; ; ; Assuming that the variance- covariance matrix of S is gi 
Treatment For Trilateration A men j = i GOES S SNenBy V 
The New Treatment Fo ateration Adjustment 1.€., V —var(S), then according to the least Squares En the 
The "one geometrical condition" connecting the lengths of the i me ON ron wi Shing by 
sides and diagonals of a plane braced quadrilateral ABCD matrix" of the measurements defined by W — &* y =! Weight 
(Fig. 1b) or the length of the sides of a triangular central polygon Y Wao'Vv C3) 
(Fig.1b) is obtained by multiplying by rows the two matrices. 
2 . . : : 
C9 being the "a periori" variance of a 
  
  
  
  
  
  
  
a = measurement of unit weight. 
[ 1 0 9 à [ o 0 0 1 Differentiating, as usual, the 
2 quadratic form partially wi*tt respect 
x?,U _2x 24 1 1 x y XS 442 to each C; , equating each result to 
1:1 1 1 1 4 4 zero, and rearranging tbe results, we 
202 .2x,.2 2 got 
x 0x 1 |AND| 1 XS +42 ^ aet 2 -l 
2105 = -) 5*3 Cw p W's TE (e) 
th t 
= em x^ +42 2x . 2 1 T x " x2 +42 If the measurements are of equal weights, then 
east-squares 3 "3 3 3 3 v3 e + BT ( BRT) k Q 
ralues of the T = 4 24 er. 3S, = — — E ,UC242,.,6 C8) 
where (x 1,y 1), r = 1,2,3,4 are the cartesian coordinates of the > (3& 3 r 
her on the points A.B.C.D respectively. In Order ia be iid to Juke a "Basic"computer program for 
lateral T ; si applying this method, we have to evaluate the determinant and 
ar Since the number oficalummis id cach matrix is jess Gian ifle its derivatives with respect to the sides, as functions of the 
uumber of rows, the determinant resulting from multiplication measured lengths. Putting AB AD=S 
must vanish identically (Ferrar,1941). . g zS,,AC-S, , es  ) 
0 common : 
ises on the Hence the required geometrical condition is: gc 5 BD. S AND CD es : 
oo! 1 1 4 : - 2 
lere will be nod d 2 oe? dk s igs 
bility of an 1 0 (AB2 (AC)2 (AD es 5 C SES : i E 23 
= 4 "eC af a S 
Sn ashe esi. epe» © + 2518, C 51 89-83 c5. £55,076) 
e 2 
1 (CA2 CBR? 0 (CDR 2 ef 2 R2 c^ 2:59 S 
rior angles (AD (m em + 255955 ( S, -S2+53 +54 -55 * 6) 
een possible 1 A2 D)? (DC)2 0 
"redundant pap m ec 542 9 2 2 8 o 2^2 2L 2 
r angles and where (AB)=(BA),etc. are the lengths of the mutual distances of 2$458584,—28, 8495 2855495 ze 9475 96 
to determine the four points À, B, C, D. Similarly, for the condition = 17274 1 
Whether or connecting the mutual distances of five points. 2.2 92 2 Jiu 2 
a matter to 35:495, S. -S 49545 +5 +5 ;- 96 
o make fne Because model (1) is implicit in the Beinen sr ‘r RE 4 eru. > oD 312 
essary for —1,2,3,4,5,6), it has to be linearized by applying Mclaurin's d... D ^ 2 EL e 
ortance due Theorem of Expansion. Noting that the corrections (S s) applied „457% 345 SS +4 5 95 Se -4 8 95 > -45, Sy 95 
except for to the measures (s) of the mutual distances of the four points to 1 1 > t 2 
made their lengths satisfy such condition are small so that the c^ c^ 2 e c ce S ) 
second and higher order terms in the expansion of (1) can be $2. 455; ( 1- Sat 545 95:6 76 
i probable neglected, the conditional equation(1) may be written : % x A 2 22 22 2.2 
rom the 24 = Sg : oc -49.5.9 
ct method Ast 2. SS 55, æ 9, -49585 449,8, 955 4955, 9.74959, MA M 
(opsesvation > Stat BY BS pap 2 
nake use o Where À,is the value of as 3A ASS AS 1.55 a8 9 + Se 0a 
plied to the estimated by the measured lengths; ga 4 34 (5 RES AS > 5 » 2 
the summation covering all sides. 32 2 2 2 2 2. 2 2 
r, : ; 34 3 
à ee 5 5 OS net eh ic ue c 
A CASS. (Sı 455-5, S 
1) Jigs long 38 eS 983 8% 3S5 396 9 MER AC 5 3 4*5 64 : ue 
1e) interior a rr n, D 2 2 2 
> have four S -A$ 85.488. Se +45, 8595-49, ey - 4985 Se 
) condition 9 4 - es 
possible to Sa aß s c? e e e se, 52) 
N, in which as $6245. 2 (S, 2:9 t =o 6 | : 
D and CD d d 2 2" 2 22 2.2 229 
D. Th a S. S, LAS 5. Scr 45.94 9, 45 54 94- 4S, 8 
D INN Ss A ESS 
puting the dg a, 2? 9217 1220 
6 2 
d P > 3.2 Bel. a TE SE a SCC US TS 
ho realized Which is the form Bc = k. ; ) —45657+ 455744455 S, S54 6 293 45) 5 
uadrilateral 
on equation Where B stands for the vector 
Our corner 24 ,25 ,22 , 22 , 24 , 24 If, as assumed, AB fixed, then 24/25, = © . 
d from the ! 2 > 4 s € The full computer program, written in Basic, is shown in the 
C stands for the vector [$s,,5s 23552) Ss, ] next section. en applied to different problems, the iterations 
were few showing a rapid convergence 
juadrilateral and K stands for (_-A,)- 
geometrical o 
uadrilateral 
parameters 
29