-
LONG GEODESICS ON THE ELLIPSOID
H. F. RAINSFORD
Senior Computer, Directorate of Colonial Surveys, Tolworth
(G.B.)
There are two main problems concerning long lines:
(a) Given the geographical co-ordinates of two points, to find
the distance and azimuths between them; commonly called the reverse
(or inverse) problem;
(b) Given the geographical co-ordinates of one point and the
distance and azimuth from it to another, to find the co-ordinates
of the second point; commonly called the direct problem.
The leaders in making a general study of the geodesic were
EVLER, LEGENDRE and BESSEL. The two main problems were solved by
relating the geodesic elements to the corresponding elements of an
auxiliary spherical triangle in which the reduced latitudes were
used instead of the geographic latitudes. The same method was given
by JORDAN, HELMERT and CLARKE. It is a solution in terms of
trigonometrical functions. However, successive approximation is
necessary to find the longitude angle of the auxiliary spherical
triangle used. This is a tedious process (particularly with 10-
figure tables) as at least three approximations are required for a
precise result. The two problems should be considered together as
each can be used to check the other by different formulae. Recently
LEVALLOIS and DuPuY have proposed a slightly different solution
using tables of 'Wallis Integrals' o r f sinZPx dx. The main
advantages are that the variation in tabular factors required is
comparatively slow and the series converge more rapidly. The tables
are now available only for centesimal arguments. It is probable
that this method would be widely used if the tables were given in
sexagesimal arguments and for several Figures of the Earth in
general use. The methods given in this article can be used to
obtain absolute precision for any length of line, without special
tables (except possibly those for the principal radii of curvature
of the spheroid).
2. Formulae for long lines should be applicable to any length of
line up to halfway round the world. To clarify one's ideas on
comparable accuracy in length, azimuth and position, suppose that
an accuracy of 1 mm is required and take the major axis of the
spheroid (0.638 107 metres) as the radius of a spherical earth.
Then i mm on the surface of the earth is equal to 0".00003 in
position: also, a variation in azimuth of 0".00001 corresponds to a
lateral shift of 1 mm at a distance equal to half round the world.
Since both problems require the solution of auxiliary spherical
triangles, it would appear that 10-figure tables (working to the
fifth decimal of a second) are necessary to obtain accuracy to the
millimetre. Such accuracy may be required for the very precise
checking of reverse and direct computations; in particular, when
the examples concerned are to be
12
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LONG GEODESICS ON THE ELLIPSOID
used for comparison with results by more approximate methods.
Accuracy to a centimetre (approximately) may be obtained with
8-figure trigono- metric tables provided that 9- or 10-figure
computat ion is used when required for the first term of any
series.
o
/
P
~0 a~
Figure 1
3. Figure 1 i l lustrates the auxil iary spherical tr iangle
used in the reverse problem. AEFG is the equator and P the pole. B
and C are the two points on the sphere corresponding to the two
given spheroidal points. The great circle through B and C cuts the
equator at A and has its point of max imum reduced lat itude at D.
BE and CF are the reduced latitudes U 1 and Uz corresponding to the
latitudes ~Pl and q~2- The spherical azimuths % and a2 are
precisely the same as the corresponding spheroidal azimuths. L is
the difference of longitude on the spheroid and X the corresponding
difference on the sphere. The azimuth of the great circle AD at the
equator is ~. The arcs (Yl (AB) and a2 (AC) are measured from the
equator. The arc BC=a2-a l =~ and 2~=at +a2. I f a and b are the
major and minor axes of the spheroid, other spheroidal parameters a
re : - -
e 2 ___ (a 2.b2)/a2
q2 = (a2_b2)/b2
f = (a-b)/a
4. Since X is unknown at first, the spherical tr iangle is
solved using >, =L as a first approximat ion to find the
correction (X -L ) . The process is con- t inued until further
repetit ion will not alter the results: usually, three
approximations are sufficient. The most convenient formulae for
machine solution are :
b b . . . . tan qo 2 tan U l -a tan ~01; tan U2- a
cosa=s in U 1 sin U 2+ cos U 1 cos U 2cosx
sin sc = cos U 1 cos U 2 sin X/sin
sin 0q cos U 1 =sin ~2 cos U 2 =sin oc
cos 2a~ = cos a -2 sin U 1 sin U2/cos 2 s~
. . . . (1 )
. . . . (2 )
. . . . (3 )
. . . . (4 )
. . . . (5 )
13
-
NOTICES SCIENTIFIQUES
cos 4am = 2 cos 2 2era - 1
cos 6~m =4 cos 3 2~m-3 COS 2~m
sin 2~ = 2 sin a cos
sin 3~ = 3 sin ~ -4 sin 3
fAocr +A 2 sin cr cos 2cry, (X-L) =fs in x L +A 4 sin 2 . cos
4*m +A6 sin 3or cos 6cr,.
. . . . (6 )
. . . . (7 )
. . . . (8)
. . . . (9 )
. . . . (10)
A 0 = 1 - ~f (I +f+f2) cos 2 ~ + 1.~f2 (1 + 9 f ) cos 4 a _
l_~sf3 cos 6 a]
A2=88 +S+f2) cos 2 ~ _if2< (1 +~f) cos4 ~ + ~-F6f3 cos6 ~ A 4
=~2-f 2 (I +9f ) cos 4 oc- ~-!3~f3 cos 6 o~ [ . . . . (11)
A 6 =7~8f 3 cos 6 ~ J
The A coefficients are given as functions o f f since they
converge more rapidly than when given as functions of e2. The
maximum value of any term in f4 (i.e.f3 in the A's) is less than
0".00001 even for a line half round the world. Thus the A 6 term
may be omitted altogether and the following simplified forms used
even for precise results:
A 0 = 1 - 88 +f ) COS 2 ~ + 1-~f 2 COS 4 0r 7 I
A 2 = i f (1 +f ) COS 2 0~ __ 88 COS 4 ~ !
A 4 =)-lz-f2 cos 4 o~ J
(12)
5. To avoid the successive approximation a method has been
produced by the Army Map Service (U.S.A.)I which gives the
correction (X-L ) directly by solution of the triangle BCP in which
the angle at P is taken as L (the spheroidal longitude).
Let x = sin U 1 sin U2 "] . . . . (13)
y = cos U1 cos Uz A Let 0 be the first approximation to cr and z
the first approximation to
sin ~, then
cos 0 =x +y cos L . . . . (14)
Z -y sin L/sin 0 . . . . (15)
Also let P= [cos 0(1 - z 2) -x]/s in 0 . . . . (16)
Then the required correction (X- L) is given by
(X - L) =fzO - 88 z{0 (1 - 5z z) - 2P (202 - sin z 0) "~ +(1-z
2) s in0cos0}
( (1 - z 2) [2cos20- 1 -zZ(2cos20 +7) ] ' ) +-l~f3z sin 0 cos 0
L "~ + 8Ps in+ 8 p2 sin 20 cos0]0 [ - 1 +Z 2 (3 +2 tan2 0)~.j . . .
. (i 7)*
- ~6f3z0 { 1 + 1 4z 2 - 3 1 Z 4 + 8P ( 1 - z 2) sin 0 cos 0 - 1
6P 2 sin 2 0} - 89 (I - 9z 2) + 89 3 { _ z 2 (1 - z 2) - 3Pz 2 cot
0
+ 2P 2} +. . . J * The terms in f 3 would not normally be used.
They are given here so that the maximum
effect of neglecting them can be found.
14
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LONG GEODESICS ON THE ELL IPSOID
This is a pecul iar formula as 0 can have all values from 0 to
,~: also P--+oo as 0--+r=. So it appears that although (X- L)
cannot exceed fr= (radians), formula (17) may be indeterminate as
some of the terms .+co as 0--+~.
6. First, it may be noted that x and z are both less than unity,
so P sin 0 will never cause any trouble. But there are also terms
in PO z, p203 and P03 cot 0. Since 0/sin 0.+1 when 0-+0, none of
these will cause trouble when 0 is small. But when 0 is large and
--+180 ~ the formula becomes indeterminate. I t is impossible to
state precisely when the formula breaks down, but it seems likely
that provided 0< 170 ~ (say), and using only the terms in fandf
z, formula (17) will give (X -L ) to within about 0".005 or better.
I t is suggested that the best method is to use formulae (1), (13)
to (17) to obtain the first approximat ion to (X -L ) and then use
(1) to (12) to obtain the precise result.
7. The computat ion of the distance is direct once the auxi l
iary spherical tr iangle has been solved with sufficient precision.
Let s be the geodesic distance and let u 2 =e12 cos 2 0r then the
distance is given by:
s/b =Boa +B 2 sin a cos 2a m +B 4 sin 2or cos 4am +B 6 sin 3cr
cos 6~., +B s sin 4~ cos 8Ore + . . . . . . . (18)
in which the coefficients are given by
Bo=I + 88 - 6-3~u4 + 2 ~ 6u'6 _ T6"5"g'~ u '175 -8
B2 = - 88 -t--11-6u4 - 31-115u6 -L'-2048-35 -8
- - 35 "8 B 4 = - T22-gu 4 +y~2zt 6 8192 u
B6 = - T~ 88 6 + r~u8 B8 = - ~ u 8
I
. . . . (19)
The max imum effect on s of any term in uS is less than half a
mil l imetre, so that all can be neglected. Hence use the following
simplified coeffi- cients:
B o = 1 + ~u 2 - 3,~u4 + 2556 u6
B2 = - 88 + qtlu4 - -5!~s ~ . . . . (20)
B4= - 118//4 +-K~2 u6 |
3 B6 = - a-~-eu6 The max imum effect on s of the term in B 6 is
only 1-3 mm so if absolute accuracy to the mil l imetre is not
required this may be neglected also.
8. I f any of the formulae (I) to (5) do not give a sufficiently
exact deter- minat ion of the required element, alternative
formulae must be used. (1) always determines U1 and U2 well, but if
either Pt or 92 are greater than 45 ~ use
cot U 1 =~ cot 91, or cot U2 =b cot 92 . . . . (21)
to avoid extra figures which provide no extra accuracy. I f a is
near 0 ~ or 180 ~ it will not be well determined by its cosine;
similarly if 0q or 0c 2 are
15
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NOTICES SC IENT IF IQUES
near 90 ~ they will not be well determined by the sine. the
azimuths first from:
cot {(oc 2 -0~1) =cot 89 cos 89 2 - Ul)/sin 89 2 + U1) cot 89
+~1) =cot 89 sin 89 z - U1)/cos { (U 2 + UA)
and then find e from
sin e/sin X =cos U1/sin oc z =cos U2/sin ~1
In either case find
. . . : (22)
. . . . (23) I f e is near 90 ~ then U1 and /-72 are small and
sin U1 sin U2/cos2 ~x appears indeterminate. In this case instead
of (5) use
cos 2e,, cos 2 ~ = cos e cos 2 0c - 2 sin U 1 sin Uz . . . .
(24)
cos 2 ~ is found sufficiently exactly from (3). 9. The part
icular solution of the direct problem which is now given was
first publ ished by McCAw2. I t does not require successive
approximation. McCaw's formulae have been recast so that arcs are
measured from the equator instead of the point of highest latitude,
and extra terms have been included for greater accuracy.
P
7' \ / ~,'/(/~>"
Figure 2
Figure 2 shows a second auxi l iary spherical tr iangle which is
used in the solution. In this triangle the longitude angle (BPC) is
precisely the same as the angle BPC in Figure 1. The relationship
between the az imuth angles of the great circle ABCD at A, B and C
is given by
k cos (~) = cos
k cos (~1) =cos ~1
k cos (~2) =cos 0~ 2
For definition of k see formula (29). 10. Being given ~I, el, s
the solution proceeds as follows:
b tan U 1 - - tan q~l - -a
sin ~ =sin el cos U 1
/ L (25) f . . . .
. . . . (26)
. . . . (27)
U2 = el2 COS 2 . . . . (28)
16
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LONG GEODESICS ON THE ELL IPSOID
k2=(1 +u2) / ( l +e12 ) . . . . (29)
tan G 1 =k tan ?l/cos ~1 . . . . (30)
K"=~/ ( l+u2) /bs in 1" . . . . (31)
y =KC o s . . . . (32)
y I=G1 -C2 sin 2G 1 +C4 sin 4G 1 -C 6 sin 6G 1 . . . . (33)
2'2 =2"1 +'f" 2Tin =Y1 +Y2 . . . . (34)
G =y + O 2 sin 2" cos 2Tm + D4 sin 2u cos 4y~ +D 6 sin 3y cos 6y
. . . . . (35)
G 2 =G 1 +G. 2Gm=G 1 +G 2 . . . . (36)
It may be noted here that formula (35) could be derived from
formula (33) to give the arc G, but the arguments would be G and
G~, still unknown. This difficulty could be overcome by solving by
successive approximation, but this is a tedious process. McCaw
solved the problem by using Lagrange's Theorem to change the
unknown arguments G and Gm to the known ones y and y,,. For McCaw's
solution of the direct problem, there- fore, successive
approximation has been entirely eliminated (see ref. 2, p.
348).
The coefficients of (32), (33) and (35) are given by:
C0=l 3U2 + 3__94U 4 -- ~T6 u133" 6--1- "i~-3--8--g u74"91" 8 "1
cz = ~u2 - 3 , ,4 _~ 11 , . 6 _1 , , . a [
. -1 -o - f ' s - - 2 -o -4~u !
C4 15 4 ~5 6~4os s ~ (37) = -z -g -6u - - - zw-6u 7- -8--f-9--s
. . . . C6 = 3 5 _ . 6 3 o 7 2 ~ _ _~9_~j ,u 8 I Ca =1 ~_~,8 D2 =
3u2 -8"-3"4 ~ 1024" _213_. 6 _ 255f fu 8 D4 = _f21gu4 Zl__,, 6 _t.
_!15 9_%, 8 L
/
-128~ - l~ss~ c . . . . (38)
The terms in C a and D a may always be omitted as inappreciable.
The terms in'uS in C2, C4, C6 and D2, D4, D 6 are appreciable if
accuracy to the milli- metre is required (maximum effect %0".0001),
otherwise they may be negIected. The maximum effect of the term in
u s in Co is 0".0006 for a line half round the world. Hence, if a
precise check to the millimetre is required between direct and
reverse formulae, all terms in u a in (37) and (38) except for C a
and D s must be included.
11. To complete the solution, we have
sin ?z = sin G 2 cos ~./k . . . . (39)
COS ~2 = k cot Gz/cot ?2 . . . . (40)
cos X = [cos G -s in p~ sin ?/]/cos ?1 cos 72 . . . . (41)
2 17
-
NOTICES SCIENTIFIQUES
(X -L ) =fs in ~ {Eoa -E2 sin G cos 2Gin +E4 sin 2G cos 4G~ - E
6 sin 3G cos 6Gin} . . . . (42)
E 0 = 1 - 88 (1 +f+f2) cos 2 ~ +1%f2 (1 +g f ) cos4 e -
~2~f~ cos ~ E 2 = 4if (3 + 5 f+ 7f 2) cos 2 ~ _ f2 (I + ~f )
cos4 0~ + 23 56 6sf 3
cos6= . . . . (43)
E4 = A f2 (1 + y f ) cos4 ~ - cos 6 E 6 =7@sf 3 COS 6 ~z
Note that E 0 is precisely the same as A 0 from (11). As before
it is found that all terms in f 3 in the E's can be omitted as the
max imum effect is less than 0".00001 and the following simplified
forms may be used:
g o =A o = I -4 i f (1 +f ) cos 2 ~ +-~f2 COS 4 ~. ~]
E 2 =~f(3 +5f ) cos 2 = _ f2 cos4 = Ct . . . . (44)
E 4 =-~2f 2 cos 4 oc J
12. In the cases when q02 is near 90 ~ or when ~2 is small or
near 180 ~ instead of (39) to (41), use Napier 's Analogies for the
triangle CBP to find the angles at B and P, i.e. 180 ~ - (0@ and X
and then find ?z from
cos q~a =sin (51) sin G/sin X . . . . (45)
I fX is near 0 ~ or 180 ~ use instead of (41)
sin ?, =sin G~v/ik2 - cos 2 ~l) /k cos q0 z . . . . (46)
13. Some confusion has often been caused in the past because
parameters of the spheroid (Figure of the Earth) have been given
which were not mutual ly consistent. Two parameters are sufficient
to specify the shape and size of a spheroid completely. I f a and b
are the major and minor axes respectively, modern practice is to
define the spheroid by specifying a and r (the reciprocal of the
flattening). Other parameters are then defined by:
1 f= (~ - b ) /~ =-
r
~2 = (42 _ b2) /42 = 2 _ _ 1"
i r 2
2 3 4 e12 = (a2 - b2)/b 2 =r +7 + r3 + . . . . . . . . (47)
1(1) n=(a-b) / (a+b)=~+ ~; + ~ + . . . .
1 b/a = 1 - -
r J
These are the simplest formulae to use for computat ion of the
parameters, in terms of ( l /r) and powers thereof, with normal
desk calculating machines.
14. As a pract ical check on the formulae given here, the
following
18
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LONG GEODESICS ON THE ELLIPSOID
examples have been computed and checked by use of both the
indirect and direct formulae. ANDOYER'S tables of the natural
sines, cosines, tangents and cotangents (15-figure) were used,
taking the tabular quantities to 11 figures and all angles to the
6th decimal of a second: results were finally rounded off to the
5th decimal of a second and the nearest millimetre. Three values of
(X- L) are given in each case: the first, rigorous from formulae
(10) and (12); the second, by SODANO'S method from formula (17),
neglects terms in f3 ; the third, also from (17), includes the
terms in f3. Unless otherwise stated, results from the direct and
reverse formulae agree to the 5th decimal of a second.
(a) McCaw's example (ref. 2, p. 158). Bessel Spheroid;
a =6377397"155 metres, r =299.1528128.
q~x +55~ 45' N; 7z -33~ 26' S; L+ 108 ~ 13' E
Rig. (X-L) + 14' 16"-62592 ~ 34 ~ 04' 48"-38630
Sodano 2 + 14' 16"-63006 ~1 96~ 36' 08".79960
Sodano 3 + 14' 16".62596 ~z 137~ 52' 22".01454
s 14110526-170 metres.
The errors of Sodano 2 and 3 are 414 and 4 in the 5th decimal of
a second. The remaining examples are all on the International
Spheroid;
a = 6378388 metres, r =297-0.
(b) q~l +37~ 19' 54".95367 N;
L +41 ~ 28' 35".50729 E
Rig. (X- L) + 05' 53".23775
Sodano 2 + 05' 53"-23741
Sodano 3 + 05' 53".23773
q~z +26~ 07' 42".83946 N;
52 ~ 25' 10"-99010
~l 95 ~ 27' 59".63089
~2 118~ 05' 58"-96161
s 4085966-703 metres.
The errors of Sodano 2 and 3 are 34 and 2 in 5th decimal.
(c) ~1 +35~ 16' 11".24862 N;
L + 137 ~ 47' 28".31435 E
Rig. (X-L) +03' 15".08310
Sodano 2 +03' 15"-08275
Sodano 3 +03' 15".08310
q~2 +67~ 22' 14"-77638 N;
12 ~ 48' 37".53647
el 15~ 44' 23"-74850
~2 144~ 55' 39"'92147
s 8084823"839 metres.
Residual longitude error is 3 in 5th decimal. The errors of
Sodano 2 and 3 are 35 and 0 in 5th decimal.
(d) q~l + 1 o 00' 00"'0 N; q~2 -0~ 59' 53".83076 S; L + 179 ~
17' 48".02997 E e 88 ~ 35' 17"'52180
19
-
NOTICES SCIENTIFI QUES
Rig, (X- L) + 36' 19"'96992 el 89 ~ 00' 00"'0
Sodano 2 +36' 20".26224 ~2 91~ 00' 06".11733
Sodano 3 +36' 20".27254 s 19960000.000 metres.
The errors of Sodano 2 and 3 are 0".29232 and 0".30262.
(e) q~l + 1~ 00' 00"-0 N;
L ~- 179 ~ 46' 17".84244 E;
Rig. (X -L ) +03' 07"-83471
Sodano 2 +02' 51".90751
Sodano 3 + 03' 11".95949
q~2 + 1~ 01' 15".18952 N;
4 ~ 59' 57".26995
el 4~ 59' 59"-99995
0~ z 174 ~ 59' 59".88481
s 19780006-558 metres.
Residual errors of latitude, longitude, az imuth are 2, 0, 1 in
5th decimal. The errors of Sodano 2 and 3 are 15".92720 and
4"-12478.
15. I t should be noted that there is one pecul iar ity of
geodesics which approach 180 ~ of arc or halfway round the world.
Take the case of two points on the equator, 180 ~ apart in
longitude. I f these two points were on a sphere, it is obvious
that there is an infinite number of great circles through both
points, and all of the same length. On the spheroid, however, there
are only two geodesic [sic] arcs, corresponding to great circles,
which pass through two points on the equator, exactly 180 ~ apart.
But the mer id ian arc is shorter than the equator ial arc through
the two points, and it is, therefore, the only true geodesic on the
shortest line definition of a geo- desic. Consequently, if two
points are situated near the equator and are separated by nearly
180 ~ of longitude there is a certain ambiguity as to what is meant
by the geodesic between them. There is a full discussion of this po
in t in an article 'The distance between two widely separated
points on the surface of the earth'.3 This article is a review of
another article of the same title by W. D. LAMBERT. 4
16. The two examples (d) and (e) show that Sodano's method
breaks down when the arc between them is nearly 180 ~ Sections 5
and 6 indicate that the main cause of the trouble is the function P
which -+Go as the arc --+180 ~ An attempt was made to get over this
difficulty by splitt ing the line into two parts at the point of
highest lat itude (where the az imuth is 90~ The function P for
each part of the line is then zero. This does not seem to help,
however, as this point is unknown: in fact, once this point is
known, the whole solution of the inverse problem follows very
simply. Sodano's method may also be used to obtain formulae,
similar to (17), for (e - 0) and for sin e. Both, however, contain
the function P and become indeterminate for arcs near 180 ~ It
would appear then that all we can do about this is to define more
precisely the useful limits of Sodano's method, by computing a
sufficient number of examples, so that the max imum error under
certain specified conditions can be stated.
17. In the past, the problem of long lines on the earth has
frequently been approached by the use of plane sections, since the
geometry was
20
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LONG GEODESICS ON THE ELLIPSOID
(supposedly) easier to understand than that of geodesic curves
in three dimen- sions. The difference in length between a geodesic
and a plane curve is very small, but the difference in azimuth can
be quite appreciable for lines over about 500 miles. If, for
example, it was known that radar waves followed a plane section
round the world, there might be some justification for plane
section computation: in the absence of much real evidence on this
or similar lines the geodesic approach is theoretically more
correct and actu- ally simpler in practice. It seems probable that
the method advocated by Levallois and Dupuy for computation of long
lines is the best, provided that tables are made for various
Figures of the Earth in sexagesimal units.
18. For lines less than 500 miles (800 km), in latitudes less
than 75 ~ formulae are available using either plane curves or
geodesicsS. The first two methods are based on plane curves; they
will give precise results but require the use of 9 or 10-figure
trigonometric tables. The other methods advocated are based on true
geodesics and precise results can be obtained with 8-figure tables.
For the inverse problem, the Mid-Latitude Formulae are undoubtedly
the best as they do not require any successive approxima- tion: the
extension to fourth order corrective terms has been given here. For
the direct problem the extension of CLARKE'S approximate formulae
may be used: corrective terms have been given up to 6th order
(spherical) and 5th order (elliptic), but many of these are
inappreciable except in high latitudes. Formulae have also been
given for the Puissant Series method up to 7th order (spherical)
and 6th order (elliptic) terms. It is not recom- mended that these
should be used for long lines as they do not converge sufficiently
rapidly, but they are useful to obtain the possible errors from
neglecting terms of any particular order.
19. The best geodetic tables available are those for Latitude
Functions on the various spheroids (Natural values of the
meridional arc; A, B, C, D, E and F factors; radii of curvature, R
and N) produced by the Army Map Service in 1944. They tabulate the
required functions to an accuracy of a millimetre at an interval of
1 minute of arc. These tables are invaluable for any form of
geodetic line computations: they are clear, well set out and very
easy to use as differences are given for 1 second. It seems strange
that the A.M.S. computed these tables for every spheroid in common
use, except their own (the Clarke 1866). If they could now see
their way to producing the 1866 tables and, also, tables for the
factors of the Mid- Latitude Formulae, while Levallois and Dupuy
produced their tables in the sexagesimal system, the geodetic world
would have available all the tables necessary for dealing with
problems of long lines on the earth.
REFERENCES
1 SODANO, E. M., 'Inverse computation for long lines; a
non-iterative method based on the true geodesic', Technical Report
.No. 7, Aug. (1950).
2 McCAw, G. T., Empire Survey Review Vol. II, 156-63, 346-52,
505-8. 3 Empire Survey Review, Vol. VIII (1943), 172-6. 4 LAMBERT,
W. D., 'The distance between two widely separated points on the
surface of the earth', J. Wash. Acad. Sci. 32 (1942), 125-30. 5
P~AINSFORD, H. F., 'Long lines on the earth: various formulae',
Empire Survey
Review, Vol. X (1949), 19-29, 74-82,
21
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LONG GEODESICS ON THE ELLIPSOID
SUMMARY
This article examines the practical application of formulae for
computing long lines on the ellipsoid. The main aim is to eliminate
the successive approximation generally required. For the inverse
problem, this is achieved by the method of E. M. SODnNO, Army Map
Service, U.S.A. An adapta- tion of a method produced by G. T. McCAw
is used for the direct problem.
Results are given of five practical examples, including two
which extend halfway round the world. Construction of further
special tables is recom- mended to simplify the computations
required by a problem which has an ever increasing application.
Rt~SUMI~
L'auteur, sp6cialiste bien connu de la question, examine et
discute les diff6rentes m6thodes utilis~es ou pr6conis~es par
diff6rents G6odfisiens, pour le calcul des lignes g6od6siques de
grande longueur/t la surface de la terre.
ZUSAMMENFASSUNG
Der Verfasser--ein bekannter Fachmann auf diesem
Gebiet--untersucht kritisch die von verschiedenen Geod~iten
angewandten oder vorgeschlagenen Verfahren zur Berechnung sehr
langer geod~itischer Linien.
R IASSUNTO
L'Autore, specialista ben noto dell'argomento, esamina e discute
i diversi metodi utilizzati o proposti dai differenti Geodeti per
il calcolo delle geo- detiche lunghe sulla superficie
terrestre.
RESUMEN
E1 autor, especialista bien conocido en la cuestidn, examina y
discute los diferentes m6todos utilizados o preconizados por
diferentes geodestas para el cMculo de lineas geod6sicas de gran
longitud en la superficie terrestre.
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