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GIMRADA Research Note No. 11

GENERAL NON-ITERATIVE SOLUTION OF THE INVERSE AND DIRECT GEODETIC PROBLEMS

By Emanuel M. Sodano

April 1963

iisBäSSiBIgSJ

DDC AVAILABILITY NOTICB

Qualified requestors may obtain copies of tiiis report from DBC.

FORT BELVOIR VA

!W^..««M.™.MWW***^

U. S, ARMY ENGINEER

GEODESY, INTELLIGENCE AND RAPPING RESEARCH AND DEVELOPMENT AGENCY

FORT BELVOIR, VIRGINIA

Research Note No. II

GENERAL NON-ITERATIVE SOLUTION OF THE

INVERSE AND DIRECT GEODETIC PROBLEMS

Task 4A011001B85002

April 1963

Distributed by

The Director

Ü. S. Army Engineer

Geodesy, Intelligence and Mapping Research and Development Agency

Prepared by

Emanuel M. Sodano

Research and Analysis Division

U. S. Army Engineer

Geodesy, Intelligence and Mapping Research and Development Agency

Fort Belvoir, Virginia

<««WMM«»mn»JK)JlMMeK^«<jj,aBi,H»i.«,a.«.^«» ..._■.. , T -— |i(iiiiiiiiiinMi»iinriwiipiniipi«Ninffni«ii»T ».^iimin mnnwiiiii M*~..,.' .iLiU^ii^W^

The views contained herein represent only the views of the

preparing agency and have not been approved by the Department of the Army.

GIMRÄDA Research Notes are published to record and disseminate

sigmficant results of research tasks or portions of overall programs during the course of the work. F ö

11

llWPJMIiyiMWWWWMIIllMiMWWIIIIIWMWMIlii^irTWrnniii Til " ' n ■ ■imi mituitmnum ■■niiuniHrnMliiwi^iniiniiiiniwiMn | ■rnu'iTT'T il^flMtlliWM^ffiliaillHlllWWIlVI'JWIIHIIIIir

GENERAL NON-ITERATIVE SOLUTION OF THE

INVERSE AND DTRECT GEODETIC PROBLEMS

I. INTRODUCTORY BACKGROUND

Earlier, at the Army Map Service, the writer published a com-

prehensive study [1] for the rigorous non-iterative inverse solution

of long geodesies, following its presentation at the XI General

Assembly of the International Union of Geodesy and Geophysics. (At

present, a translation by the German Geodetic Commission and an

abstract by the U. S. S. R. Academy of Sciences, complete with final

formulas, are also available.) The method received favorable commen-

taries from several authoritative writers on the subject. A com-

parative evaluation study published in [2] indicated that the method

was as accurate as contemporary solutions, yet simpler and shorter

to compute. The procedure does not require any special geodetic

tables and calls for relatively few trigonometric interpolations.

Of greatest interest was the fact that it was the first rigorous

non-iterative inverse solution to go not only beyond the first power

of spheroidal flattening but through all the cubic terms of flatten-

ing. Yet, it was practical.

II. THEORETICAL REFINEMENT OF NON-ITERATIVE INVERSE

Since [1] represented only the formative stage of development,

the next phase consisted of making a thorough mathematical analysis

of the formulas in order to determine whether there were any con-

cealed intrinsic properties or relationships which could be used to

obtain an optimum solution. This analysis resulted in the uncover-

ing of three basic quantities, a, m, and 0, the exclusive substi-

tution of which promised a concise, orderly pattern for the tenns

of the two main spheroidal power series, x and S, that were origi-

nally given in [1], at the top of page 18 and the bottom of page

19, respectively.

III. DEVELOPMENT OF CORRESPONDING DIRECT

Since there were indications that a, m, and 0 were a rather

unique set of quantities (as will be shown later), an attempt was

also made to introduce their equivalent into the formulation of a

corresponding Direct solution, which so far was lacking. As if by

design, two spheroidal power series resulted, again with a simple,

orderly pattern of terms. Moreover, the format was identical to

that of the Inverse, The three corresponding basic quantities were

denoted as a1, m. , and "0 . This Direct solution was derived, from

the formulas on pages 14 and 15 of [1]. Essentially, the power

series of the quantity S (now appearing also in Appendix E) was used

ft»-.V«"''«W.*l««tw«M««W^iBU((U*PW««*'«r " xamaas^stiutiiiisiisstn'ih

to solve for 0O; then the expression of 0O was in turn placed into the X power series, A critical factor in the mathematical determi-

nation of aj and lo^ was the proper choice of their common smaller

order term .Se'3 sin3 ß, , which is probably one in a series of others

required for the orderly theoretical extension of the solution to

higher degree.

JV. NOTES ON COMPUTATION FORMS

The above efforts led to the Inverse and Direct computation forms

now shown in Appendices A and B, respectively. The main Inverse

spheroidal expressions are indicated by (S T b-) and {\ - L) T c,

while the Direct by 0« and (L - \) r cos ß0. It is to be noted that

their outer coefficients are simple product combinations of a and m

or a^^ and n^ , while their bracketed expressions are functions of only

the variable 0 or 0 and powers of the spheroidal parameter f or e'2.

By tabulation of the functions of 0 and $„ so that they may be rapid-

ly obtained by interpolation, it would then be very simple to multi-

ply them by the easily determined outer coefficients. Since con-

ventional trigonometric tables can be used to obtain the first-order

term 0, and no tables are required to obtain the first-order term

0 , only short tables for the second- and third-order terms need be

drawn up. The third-order terms, which are small, vary sufficiently

slowly for visual interpolation.

In addition to their two pairs of main power series and the

basic quantities necessary to numerically evaluate them, these In-

verse and Direct computation forms generally contain one simple

closed trigonometric expression for each required final quantity ex-

cept when, for example, the trigonometric cofunction is given as an

alternative for occasions when a weak determination or unlinear in-

terpolation may otherwise result. The forms may appear cluttered

with rules for choosing signs, trigonometric quadrants, and so forth.

Actually, these do not entail any added calculations but simply de-

fine the problem without ambiguity. In addition, the choices pro-

vide for greater generality and more varied applications. For ex-

ample, one may calculate either the shorter geodesic between two

given positions, or the longer geodesic around the spheroid's back

side. Also, the subscripts 1 or 2 may be assigned to either position

without fear of ambiguity. For less accuracy, terms in f and e'4

may be omitted.

V. FORMULAS FOR VERY SHORT AND LONG GEODESICS

From its inception, the non-iterative derivation was developed

primarily for very long geodesies; therefore, the auxiliary trigo-

nometric functions which were correspondingly designed provided, the

greatest simplicity at the expense of generality. Recently, how-

ever, this Agency as well as the Army Map Service placed a requirement

i.-^IW.^rwMiWffWWl^NWWIMr^Jlf«**^«^

for a single set of formulas applicable to very short: as well as very

long lines, since they were to be used also in the adjustment of tri-

angulation and trilateration ground nets. It was felt that the basic

long line formulas given in Appendices A and B appeared particularly

convenient for the electronic computers which were to make the calcu-

lations. Howeverj in order to obtain the same or even greater accu-

racy for very short geodesies, the alternate formulas presented in

Appendix C were provided. The reason for the increased accuracy re-

quirement is easily understood when one considers, for example, that

if the length of a line is decreased a thousandfold, the positions

of the new endpoints must be known a thousand times more accurately

to maintain a constant azimuth accuracy. This means that the lati-

tude and longitude will have a greater number of decimals and, there-

fore, additional significant digits. In order to avoid the carrying

of too many fixed places, which are more apt to be affected by round-

ing errors if there is no spare digit capacity, Appendix C provides

formulas whose terms are generally very small when a geodesic is

very short, so the computations can be done conveniently by floating

point for greater decimal accuracy. To obtain the full required

accuracy, no additional terms need be added to the power series

formulas, because (as shown in the second paragraph of Appendix F)

they converge to more good decimal places for shorter geodesies.

Since many of the small quantities in Appendix C consist of sines of

small angles, their evaluation is especially adaptable to electronic

computers, which by means of floating point can readily calculate

trigonometric series that inherently converge to additional decimals

for such small angles. Actually, the formulas in Appendix C are

equally applicable to short and long lines, so only one set of

equations need be programmed into an electronic computer. A floating

point formula for a more accurate cosine of large absolute latitude

is also included in Appendix C.

VI. CONCLUDING REMARKS

Appendix D provides the complete numerical calculations for a

very short and a very long geodesic--! mile and 6,000 miles, re-

spectively. In each case, the Direct solution provides a check on

the Inverse. The discrepancies between the two types of solutions are given at the end of Appendix D. The better positional accuracy

provided for the short line by the formulas of Appendix C is con-

vincingly shown.

Appendix E provides a non-iterative inverse solution of higher

order accuracy (that is, through f3 and e'6 terms) for use as a

theoretical check on Direct or other Inverse formulas. In Appendix

F, several interesting types of inter-relations of the terms of the

power series are discussed and illustrated. These include relation-

ships between numerical coefficients as well as algebraic terms. In

Appendix G, meridional arc formulas are derived as special cases of

the Inverse and Direct,

.w*^l*liMi'iW»wwn«^;t«MwH»Whlkrf:i.Kll..w*Mii*t««M«i«..^^

In conclusion, it should be noted that the Inverse case of almost antipodal positions, treated on pages 24 through 25 of [1], is omitted here because of its rare practical occurrence. Also, the elimination of ß by substitution in terms of the given B is not undertaken because simple closed functions herein would become series expansions.

I

BIBLIOGRAI'HY

1. Sodano, E. M.: "A Rigorous Non-Iterative Procedure for Rapid

Inverse Solution of Very Long Geodesies,"

Bulletin Geode'sique, Issues 47/48, 1958,

pp. 13 to 25.

2. Bodemuller, H.: Beitrag zur Läsung der zweiten geodätischen

Hauptaufgabe für lange Linien nach den

direkten Verfahren von E. Sodano und H.

Moritz sowie nach dem sog. Einschwenkverfahren,

Braunschweig, im Juni 1960.

Appendi x.

B

c

0

E

F

G

APPENDICES

I tem

INVERSE COMPUTATION FOHM

DIRECT CC"rlPl:TI.TION FO .~

ALTERNATE I l\VERSE AA'TI DIRECT FORMULAS

NUMERIC.\L ILLUSTRATIONS OF INVERSE AND DIRECt

THEORETICAL FCHMULAS FOR HIGHER ACCURACY

INTER-REUT: ~ii.JS OF TiiE Tt: l<L';~ 0£ TilE POWER SERIES

MERIDIONAL ARC AS SPECIAL CASE OF NON­ITERATIVE INVERSE AND DIRECT

6

7

9

12

14

i9

21

^^«Mriu^^iiMrf^MIMtlM^iH^tKriWMUattaHWWIMrU

APPENDIX A

INVERSE COMPUTATION FORM

Given: B, , 1^ = Geodetic latitude and longitude of any point.

Bj, Lg = Latitude and longitude of any other point.

(South latitudes and west longitudes considered negative.)

Required: a, S = Geodetic azimuths clockwise from north and distance,

a«, b. = Semimajor and semiminor axes of spheroid,

f = Spheroidal flattening = 1 - H—

L = (L, - 1^) or (L, - Lj) + [sign opposite of (L, - L,)] (360°)

Use whichever L has an absolute value < or > 180°, according

to whether the shorter or the back-side's longer geodesic is

intended. However, for meridional arcs (II' =0° or 180 or

360°), use either L but consider it (+) for the shorter and

(-) for the longer.

tan ß = (tan B) (1 - f) when Ißl S 45°

or cot ß = (cot B) T (1 - f) when IBI > 45°

a = sin ß,. sin ßs ; b = cos ß^ cos ß3 ; cos 0 = a + b cos L.

+ sin 0 = - „/(sin L cos ß3)* + (sin ßg cos ß^ - sin ß, cos ßa cos L)'

The sin p is (+) for the shorter arc and (-) for the longer,,

Compute the radical entirely by floating decimals to prevent

loss of digits, especially for very short geodesies,

0 = Positive radians in proper quadrant, reference angle being

determined from sin 0 or cos 0, whichever has the smaller

absolute value.

c - (b sin L) T sin 0; m = 1 - c 3

W*»WwW.bi4**«M»WawWW»*l»WMWMI«W«^»*«>^»W^^ MaiMrmcn'w^<*»MNiavp)riMiMlWMi«iMklii-^:^^lMiUH^^^J*iMM(H(MilitiMl ummmamnatuvLtuiimiiMKMi

[(1 + f + f3) 0]

+ a [(f + fa) sin 0 - (i,_) 03 esc 0] fS M3 .O.

fS. + m [- (1±-L) 0 - (Li-L) sin 0 cos 0 + (1.) 02 cot 0]

+ as [- (JL) sin 0 cos 0]

„a r/f* •fSv „.- + ma [(i_) 0 + (£-) sin 0 cos 0 - (JL) 0s cot 0 - (£_) sin 0 cos3 0] 16 16 I 8

■c2 r2 + am [(£-) 02 esc 0 + (£_.) sin 0 cos3 0]

2 2

4^ = [(f + f3) 0] f2

+ a [- (y-) sin 0 - (f3) 03 esc 0]

5 f2 f3 -f m [- (---—-) 0 + (—-) sin 0 cos 0 + (f3) 03 cot 0] radians

cot «! a = (sin ßs cos ß1 - cos \ sin ßj cos ß2) T sin \ cos ß3

cot a3 ]_ = (sin ßa cos ß1 cos \ - sin ß1 cos ßa) T sin \ cos ß.

For meridional arcs, consider a as having 0° reference angle, and ob- tain only the signs of the cotangents by disregarding the denomina- tors. For other geodesies, replace cotangent by tangent when I cot aj >l, by taking the reciprocal of the quotient's value.

Quadrant of c^ _- 'Quadrant of o-g _1

If L is (+) ...and cot (tan) of o^g is (+) or (-), Q'^JJ is in quad I or II, respectively.

..„and cot (tan) of cfg. is (+) or (-), a3_l is in quad III or IV, respec- tively.

If L is (-) and cot (tan) of a, _g

is (+) or (-), Q'i.g is in quad III or IV, respec- tively.

...and cot (tan) of o-g,-, is (+) or (-), Og,! is in quad I or II, respec- tively.

■••WMfl-M-M^IUH/W-UIHM-llMaH./k-.fb. ■iaV^I|N'mii*yn«^«MM«r^Mjna«UMWMiitUiUM^[>l)||MIMWMM^IAMIMbk««IM^

APPENDIX B

DIRECT COMPUTATION FORM

Given: B1 , 1^ - Geodetic latitude and longitude of any point 1,

^i-a' ^ ~ Azimuth clockwise from north and distance to any point 2.

Required: Geodetic cfa i, B2 , and 1^.

(South latitudes and west longitudes considered negative.)

a0, bc = Semimajor and semiminor axes of spheroid. bo

f = Spheroidal flattening = l - —

e = Second eccentricity squared = (a* - b«) -f b*

tan ß = (tan B) (1 - f) when 'BI <: 45°

or cot ß = (cot B) T (1 - f) when Ißl > 45°

cos ß0 = cos ß1 sin c^ _2 ; g = cos ß1 cos ^ ..2 ;

ir^ = (1 + £l. sin3 ß^ (1 - cos2 ß0); 0g = (S T b0) radians:

a1 = (1 + £-_ sin3 ßj) (sin3 ß1 cos 0S + g sin ß1 sin 0g).

0o = [051

p l2

+ a [- ™ sin 0 ] L 1 o

e.2 el2 + ^ [- — 0S + — sin 0s cos 0s]

+ a3 [ ^ll sin 0 cos 0 ] Ü So

- r]Te14 13e'4 e'4 a Se'4 a + m^ [-^—- 0S " -54— sin0scos0s - — 0scos<s0s + -y~ sin0scos ^

r Be'4 ■ e'4 5e14 • 2 1 , • + a, 111, _— sm 0 + 0 cos 0 - _ sin 0 cos 0 radians 1 lL 8 S 4 S ^S 8 S VS-J

MW r™ ma <--*■•■ n« DWM w v tttf rftftVJu # »(« ■ i i*«»t w >. JI J >• ■—IM—. ^i,) ■> a M-"—"■ «• "ü "n ■ n "•» •»M H» t.^U ,i V^' ^ •><•■•■••« i' n HC«MlalU*tMiWIUU«flillM>«MMWIUI^lLi'iMwl Uai^UIB«

cot »;,_! = (g cos 0O - sin ßj sin 0O) -;- cos ß0

For meridional arcs, consider aa .■, as having 0° reference angle,

and obtain only the sign of the cotangent by disregarding the de-

nominator. For other geodesies, replace cotangent by tangent

when 'cot aa.jl > 1, by taking the reciprocal of the quotient's

value.

Quadrant of ^ _1

If (00^ai.8 s 180°) ....and cot (tan) of cfg^ is (+) or (-), as ! is in quad III or IV, respectively.

If (1800< CVLJ, < 360°) and cot (tan) of Qg_1 is (+) or (-), ffg_1 is in quad I or II, respectively.

cot X = (cos 01 cos 0O - sin ßj sin 0O cos o^ .2) r sin 0Q sin ^.3

For meridional arcs, consider \ as having 0° reference angle, and

obtain only the sign of the cotangent by disregarding sin a, a.

For other geodesies, replace cotangent by tangent when.

| cot \ i > 1, by taking the reciprocal of the quotient's value.

(Quadrant and Sign of \

When Qa<0o^l8O0

(sin 0O considered

positive)

When 18OO<0O<;36O0

(sin 0» considered

negative)

and

(O0^, .^180°) ...then if cot (tan) of \

is (+) off(-)) X is in quad I or II, respect- j irp 1 1.7

...then if cot (tan) of \

is (+) or (-), X is in quad III or IV, respect-

ively.

and

(180°^ 5<360o; .,.then if cot (tan) of X

is (+) or (-), the associ-

ated angle is in quad III

or IV, respectively, and

X is obtained by sub-

tracting 360°,

...then if cot (tan) of X

is (+) or (-), the associ-

ated angle is in quad I

or II, respectively, and

X is obtained by sub-

tracting 360°.

10

^MIH[l»M^«lMfM|l«M()#IMMlMHMl.'ikMM'Mhi(^ll>U»V>»^>^». **t**mmm*»iUiVM.w^*Vwn*i^Hnmt^ltMSdmlue„22,W™^Hrw

L - k

cos 0C

[- f 0J

+ c^ [_3|1 sin 0s]

+ ral [ML 0 - 1L. sin ^^ cos 0_] radians

Lg = LL +L

[If ILJI >1800, modify L., by adding _or subtracting 360°, according to whether it is initially negative o£ positive.]

sin ß3 = sin ^ cos 0O + g sin 0O

cos ß3 = V(cos ßo^2 + (§ cos 00 " sin h sin 0o)2

Compute the radical entirely by floating decimals to prevent loss of digits, especially for large absolute latitudes.

tan 3S = (sin j32 -s- cos ßs) Use whichever has the

or cot ß3 = (cos ß2 -I- sin ß2) smaller absolute value.

Obtain tan (or cot) of g from earlier defined relation of B to R. s

Determine (-900s;ß,ä905), applying sign of its tan (or cot).

II

i»i*«t»..'*im("*i"«i*iwt«.<wwiij<*lW*"M«^»».*MwaM^-i^i^^

APPENDIX C

ALTERNATE INVERSE AMD DIRECT FORMULAS (For very short: as well as long geodesies)

The following alternate formulas for corresponding ones in Appendices A and B are designed to maintain or appropriately in- crease the accuracy of various elements of short geodesies, without decreasing the accuracy of long geodesies. The formulas specifically take advantage of inherently small quantities and of small differ- ences of given large quantities, so as to provide--through the application of floating point calculations--increased decimal place accuracy without requiring additional operational digits. The small angles involved are especially adaptable to electronic computers, which by means of floating point can readily obtain greater decimal accuracy inherent in trigonometric power series of such small angles.

1. FOR INVERSE SOLUTION:

sin 0 = - yCsin L cos ßs)2 + [sin(ß8- ß^ + 2 cos ßg sin ßj sin2 T]2

cot aj a = [sin(ß3- ß1 ) + 2 cos ß2 sin ßj sin3 j] -f cos ß2 sin \

cot a3 l ~ [sin(ßa- ß1) - 2 cos ß1 sin ß8 sin3 1] H- cos ß1 sin \

where

(ßg-ßi) - (B2-B. ) + 2 [sinCBg-Bj] [(n + n3 + n3) a - (n - n3 + n3) b]

2. FOR DIRECT SOLUTION:

Bs = Bi + (ßs-ßi) + 2 [;3in(0a-0i)] [> + n3) cos (Bs+Bi) + n3cos(a3-.B1)]

Äere sm^ßg-ß^ - Si.n .-- ^os .„^-2 - £ =>j.n j sxn p:L cos ß3

ind the required approximate Ba and cos ga are obtained in Appendix B,

12

■ .OMftl^Vnr-^-v^r^Bi^aniioaiHMMMIrtUM^HAnM"»^

3. FOR INVERSE AND DIRECT AT GIVEN ABSOLUTE LATITUDES > 45°:

cos 0 = sin [(90 + P.) t 2 [sin (90 + B)] (n + n2 -I- n3) sin ß}

the upper and lower signs of which are applied for the northern and

southern hemispheres, respectively.

In the preceding three sets of formulas, n = (a,. - b.) -f

(a0 + b0). Some smaller coefficients of the almost negligible n3

have been removed because they are unsymmetric, and because they

become even smaller in Parts 1 and 2 for short geodesies and in

Part 3 for large absolute latitudes. It should be noted that terms

containing powers of n are in radians.

The accurate floating point calculations for short geodesies

should be applied not only to the formulas of this appendix but, in

turn, also to associated formulas in Appendices A and B, as illu-

strated numerically in Appendix D. The prescribed increase in deci-

mal accuracy in the sine of a small angle, for example, can be ob-

tained not only from the sine series, but also from trigonometric

tables by taking the reciprocal of the large interpolated cosecant

of the angle. However, in addition to sufficient significant

digits, the table should have intervals small enough for accurate

linear interpolation. Even better, of course, is a table of high

decimal accuracy for the small sines themselves.

13

!.PPENDIX D

NUNERICAL ILIJJSTRATIONS OF INVERSE AND or .t.CT

(GPodesics of approxi.rnately 1 <-' n~ 6,000 ml.le s [or t.:ac h)

111~ two extr~u1e cest oi:>!:ances notc:d above are chos en to il l •Js­trate b~· calculation not or.!y the basic computation forms of Appe n­dices A and B but also the a '.ternate fortrulas of Appendix C. The degree of !;Onsistency of the answers has been determined below by checking each Inverse solution a r ainst the correspor~ding Direct. The resulting discrepancies, which for each geode s i c arc :>ummarized at the end of this appendix, therefore represent the comb ined e rrors of the Inverse and Direct.

Inverse Solution Long Ceod~_ili Short Geodes ic

~ +2cP

~ +12° 11 I 18"

~ +45° OQ I 3611 • 5

~ +12° 12 I 09" o 5

8o (meters) 6 378 J8ij.OOO 6 378 :;ss .ooo

ho (meters) 6 356 911. 94f ~ ...: .3E 911. 946

f .oo 33670 03367 • oo 336 70 o:n6J

n • 00 16863 40641

L 51" .5

tan 8 1 .36274 47453 • 99663 2 9966

tan 62 • 99663 29966 • 0~698 ~ 7825

cos 61 . 94006 2327) .7082<": 81969

cos ~ . 7 08 2 9 81 96 9

~tr, 61 .34100 26695 . 705~1 33545

.; ir. 82 .70591 33545 • 70603 868L

14

L,,_.,1.***-».*.H.-I"«»'»"—'^.■'"••l''*-'(J

mm»mm*H*M*w™*w*'miWMmtMU»*uwM**u**'u*m»***mm»n**mm»m*tmHmt3ä tu&.nii.»iutiliat3»ll

Inverse So .ution Long Geodesic Short Geodesic

a .24071 83383 .49840 21342.

b .66584 44515 .50159 78502

sin L .96126 16959 .000 24967 90432

cos L -.27563 ■73558 .99999 99688

COS 0 .05718 67343 .99999 99687

sin (Bg - Bj^)

(ßs - ß1) radians

sin (02 •■ ßj

sin" 2

sin 0

0 (radians)

m

(S v b0)

S (meters)

(A - L) ^ c

A. (radians)

sin X

cos X

.99836 34996

1.51357 83766

.64109 99269

.58899 08837

+1.51869 17590 + .00080 87665 - „00156 22320 - .00000 00188 + „00000 01279 + ,00000 18468

9 649 412.505

+ .00511 33825 - .00000 76243 - .00001 16616

1,85331 48325

.96035 63877

-.27877 5192/

s in X

.000 17695 69927

.000 17695 60928

.000 17695 60919

.000 00001 55849

,000 25016 57049

.000 25016 57075

.50062 20631

.74937 75499

+.000 25101 08523 +,000 00042 05152 ■■,000 00063 22699 -.000 00000 03522 -.000 00000 07962 +.000 00000 10592

1 594,307 213

'+,000 00084 51448 ~,000 00000 21203 .000 00000 00000

.000 25010 10825

.000 25010 10799

,99999 99687

.000 00001 56376

._

15

Mttiw»^(i44t»«"*itrifwi»ww»icmiWlr;»i»wMt'Wiw»a4m«r.fw"iM»M«»»i;«'^i'ifla'« »•* W«U« n MUM W >< f'l« ht 1 • M ) I f >»)« ti «i^rf«lllliW»ifcWWttlti5t^lW«iM»»*)HIM*l*»lM*llilWil(lMl«l*«miM«WiliilIW«B^ tHfllinVHIMhlllMM

Inverse So iution Long Geodesic

cot Qf1-a 1.07455 96453

cot »3-! -.47245 22960

»1-2 42056,30".03503

Cfo 295017,18".59981

Short; Geodesic

.99919 16383

.99883 88553

45001,23".40210

225001,59".82121

Direct Check

h

Q'l.s

S (meters)

a0 (meters)

b0 (meters)

f

e'2

n

tan ^

cos h sin ßi

sin al„2

cos «1 -S

cos ßo

g

l\

08 ^radians )

Long Geodesic

+20°

0°

42056,30".03503

9 649 412.505

6 378 388.000

6 356 911.946

.00 33670 03367

.00 67681 70197

36274 47453

94006 23275

.68125 35334

.73204 75552

.64042 07822

.68817 03286

.59009 33386

1.51794 02494

Short: Geodesic

+45°

+12°11'18"

45o01,23".40210

1 594.307 213

6 378 388.000

6 356 311.946

„00 33670 03367

.00 67681 70197

„00 16863 40641

.99663 29966

.70829 81969

.70591 33543

.70739 26381

,70682 08083

.50104 49301

.50063 99041

.25146 93699

.000 25079 90085

16

n rl«l>in«NR«l««« Ml MUlitJfoiW WMS4.I. *••<*-• «.-mH.MiJ^M.« «i-^^™*». >MU I »»■•,*<'to >'|l Mi<«< '""^"•^^"•"^^«HtWft'rtÄi**«^™*^»«»«"»'»^!*««.«^»!»^ •«UUll»iMM|MMM^uiMWHIMr>*«tMWMaUIUU||IMIiCVIUWMUIIIIUWli(t^lUU^^^ihKi)

Direct Check

sin 0

cos 0

0O (radians) =

sin 0O

cos 0O

cot afg»!

cot. X =

X

(L - \) T cos

L (radians)

sin ß3

cos p2

tan ßs

tan B 3

Long Geodesic

.99860 34425

.05283 14696

.24057 82171

+1.51794 02494 .- .00081 30002 - .00146 29306 + .00000 00374 + .00000 39825 + .00000 25543

+1.51567 09428

.99848 09807

.05509 74693

-.47245 22450

295017,18".59121

Shojrt Geodesic

.000 25079 90059

.99999 99685

.49924 27565

+.000 25079 90085 -.000 00042 37199 .000 00000 00000

+.000 00000 17897 .000 00000 00000 .000 00000 00000

+.000 25037 70783

.000 25037 70757

.99999 99687

.99883 88542

P^Ol^".82132

-.29028 29979 tan X = .000 25010 10884

1060ll'13n.61256 51".58705 146

-.00511 09099 +.00000 40853 +.00000 73513

1.85004 89647

105059,59".99117

,70591 33687

7 0090 B1,Q9Q

.99663 30364

1.00000 00399

45°00'00".00411

-.000 00084 44411 +.000 00000 21292 .000 00000 00000

.000 24967 90471

12° 12 W.50000 027

.70603 86812

.70817 32700

.99698 57817

45o00l36".49992

17

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Direct Chec k

sin" 2

sin % - ßl)

(ß3 -ßl) radians

cos (Ba + \)

COS (B8 - V \ i improved value)

Long Geodesic Short Geodesic

.000 00001 56376

.000 17695 60922

.000 17695 60931

-.00017 69566

.99999 99841

45o00,36".50000 005

Discrepancies between Inverse and Direct

^s-i

Long Geodesic

0".00411

0".00883

0".00860

Short_Geodesic

0".00000 005

O".00000 027

0".00011

In addition, the preceding Inverse and Direct illustrative examples contain several common intermediate and secondary components whose values can be compared. Also, since the solutions of the long geodesic are illustrated by the same numerical problem that was used in reference [1] for the earlier form of the Inverse method, oppor- tunities for other comparisons are available. It is apparent that the extremely high positional accuracy for the short geodesic is due to the use of alternate formulas given in Appendix C. The azimuth error is consistent with this positional error, in view of the line's shortness. Comparable accuracies are also obtainable at large absolute latitudes, but only if interpreted relative to the in- creasing convergence and closeness of the meridians in polar

regions.

18

vmtZUUiwitimviUiAtoMt* t'f*f?i[3W'"M»i'ywti -■ wiiwirnMimii

APPENDIX E

THEORETICAL FORMULAS FOR HIGHER ACCURACY

The results of ehe illustrative numerical examples given in Appendix D indicate that the formulas in Appendices A through C provide sufficient practical accuracy. For theoretical purposes, however, the formulas could be extended through f3 and e'6 terms or beyond. The outer coefficients of the formula for (S 4- b-) in Appendix A would then include, for example, the higher order combi- nations a3, m3, a8m, and am3. Similar orderly extensions should be expected for the (A. - L) T c formula in Appendix A and the 0O and (L - A.) T cos ß in Appendix B, except that in the case of the latter two their outer coefficients will bear the subscript 1, and their components a1 and n^ would have to be properly defined to higher powers of e'2. If necessary, appropriate formulas in Appendix C can also be extended.

In the present appendix, only the (X - L) T c power series of Appendix A will be given to the next higher order terms, since it provides a non-iterative rigorous solution for the quantity X which is required in most of the classical methods for calculating the. Inverse of long geodesies. The unique form of the extended (\ - L) - c power series given below has been derived from the top of page 18 of [1], by substitution in terms of a, m, 0, and f. The series is followed by accurate Inverse distance and azimuth formulas taken in large part from pages 14 and 15 of reference [1]. The resulting method of solution can be used for precise computation of Inverse problems, especially as a. theoretical check on Direct or other In- verse formulas.

X - L [(f + f2 + f3) 0]

-h a [ - (-- + f3) sin 0 - (f2 + 4f3) 02 esc 0

3f3 4- (-7-) 03 esc 0 cot 0]

.'if2 , f3 f3 H- m [- (-~- + 3f3) 0 + (-- + -—) sin 0 cos 0

+ (f3 + 4f3) 03 cot 0 - (-?-) 03 esc2 0 - (f3) r CötJ

+ a3 [(f3) 0 + (y~) sin 0 cos 0 + (f3) 03 csca 0]

19

■■*^W**>«1-~W^*'>|[toM^M*WWWH^M«M<.*«^^ ''?***^V*<*'f***>*rwwtn»*tumiuwtilmmt™&iiw*m%w

„a r,31f3 .9f2 , f + m2 [(.^±-) 0 - (ii-) sin 0 cos 0 + (---) 0 cos2 0

- (~~) 03 cor. 0 + (.L) sin 0 cos'3 0 2 8

+ (L.) 03 esc13 0 + (2f3) 03 cot2 0]

'3fS ^ _. . , /9f3 + am [(f3) sin 0 - (££_) 0 cos 0 + (Z£-) 03 esc 0

.p3

(—-) sin 0 cos3 0 - (—T-) 03 esc 0 cot: 0] radians

where the component quantities are again defined in Appendix A, while some alternate definitions are found in Appendix C.

Next, 0« is obtained in the same manner as 0, except that the value of A. obtained from above is now to be used in place of L. Then continue as follows:

cos ß0 = (b sin \) -r sin 0O ; cos 2a = (2a -f sins ß0) ■- cos 0Q.

18 AQ - 1 + i-1 sin2 ß0 - hl sin4 ß0 + ^ ^in' |](

64 256 16

o'2 ,s ' e'4 A l^e' ■ 6 n

30 = Y sin to - frsin ß°+ "512"sin 0O

0'4 3e 16

Co = 128 Sin ßo " sTT Sin ßo

16

D0 = 1536 Sin ßo

S = b0 (A. 0O + B0 sin 0O cos 2a - C0 sin 20o cos 4a

+ Dn sin 30o cos 6a)

To complement the above geodetic distance, S, the azimuths Q'L _ and a3 , are obtained from formulas given in Appendix A or

20

■■■.wwiM>umww..«w»..MM»»»t«**»^»»*fc*Mi"««M^^^

APPENDIX F

INTER-RELATIONS OF THE TERMS OF THE POWER SERIES

As noted earlier, the coefficients a and m in the (S T bö) and

(X - L) f c Inverse power series in Appendices A and E display a

unique set of product combinations. The identical simple pattern

is also repeated in the two Direct power series in Appendix B, ex-

cept that it occurs instead with the subscripted a, and n^ , Al-

though not shown in this paper, even the higher degree combinations

(such as a2m, m2a, a3, and m3) appear to enter in orderly fashion

in the further extension of the power series. It is of significant

importance that the a and m (or a1 and m, ) combinations are com-

pletely factorable from the power series terms, since this permits

the latter to be tabulated as a function of only the variable 0 or

0g and the parameter f or e'3. Electronic computer programming and

calculations also become simpler, whether for producing just the

table or for calculating the entire Inverse or Direct.

Another interesting inter-relation of the terms of the series concerns the numerical coefficients of the powers of f and e'2.

It should be noted, for example, that in Appendix B the numerical

coefficients related to the TX% terms of the 0- power series are:

11 13 ' 1 5

64' " 64' " 8' 32"

The total of the above four numbers is found to be exactly zero.

Upon closer inspection, it is found from the power series in Appendi-

ces A, B, and E that the zero sum occurs with all sets of terms

having m or n^ as one of the factors, even for the (S -r b0) series

in Appendix A, if it is modified as shown later. When different

powers of f are present, the sum is zero separately for the numeri-

cal coefficients of the f terms, f2 terms, and so forth, such as

in the (\ - L) -r c series in Appendix E. In all instances described,

the sum is zero by virtue of the fact that each term---which is a

function of 0 (or 0o)--is first put into a form which satisfies the

following condition: The algebraic sum of the exponents of 0 and

sin 0 (after all trigonometric functions of 0 are converted to sines

and cosines) is unity. Actually, the above condition can be (and

has been) satisfied even for the non-m and the non-m, series terms.

For very short geodesies (which of course have a small arc value 0

and, therefore, sin 0 approaches 0 and cos 0 approaches unity), the

resulting unity exponent implies that every term is of the small

order of 0, times its numerical coefficient and the proper power of

f or e'2. Since even the omitted terms of the series contain that

small order of 0 (or An), the power series converge to a greater

number of decimals for short geodesies. This is shown by the much

better positional consistency obtained from the numerical example

for the short geodesic in Appendix D. For terms which have m or m.

as one of the coefficients, the convergency for short geodesies is

even greater because (as noted above) the. sum of the numerical coef-

ficients is zero separately fo1* each power of f or e , and ^ or 0„

is practically a common factor.

As for the (S -r b.) series mentioned in the preceding para-

graph, the expression given in Appendix A can be reduced to the

following form:

~= [(l + f+f2)0]

+ (m cos 0 - a) [- (f + f2) sin 0 + (i~) 0a esc 0]

+ m r ,f + f2 A , ,f + t\ . , , [- (—2 ) 0 + (~—2 ) sin 0 cos 0]

+ (m cos 0 - a)2 [■- (y-) sin 0 cos 0J

+ m jr8 rS cS

[(jg) 0 + (jg) sin 0 cos 0 " (3-) sin 0 cos3 0]

c2 ^2 r

+ m(m cos 0 - a) [(7-) sin 0 coss 0 - (y-) 02 esc 0]

The compound coefficient (m cos 0 - a) is an expression which

appeared extensively in the course of the original derivation of

the Inverse solution. As used above, it causes the numerical coef-

ficients of the terms with the factor m to add to zero, just like

the other power series. It is interesting to note that the next

higher order extension of (S -f b0) continues to give the proper

zero sum for the numerical coefficients of applicable terms, when

the additional prescribed product combinations of the same

(m cos 0 - a) and m are used.

In conclusion, it is worth noting that, of the four main power

series given in Appendices A and B, only (S -f b.) does not lend it-

self to completely factoring out the ellipsoidal parameter from each

series of terms. The capability of factoring for all four power

series (at least to the extent of the number of terms given) may be

important. It would mean, for example, that the total value of

each series of terms could be tabulated independently of any specific

spheroid flattening or eccentricity. (Of course, the parameters

would then be made a part of the external coefficients instead.)

In the (S -f b0) formula given in the present ippendix, only the

terms whose coefficient is (m cos 0 - a) do not lend themselves to

factoring out the function of flattening. Those terms, however,

can be represented as in the following:

22

^«»»««IWM-MM.WWW^MW^MWÄMl^MM^

[ (in cos 0 - a) (1. - £ f ) ] [ - (f + f2) sin 0 ] z s in 'li

where the unwanted portion of flattening has been transferred to the external coefficient. This new compound coefficient may be used in place of the previous (m cos 0 - a) throughout the (S T b0) expression for consistency, since the extraneous f3 terms which are introduced are negligible.

23

Mtt(Wm*Nit|HWE;iMMt^1MiMllteMMilMMMk-MIU4M(IM.Mi«.^

APPENDIX G

MERIDIONAL ARC AS SPECIAL CASE OF NON-ITERATIVE INVERSE AND DIRECT

An interesting indication of the simplicity and rapid con- vergence of the non-iterative inverse is to reduce it to the special

case of meridional arc distances, for northern latitudes up to 90°

from the equator. Since ß. and L are then 0°, the following result:

a=0, m=l, 0=ßs radians.

Therefore, such meridional distances, S , become:

S = b0 [( 1 + 1 + ||- ) ß2 - ( | + [f ) sin ß, cos ß2

( L ) sin ß3 cos3 ß ]

Similarly, ß2 can be derived for the corresponding S by

letting ß1 and a. _,, equal 0° in the Direct solution, whence:

a1 = 0, i^ = 1, ßs - 0O.

Then by substitution into the 0O power series, there results:

P'2 UP'4 P'

3 np'4

M

, , , , rflH i An 32 ' yM ^M v 8 M yM

5P ,4 , , p,4

( -rrr-- ) sin 0 cos3 0 - ( —-— ) 0 cos3 0 radians.

where 0 = ( S T bn ) radians. M M 0

As a check (the complete details of which need not be shown),

the above Inverse and Direct meridional arc solutions were compared

mathematically and found to be fully consistent with each other.

Essentially, dividing the Inverse meridional formula by hQ produced

0 as a function of ß2 , from which sin 0M and cos 0^ were then ob-

tained by expanding in series around sin ß2 and cos ß3, respectively.

Substitution into the. Direct meridional formula finally made the

right side identically equal to the left side's ßg, up through all

e'4 terms.

24 Vlt.MY - l;Uh 1 HI.I.UJIU,

.,i<.,.mm.*M,M.*.,*~***.'.*.m*~~~*,:>~-,m\Mi~'~~-.:.~.~*.~. MMM^^Pl^^MW^'r^^WM^^tMWMlMWkWH^V^UMWt^bUUMklMMMMMMIUgttrtMaMMMraWlWUtiWl

Section

1

II

III

IV

V

VI

CONTENTS

Title Page

LV SUMMARY

INTRODUCTORY BACKGROUND 1

THEORETICAL REFINEMENT OF NON-ITERATIVE INVERSE I

DEVELOPMENT OF CORRESPONDING DIRECT 1

NOTES ON COMPUTATION FORMS 2

FORMUIAS FOR. VERY SHORT AND LONG GEODESICS

CONCLUDING REMARKS

BIBLIOGRAPHY

APPENDICES

2

3

5

6

in

SUM.'-IARY

I:np1.·oved pr .1~ tic<tl a·!d theor ..., t i ~.: a l fo r111u l.us are f::=- ...: ;ent c d fo~ th e. (" ::lLcuiation of g~od ~tic dist..lnr. s, o·.: :!lUth s , ."l,rl rus iti0::1S 0"

sph(~ roic. the fo::•nul a:.; ax e designed for us·~ •.dth e i t h~ r e l ec troni c:..;mp·•ter s or desk ca l c- '..l .. <J tors. For the latter, the f o r mulas 1 e nd r:hen •. :;elves to ::h e .·ons cr.uction of u :·;aful inter ,).:>lation tables .

Tite r eport i~;; ~ud r~:; ::om enient c omp'.Jtat ion :: lJ r ,ltS and a•1xil iar equa: ions \"ltich n!'::-;ure t . high de gree of a-::n1ra·: y fot· a ny geodetic line, no matter ho'A shorl: or l o:tg (up to h.1lf o r. fully around the eartt!) and regardl€:s~: of 1.ts nrientation or lo:?..tio:t. Num:!rica l examples illustrate the compl Pte calculation p roc~d~rc .

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