowledge enables to implement geodesics
basedcomputationswhichpresenthigheraccuracy.It
also lets to assess the quality of contemporary
algorithms used in practical marine applications.
It
shouldbenotedthatanimportantstepinthesolution
issimplificationbytheomissionoftheexpansionpart
into power series of mathematical solutions,
previously known from the literature, i.e., [Torge,
2001]and[Veis,1992],andrelianceintheexplanatory
memorandum of application, in particular, on the
amountof
theavailableprocessingpowerofmodern
calculating machine (processor). In the authorʹs
opinionthiscriterionisrelevantfromapracticalpoint
of view, but temporary, given the growth and
availability of computing power, including GIS
[Pallikaris et al., 2009], [Weintrit & Kopacz, 2011,
2012].
272
Scientific workshop employed to solve the
problemmakesuseofvarioustools,i.e.ofdifferential
geometry, marine geodesy (marine navigation),
analysisofmeasurementerror,approximationtheory
and problems of modelling and computational
complexity, mathematical and descriptive statistics,
mathematicalcartography.Geometricalproblemsare
importantaspectofthetestedmodelswhichare
used
asthebasisofcalculationsandsolutionsimplemented
in contemporary navigational devices and modern
electronicchartsystems.
Thispaperwaswrittenwithavarietyofreadersin
mind, ranging from practising navigators to
theoretical analysts. It was also the author’s goal to
present current and uniform approaches to sailing
calculationshighlightingrecentdevelopments.Much
insight may be gained by considering the examples.
Thealgorithmsappliedfornavigationalpurposes,in
particular in ECDIS, should inform the user on
actuallyusedmathematicalmodelanditslimitations.
The shortestdistance (geodesics) between the points
depends on the type of metric we use on
the
considered surface in general navigation. It is also
important to know how the distance between two
pointsonconsideredstructureisdetermined.
Anattempttocalculatetheexactdistancefromthe
equatorto the polewasjust an excusetolook more
closely at the methods of determining the
meridian
arc distance and the navigation calculations in
general.
The navigation based on geodesic lines and
connected software of the ship’s devices (electronic
chart,positioningandsteeringsystems)givesastrong
argument to research and use geodesic‐based
methods for calculations instead of the loxodromic
trajectories in general. The theory is
developing as
well what may be found in the books on geometry
andtopology.Thisshouldmotivateustodiscussthe
subjectandresearchthecomponentsofthealgorithm
ofcalculationsfornavigationalpurposes.
Algorithmsforthecomputationofgeodesicsonan
ellipsoid of revolution are given. These provide
accurate,
robust,and fast solutions to the direct and
inversegeodesicproblemsandtheyallowdifferential
and integral properties of geodesics to be computed
[Karney,2011]and[Karney,2013].
REFERENCES
Adams, O.S., 1921. Latitude Developments Connected with
Geodesy and Cartography. Washington, United States
CoastandGeodeticSurvey,SpecialPublicationNo.67,
p.132.
AMN, 1987. Admiralty Manual of Navigation, Volume 1,
General navigation, Coastal Navigation and Pilotage,
Ministry of Defence (Navy), London, The Stationery
Office.
Banachowicz, A., 1996. Elementy
geometryczne elipsoidy
ziemskiej (in Polish). Prace Wydziału Nawigacyjnego
AkademiiMorskiejwGdyni,No. 18,p.17‐31,Gdynia.
Bomford,G.,1985.Geodesy.OxfordUniversityPress.
Bowditch,N.,2002.AmericanPracticalNavigator,Pub.No.9,
The Bicentennial Edition, National Imagery and
MappingAgency.
Bowring, B.R., 1983. The Geodesic Inverse Problem,
Bulletin
Geodesique,Vol.57,p.109(Correction,Vol.58,p.543).
Bowring, B.R., 1984. The direct and inverse solutio ns for the
great elliptic and line on the reference ellipsoid, Bulletin
Geodesique,58,p.101‐108.
Bowring, B.R., 1985. The geometry of Loxodrome on the
Ellipsoid,TheCanadianSurveyor,Vol.
39,No.3.
Dana, P.H., 1994. WGS‐84 Ellipsoidal Parameters. National
GeospatialIntelligenceCollege,9/1/94.
Deakin,R.E.,2012.GreatEllipticArcDistance.LectureNotes.
School of Mathematical & Geospatial Science, RNIT
University,Melbourne,Australia,January.
Deakin,R.E.,Hunter,M.N.,2010.GeometricGeodesy,PartA.
School of Mathematical &
Geospatial Science, RNIT
University,Melbourne,Australia,January.
Dresner,S.,1971.UnitsofMeasurement.JohnWiley&Sons
Ltd.May6.
Earle, M.A., 2000. Vector Solution for Navigation on a Great
Ellipse,The JournalofNavigation,Vol.53,No.3.,p.473‐
481.
Earle, M.A., 2011. Accurate Harmonic Series for Inverse
and
Direct Solutions for Great Ellipse. The Journal of
Navigation,Vol.64,No.3,July,p.557‐570.
Karney,C.F.F.,2011.Geodesicsonanellipsoidofrevolution.SRI
International, Princeton, NJ, USA, 7 February, p. 29‐
http://arxiv.org/pdf/1102.1215v1.pdf.
Karney, C.F.F., 2013. Algorithms for geodesics. Journal of
Geodesy,January,Volume87,
Issue1,p.43‐55.
Kawase,K.,2011.AGeneralFormulaforCalculatingMeridian
ArcLengthanditsApplicationtoCoordinateConversionin
the Gauss‐Krüger Projection. Bulletin of the Geospatial
InformationAuthorityofJapan,No.59,pp.1–13.
Michon, G.P., 2012. Perimeter of an Ellipse. Final Answers.
www.numericana.com/answer/ellipse.htm
Pallikaris,
A., Latsas, G., 2009. New Algorithm for Great
EllipticSailing(GES),TheJournalofNavigation,Vol.62,
p.493‐507.
Pallikaris,A.,Tsoulos,L.,Paradissis,D.,2009.Newmeridian
arc formulas for sailing calculations in GIS, International
HydrographicReview.
Pearson,F.,II,1990.MapProjection:TheoryandApplications.
CRC
Press,BocaRaton,AnnArbor,London,Tokyo.
Phythian,J.E.,Williams,R.,1985.Cubicsplineapproximation
to an integral function. Bulletin of the Institute of
MathematicsandItsApplications,No.21,p.130‐131.
Snyder,J.P.,1987. Map Projections:A Working Manual.U.S.
GeologicalSurveyProfessionalPaper1395.
Torge, W.,
2001. Geodesy. Third completely revised and
extendededition.WalterdeGruyter,Berlin,NewYork.
Veis, G., 1992. Advanced Geodesy. Chapter 3. National
TechnicalUniversityofAthens(NTUA),p.13‐15.
Walwyn,P.R.,1999.TheGreatellipsesolutionfordistancesand
headings to steer between waypoints, The Journal of
Navigation,
Vol.52,p.421‐424.
Weintrit,A.2009.TheElectronicChartDisplayandInformation
System (ECDIS). An Operational Handbook. A Balkema
Book.CRCPress,Taylor&FrancisGroup,BocaRaton–
London‐NewYork‐Leiden,p.1101.
Weintrit, A., 2010. Jednostki miar wczoraj i dziś. Przegląd
systemów miar
i wag na lądzie i na morzu (in Polish)
AkademiaMorskawGdyni,Gdynia.
Weintrit,A.,Kopacz,P.,2011.ANovelApproachtoLoxodrome
(RhumbLine), Orthodrome(GreatCircle)andGeodesicLine
in ECDIS and Navigation in General. TransNav‐
International Journal on Marine Navigation and Safety
ofSeaTransportation,
Vol.5,No.4,p.507‐517.
Weintrit,A.,Kopacz,P.,2012.OnComputationalAlgorithms
Implemented in Marine Navigation Electronic Devices and
Systems. Annual of Navigation, No 19/2012, Part 2,
Gdynia,p.171‐182.
Williams, E. 2002. Navigation on the spheroidal Earth,
http://williams.best.vwh.net/ellipsoid/ellipsoid.html.
Williams, R., 1996. The Great Ellipse
on the surface of the
spheroid, The Journal of Navigation, Vol. 49, No. 2, p.
229‐234.
