The Three-point Problem of The Median Line Turnina Point: on the Solutions for the Sphere and Ellipsoia

Transcription

1 The Three-point Problem of The Median Line Turnina Point: on the Solutions for the Sphere Ellipsoia Lars E. Sjôberg, Royal Institute of Technology, Division of Geodesy, Sweden The problem of determining turning points of median lines between states separated by sea is considered. The turning point is defined as the point with equidistance lines to two basepoints along the shoreline of one state one basepoint in the adjacent state. For the sphere the equidistance lines are parts of great circles, the problem is solved by closed formulas. For the ellipsoid the lines are defined along geodesics, an iterative solution is presented. Introduction The median line is a line every point of which is equidistant from nearest points on the baselines of two states (IHB, 1993). This line is crucial for the maritime delimitation between opposite coasts of two states separated by sea. In general, the median line runs smoothly as the straight line' equidistant between two distinct baselines, each belonging to one of the states. However, whenever points belonging to another baseline along one coastline get closer to the median line than points on the previous baseline, the median line turns. As suggested by Carrera (1987) the median line turning point can be defined by a three-point method, i.e. the turning point is the point equidistant to three baseline points (belonging to different baselines). If the scale of the baselines (between defined basepoints) is small compared to the scale of the coastline separation, the points along baselines can be approximated by the discrete basepoints. This approximation will be used here to determine median line turning points. This problem was also treated by Carrera (1987), Horemuz (1999), Horemuz et al.(1999) Fan (2001). Carrera(1987) solved the problem by classical formulas for solving the geodetic direct indirect problems for geodesics on the ellipsoid. Horemuz (1999) Horemuz et al. (1999) solved the problem explicitly by rectangular coordinates for the spherical surface of reference, while an approximate method was used for the ellipsoidal surface of reference. In this paper the three-point turning point problem is solved explicitly for the sphere by spherical co-ordinates. For an ellipsoidal surface of reference the problem is formulated by integral equations along geodesics. Solutions for the Sphere Proposition: Given the points P, ((pi h) with latitudes tp, longitudes X,; i = 1,2,3, the three-point problem has the solutions

2 Case I : 2): where S 23 cos (p, co s À, - S j 3 cos <p2 cos X 2 + S12 co s cp3 cos t^n A - - S 23 cos cp, sin / -, + 8,3 cos <p2 s in A 2 - S 12 co s (p3 sin A 3 _ co s cp2 cos A X 2 - cos <Pj cos A X, tancp = S;j = sin<p; -s in c p j ; i,j =1,2,3 A X i = A - À, ; i =1,2. s 12 (la ) (lb ) Case II : (q)i= 925* 93):, c o s À, cosx 9 ta n X = » r 2- s in À 2 - s i n A, coscp, cosaà, -coscp, cosaà, t a n < p - -Î2 2 L (2b) S 13 Proof: The point P(tp,A,) is defined by equal geocentric angles to the known points Pi ((pi,à,);i=l,2,3. From the spherical cosine theorem one obtains the following relation between \ /i the spherical co-ordinates of the points P Pi : cos \ /, = sin cp sin<p; + cos cp cos cp; cos AÀ, ; i =1,2,3. (3) (2 a) As P (cp,à) is defined by \ /i=v /2=\j/3, it follows that sincpsincp, + coscpcoscp, cos A À, = sincpsincp2 + coscpcoscp 2 c o s A À,2 = (4) or = s in <p sin<p3 + cos cp cos cp3 co s A A 3 ta n cpsin cp,+ coscp, c o s A À, = tancpsincp2 + cos< p 2 c o s A À 2 = (5) = tan cp sin cp3 + co s cp3 co s A A 3 The first equation of formula (5) (including points Pi P2) can be rewritten: tan cp (s in cpj - sin cp2 ) = cos cp2 cos A A 2 - cos cp, c o s A A, Same treatment of the second equation of formula (5) yields: tan cp (s in cp, - s in cp3 ) = cos cp3 co s A X 3 - cos cp, c o s A À, Let us now consider the different cases of the proposition. (6 a) (6 b) Case I (91 * <pi): In this case (p is eliminated by dividing each member of Eq. (6 a) by (6 b). For (pi * <f>3the result is Inserting into (7) 5. 2 _ sin cp, - s in cp2 _ co s cp2 cos A X 2 - cos cp, co s A À, 5. 3 s in cp, - sin cp3 co s cp3 cos A À 3 - cos cp, co s AA., cos A A j = co s À cos À, + sin À sin À j one easily arrives at the solution (la ) for X, (lb ) is directly obtained from Eq.(6 a). If (pi = <ps, the left h side of (6 b) vanishes. Together with (8 ) it yields t a n ^.sinx.-sma, C0 S À, - c o s X 3 (8)

3 which formula agrees with (la ). Case II (q>i = <p2 * cp3): Formula (8 ) inserted into Eq. (6 a) with Su= 0 yields cos X co s X2+ sin X sin X2= cos X co s X l + s in X s in A, which can be rewritten on the form (2a). The solution (2b) follows directly from (6 b). Q.E.D. The solution for Case I was also derived by Fan (2001). Although the above solutions of the proposition are mathematically exact, they may suffer from numerical instability in the practical application. One improvement is gained by substituting the differences of sines of St by Sy = 2 cos cpy s in (A (p ;j / 2 ) (10) where cp*, = (cp* + <pj)/2. Acp = <p, - cpj A similar improvement can be achieved for the differences of cosines of (9). To avoid the division by near zero for small latitude /or longitude differences among the known points the following corollary may be useful. Corollary 1: The coordinates À <p of the median line turning point is given by s i n À = ±, c o s À = ± ( i l ) D[ D, (1, A,, B, (12) sincp = ± -, coscp = ± D2 ^ D2 where Ai Bi are the numerator denominator, respectively, of the right h side of E q.(la), D j = y j A,2 + B f = ^ S 23 c o s 2 cp, + S f3 c o s 2 (p2 + S,22 c o s 2 (p3 A 2 = cos (p2 cos A X 2-2 cos (p, co s A À, + co s (p3 c o s A À 3 B 2 - S 12 + S i3 *14a^ (14b) d 2 = V a 2 + b }2 *2 2 (14c) The proof is given by simple trigonometric manipulations of Eqs. (la ) (6 a,b). Unfortunately, there is still the possibility that for small baselines the denominators D1 D2 may be small, making the solutions for X <pnumerically unstable. This problem must be further studied. Corollary 2: The azimuth» at point Pi (cpi,x.i) along the great circle towards P (,¾.) is given by sin (X X: ) MG') tanocj = ^ ; i = 1,2,3 (15) cos ^ tan cp - sin <Pj cos (A - Xi ) The proof is given e.g. in Sjôberg (2002). Solution for the Ellipsoid In the case of an ellipsoidal surface of reference it is convenient to introduce the reduced latitude P related by the geodetic latitude <pby the relation (16) tan (3 = V I - e 2 tancp where e = \ja 2 b 2 /a is the first excentricity of the ellipsoid defined by the semi-major -minor axes a b. The distance (s>) the ellipsoidal longitude difference (AL) from the point Pi ([}i, L) to the wanted turning point P ( 3, L) along the geodesic are given by

4 4 (17a) si = j f ( P, h i ) d P = F ( P, h i ) - F ( p i, h i ) where M., = L - L, = / g ( P,h i ) d p = G ( P, h, ) - G ( P 1, h i ) (18a) ç fn, \ 1 - e cos P Q f ( p,h ) = a j ; cosp v y cos P - h f (17b) g ( P, h f ) = ± 1 - e 2 c o s 2 P, ( i 8 b) c o s p \ c o s 2 P - h f Here hi = cos W, where Pm«is the maximum (or minimum) latitude of the geodesic. Alternatively, using the variable substitutions s in P sin v = V l - h 2 (19a) h q sin w =. tan p (19b) V l - h 2 the integrals (17a) (18a) can also be written (Klotz ; Schmidt ): v (2 0 a) Si = J f *( v, hj ) d v = F* ( v, h; ) - ( V j, h j ) W A L j = J g* ( w, h j ) d w = G* ( w, h j ) - G * ( w ^ h j ) (2 1 a) where f * ( v, h i ) = a ^ l - e 2 { l - ( l - h 2 ) s in 2 v } (2 0 b) g V ( w, hu\ i ) = ± + /i e2h' (21b) h? + ( l - h f ) s in 2 w The three-point problem can now be defined by the equations Sj S2 S3 (2 2 a) L = L 1 + A L 1 = L 2 + A L 2 = L 3 + A L 3 (22b> where si A L are given by Eqs. (17a) (18a) or by Eqs.(20a) (21a). The longitude L can be eliminated from (22b), yielding four independent equations with four unknowns ( 3, hi, h2, ha). The system of equations can thus be written

5 s, s2 (23) s, = s 3 Lj + AL, = Lj + AL2 Lj + ALt = L 3+ AL>3 (24a) X T = (A P,A h,,a h2,ah3) (24b) vt = { ( s =) - (s, ).( ¾ ) - (s, ), l 2- ( L 2 ) - Li -t-(l, ), -<l 3) - L, + (L, )} Here the bracket ( ) denotes an approximation to the quantity within the bracket, determined by p HÎ. The vector of unknowns X contains improvements to the approximate values p, hs, h9 hs. The elements of the (4x4) design matrix A are presented in the Appendix. In order to determine the elements of Y A the integrals (17a) (18a) or (20a) (21a) must be employed, e.g. by series expansions or direct numerical integrations (Klotz ; Schmidt ). Starting values for (3, v hpare preferably given by the spherical solutions of Section 2. As the equations are linearised, the solution should be iterated. Concluding Remarks The solution of the position of a median line turning point from the three-point problem was derived explicitly for the sphere as an iterative vector solution for the ellipsoid. The problem with possible unstable solutions for small baselines deserves further attention. However, numerical examples are left for a forthcoming paper. A future challenge is to avoid the approximation by the three-point problem to determine the position of the turning point directly from all points along the baselines. References Carrera, G- (1987): A method for the delimitation of an equidistant boundary between coastal states on the surface of a geodetic ellipsoid. International Hydrographic Review, 64(1), Fan, H. (2001): Geodetic determination of equidistant maritime boundaries. (In preparation.) Horemuz, M. (1999): Error calculation in maritime delimitation between states with opposite adjacent coasts. Marine Geodesy, 22, 1, 1-17 Horemuz, M L. E. Sjoberg H. Fan (1999): Accuracy of computed points on a median line, factors to be considered. Proc. Int. Conference on Technical Aspects of Maritime Boundary Delineation Delimitation, International Hydrographic Bureau, Monaco, 8-9 September, 1999, pp IHB (1993): A manual on technical aspects of the United Nations convention on the law of the s e a Special publication No. 51, International Hydrographic Bureau, Monaco Klotz, J. (1991): Eine analytische Lôsung kanonischer Gleisungen der geodatischen Linie zur Transformation ellipsoidischer Flachenkoordinaten. Deutsche Geodàtische Kommission, Muenchen, Reihe C, Nr 385

6 Klotz, J. (1993): Die Transformation zwischen geographischen Koordinaten und geodàtischen Polar- und Parallel- Koordinaten. Zeitschrift fuer Vermessungswesen, 118, 5, Schmidt, H.(1999): Lôsung der mittels geodàtischen haupaufgaben auf dem Rotationsellipsoid numerischer Integration. Zeitschrift fuer Vermessungswesen, 124, , 169 Schmidt, H (2000): Berechnung geodâtischer Linien auf dem Rotationsellipsoid im Grenzbereich diametraler Endpunkte. Zeitschrift fuer Vermessungswesen, 125,2, Sjôberg, L. E. (1996): Error propagation in maritime delimitation. In: Proc. Second International Conference on Geodetic Aspects of the Law o f the Sea, Denpasar, Bali, Indonesia, 1-4 July 1996, pp Sjôberg, L. E. (2002): Intersections on the sphere ellipsoid. (Journal of Geodesy, in press), 76: Appendix The elements of the design matrix A are as follows: A = 9s, ^13 a I4 = o ' V * 3f _ ah 'f f ( v.h f ) = 1 T^(P h )cospdp = 2 f, <lv v 1 y - ft 1ah- l^ hi J l-(h ) v, cos v ds2 fa s,] A2 V ap r J ^22 _ ^12 A23 =0 A 24 A,i = ^32 ds3 f 3s0 1 - ( ¾ ) I ^ ap ^ J cos2 V _. ~m -ah 3 r f* (v,h ) dv i - ( h? ) 2i dal, j [ 0AL. 3P 0AL, 9h, ap - g ( e ".h r )-g (p,h ;) A 33-3AL2 h fg * (v.h ) 2 f l - ( h )2 { COS V dv A4,= A 42 - A 32 a 43= o f dal, f 0A L 3 ^ _ h? f g* (v.h? )d.. dh, \ 1 J k 9h3 J l - ( h? ) v, cos2v 1 i - ( h! ) '! dv h j _ J ( v X ) dv

7 Biography Lars E. Sjôberg got his Ph.D. ir Geodesy in 1975 at the Royal Institute of Technology (KTH) in Stockholm. After working with professor R.H Rapp at The Ohio State University during about four years work with the National L Survey of Sweden, he returned to KTH in 1984 to succeed his old professor on the chair of Geodesy. Through the years he has been a member of several IAG special study groups (chairing three of them) commissions, he is an IAG Fellow since He is a presidium member of the Nordic Geodetic Commission since 1982, a Fellow of the Alexer-von-Humboldt Foundation since 1983 a corresponding member of the German Geodetic Commission since His research interests are in the fields of geodetic theory of errors, physical geodesy, GPS positioning deformation analysis. From 1989 to 1995 he served in the editorial boards of Manuscripta Geodaetica Bulletine Geodesique, he has published about 200 scientific papers reports. sjoberg@geomatics.kth.se