The Three-point Problem of The Median Line Turnina
Point: on the Solutions for the Sphere and Ellipsoia
Lars E. Sjôberg, Royal Institute of Technology, Division of Geodesy, Sweden
The problem of determining turning points of median lines between states sepa­
rated by sea is considered. The turning point is defined as the point with equidis­
tance lines to two basepoints along the shoreline of one state and one basepoint
in the adjacent state. For the sphere the equidistance lines are parts of great cir­
cles, and the problem is solved by closed formulas. For the ellipsoid the lines are
defined along geodesics, and an iterative solution is presented.
Introduction
The median line is a line every point of which is equidistant from nearest points
on the baselines o f 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 dis­
tinct 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 approxi­
mation will be used here to determine median line turning points. This problem
was also treated by Carrera (1987), Horemuz (1999), Horemuz et al.(1999) and
Fan (2001). Carrera(1987) solved the problem by classical formulas for solving
the geodetic direct and indirect problems for geodesics on the ellipsoid. Horemuz
(1999) and Horemuz et al. (1999) solved the problem explicitly by rectangular co­
ordinates 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 prob­
lem is formulated by integral equations along geodesics.
Solutions for the Sphere
Proposition: Given the points P, ((pi h ) with latitudes tp, and longitudes X,; i = 1,2,3,
the three-point problem has the solutions
Case I :
¢ 2):
t^ n A
and
S 23 co s (p, c o s À , - S j 3 co s <p2 co s X 2 + S 12 c o s cp3 co s
——
--—
—S 23 cos cp, s in / - , + 8 ,3 co s <p2 s in A 2 - S 12 c o s (p3 s in A 3
(la )
(lb )
_ c o s cp2 co s A X 2 - c o s <Pj c o s A X ,
tancp =
s 12
where
S;j = sin<p; - s in c p j ; i,j =1,2,3
and
A X i = A - À , ; i =1,2.
Case II : (q)i= 925* 93 ):
,
c o s À ,—c o s X 9
t a n X = -------- - » ------------- r 2s in À 2 - s i n A ,
and
ta n < p -
( 2 a)
coscp, cosA À , -coscp, cosAÀ,
-Î 2
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 and the spherical co-ordinates
of the points P and Pi :
co s \|/, = sin cpsin<P; + cos cp cos cp; cos AÀ, ; i = 1 ,2,3.
(3 )
As P (cp,À) is defined by \|/i=v|/2=\j/3, it follows that
sincpsincp, + co scp co scp , co s A À , = sincpsincp 2 + coscpcoscp 2 c o s A À ,2 =
(4)
= s in <p sin<p3 + c o s cp co s cp3 c o s A A 3
or
ta n c p s in c p ,+ c o s c p , c o s A À , = ta ncpsincp 2 + co s < p 2 c o s A À 2 =
(5)
= ta n cp s in cp3 + c o s cp3 c o s A A 3
The first equation of formula (5) (including points Pi and P2) can be rewritten:
ta n cp ( s in cpj - s in cp2 ) = c o s cp2 c o s A A 2 - cos cp, c o s A A ,
( 6 a)
Same treatment of the second equation of formula (5) yields:
ta n cp ( s in cp, - s in cp3 ) = c o s cp3 c o s A X 3 - co s cp, c o s A À ,
( 6 b)
Let us now consider the different cases of the proposition.
Case I (91 * <pi): In this case (p is eliminated by dividing each member of Eq. (6 a) by (6 b). For (pi * <f>3 the
result is
5 . 2 _ s in cp, - s in cp2 _ c o s cp2 c o s A X 2 - c o s cp, c o s A À ,
5 .3
s in cp, - s in cp3
c o s cp3 c o s A À 3 - co s cp, c o s AA.,
Inserting into (7)
c o s A A j = c o s À co s À , + s in À s in À j
(8 )
one easily arrives at the solution (la ) for X, and (lb ) is directly obtained from Eq.(6 a). If (pi = <ps, the left
hand side of ( 6 b) vanishes. Together with (8 ) it yields
t a n ^ . sinX.-sma,
C0 S À , - c o s X 3
which formula agrees with (la ).
Case II (q>i = <p2 * cp3): Formula ( 8 ) inserted into Eq. (6 a) with Su= 0 yields
2
co s X c o s X + s in
X s in X 2=
co s
X cos 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 numeri­
cal instability in the practical application. One improvement is gained by substituting the differences of
sines of St by
S y = 2 co s cpy s in (A (p ;j / 2 )
(10)
where cp*, = (cp* + <pj)/2.and 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 and/or longitude differences among
the known points the following corollary may be useful.
Corollary 1: The coordinates À and <p of the median line turning point is given by
s in À = ± —
,
cosÀ = ± —
D[
D,
, A,
sincp = ± — - ,
, B,
( il)
(1
and
D2
coscp = ± —
^
(12)
D2
where Ai and Bi are the numerator and denominator, respectively, of the right hand side of E q .(la ), and
D j = y j A ,2 + B f = ^ S 23 c o s 2 cp, + S f 3 c o s 2 (p2 + S ,22 c o s 2 (p3
A 2 = co s (p2 c o s A X 2 - 2 co s (p, c o s A À , + c o s (p 3 c o s A À 3
*14a^
B 2 - S 12 + S i 3
(14b)
d
2 = V a 2 + b}2
*22
(14c)
The proof is given by simple trigonometric manipulations of Eqs. (la ) and (6 a,b).
Unfortunately, there is still the possibility that for small baselines the denominators D1 and D2 may be
small, making the solutions for X and <p numerically 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
tanoCj =
sin
(X —X: )
^
; i = 1, 2, 3
co s ^ tan cp - sin <Pj cos (A - X i )
MG')
(15)
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 relat­
ed by the geodetic latitude <p by 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 and -minor axes
a and b. The distance (s>) and the ellipsoidal longitude difference (AL) from the point Pi ([}i, L) to the want­
ed turning point P (|3, L) along the geodesic are given by
4
si = j f ( P , h i ) d P = F ( P , h i ) - F ( p i , h i )
(17a)
and
M . , = L - L , = / g ( P ,h i ) d p = G ( P , h , ) - G ( P 1, h i )
(18a)
where
\
1 - e cos P
Q
f ( p , h | ) = a j ------ ;-------- co sp
v
y cos P - h f
ç fn ,
(17b)
and
g (P ,h f ) = ±
,
1 - e 2 cos2 P
( i 8 b)
c o s p \ cos2 P - h f
Here hi = cos|W , where Pm« is the maximum (or minimum) latitude of the geodesic.
Alternatively, using the variable substitutions
(19a)
s in P
s in v = •
V l-h 2
and
s in w =
.
h
q
ta n p
(19b)
V l-h 2
the integrals (17a) and (18a) can also be written (Klotz 1991 and 1993; Schmidt 1999 and 2000):
v
( 2 0 a)
Si
f * ( v , h j ) d v = F* ( v , h; ) ( V j, hj )
and
(2 1 a)
W
=J
ALj
= J g* ( w , h j ) d w
= G* ( w , h j ) - G * ( w ^ h j )
where
( 2 0 b)
f *( v , h i ) = a ^ l - e 2 { l - ( l - h 2 ) s in 2 v }
and
+ /i1 - g V( w , hui\) = ±
e2h'
(21b)
h? + ( l - h f ) s i n 2 w
The three-point problem can now be defined by the equations
Sj — S2 — S3
( 2 2 a)
and
L = L 1+ AL1= L 2+ AL2 = L3+AL3
(22b>
where si and A L are given by Eqs. (17a) and (18a) or by Eqs.(20a) and (21a). The longitude L can be elim­
inated from (22b), yielding four independent equations with four unknowns (|3, hi, h2, ha). The system of
equations can thus be written
s, —s 2
(23)
s, = s 3
Lj + AL, = L j + A L 2
Lj + A Lt = L 3+ AL>3
(24a)
(24b)
X T = (A P ,A h ,,A h 2,A h3)
v T = { ( 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° and
HÎ. The vector of unknowns X contains improvements to the approximate values p°, hS, h9 and hS. The ele­
ments of the (4x4) design matrix A are presented in the Appendix. In order to determine the elements of
Y and A the integrals (17a) and (18a) or (20a) and (21a) must be employed, e.g. by series expansions or
direct numerical integrations (Klotz 1991 and 1993; Schmidt 1999 and 2000). Starting values for (3°, v°
and 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 and 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 and 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), 147-159
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 and H. Fan (1999): Accuracy of computed points on a median line, factors to
be considered. Proc. Int. Conference on Technical Aspects o f Maritime Boundary Delineation and
Delimitation, International Hydrographic Bureau, Monaco, 8-9 September, 1999, pp.120-132
IHB (1993): A manual on technical aspects of the United Nations convention on the law of the s e a -1982.
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
Klotz, J. (1993): Die Transformation zwischen geographischen Koordinaten und geodàtischen Polar- und
Parallel- Koordinaten. Zeitschrift fuer Vermessungswesen, 118, 5, 217-227
Schmidt, H.(1999): Lôsung der mittels geodàtischen haupaufgaben auf dem Rotationsellipsoid
numerischer Integration. Zeitschrift fuer Vermessungswesen, 124, 121-128, 169
Schmidt, H (2000): Berechnung geodâtischer Linien auf dem Rotationsellipsoid im Grenzbereich diametraler Endpunkte. Zeitschrift fuer Vermessungswesen, 125,2, 61-64
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. 153-168
Sjôberg, L. E. (2002): Intersections on the sphere and ellipsoid. (Journal of Geodesy, in press), 76: 115-120
Appendix
The elements of the design matrix A are as follows:
9s,
A„ =
¥
' V
*• 3 f
3h,
v 1y
_
ah°
= 1 T^(P’h°)cosPdP =
- 1l^ahft
hi
'f f (v .h f )
<lv
,
‘ 2f
J l-(h °) v, cos v
ds2
•^13 “
3h,
1-(¾ ) I
cos2 V
a I4 = o
f a s ,]
A2|—
V ap
r J
^22 _ ^12
f 3s0
^ ap
^J
A23 =0
r f* (v,h° )■dv
- a h °3
ds3 _. ~m
A 24 —
3h,
i - ( h ? ) 2i
dAL, j [ 0AL.
A ,i =
3P
- g (e " .h r )-g (p ”,h ;)
ap
0AL,
^32 “
9h,
3 AL 2
A 33 -
A4,=
3h ,
f dAL,
dh,
\
1 J
h°2
f g * ( v . h° )
l - ( h “ )2 {
f 0A L 3 ^ _
k 9h3 J
h?
dv
COS V
f g* (v .h ? )d..
l - ( h ? ) v, cos2v
A 42 - A 32
a 43= o
h j _ J £ ( v X ) dv
dv
1i - ( h
! ) '!
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 1977 and 1978 and about four
years work with the National Land Survey of Sweden, he returned to KTH in 1984 to succeed his old pro­
fessor on the chair of Geodesy.
Through the years he has been a member of several IAG special study groups (chairing three of them) and
commissions, and he is an IAG Fellow since 1991. He is a presidium member of the Nordic Geodetic
Commission since 1982, a Fellow of the Alexander-von-Humboldt Foundation since 1983 and a corre­
sponding member of the German Geodetic Commission since 1989.
His research interests are in the fields of geodetic theory of errors, physical geodesy, GPS positioning
and deformation analysis. From 1989 to 1995 he served in the editorial boards of Manuscripta
Geodaetica and Bulletine Geodesique, and he has published about 200 scientific papers and reports.
E-mail: [email protected]