A GRAPHICAL METHOD OF ADJUSTING
A DOUBLY-BRACED QUADRILATERAL
OF MEASURED LENGTHS WHEN THEIR WEIGHTS ARE
ALSO TAKEN INTO CONSIDERATION
J.C.
by
B
h a t ta c h a r ji,
M.A., F.R.A.S., F.I.G.U.
S u rvey o f In dia
A B STR A C T
A n attem pt has been made to develop a graphical m ethod o f precisely
adjusting a doubly-braced qu adrilateral o f measured lengths b y fittin g the
least squares method to differen t systems o f w eighting, w ith ou t recourse
to elaborate com putations and trigon om etrical tables. The suggested method
is m ore simple and less tim e-consum ing than the usual m ethods.
IN TR O D U C TIO N
Due to the advent o f m odern electronic devices for distance measure­
m ent it has become incum bent upon us to develop new techniques in our
methods o f figurai adjustm ent which m ay be carried out w ith com parable
speed and accuracy.
B.
1957), A.
and G. T . T h o r n t o n - S m i t h (E.S.R., X IV , 106, October
(E.S.R., X V , 111, January 1959) and G. T.
T h o r n t o n - S m i t h (E.S.R., X V I, 124, A p ril 1962) have dem onstrated relevant
methods o f adjusting a doubly-braced qu adrilateral o f m easured lengths
and o f computed spherical angles, expressed in the form o f a condition
equation satisfyin g the o n ly geom etrical relation requ ired to close the
figure exactly, according to the least squares principle. In actual practice
how ever the methods prove to be v e ry tim e-consum ing as th ey in volve a
considerable am ount o f num erical calculations, including the evaluation
o f a number o f trigon om etrical functions, especially when the w eights o f
measured lengths are also taken into consideration.
T.
T
M
u rph y
ar czy
-H
orn och
A n attem pt has th erefore been made to m ake use o f the elem entary
properties o f vector elem ents fo r the solution o f w eighted con d ition equa­
tions on the least squares principle, thus provid in g a straigh tforw ard
graphical method o f figurai adjustm ent which is m ore simple and less
tim e-consum ing than the existing methods.
D ESCRIPTION OF METHOD
In Fig. A o f the diagram , let ABCD be a doubly-braced quadrilateral
o f w hich the four sides a, c, e, f and the two diagonals b, d have been
measured by means o f a Tellu rom eter.
1
1
1
L et us compute the spherical angles Cx -)------ e 1( C2 H------- £2. C3 H------- £3
3
at a station C by the usual cosine form u la and
3
L e g e n d r e ’s
3
theorem, w here :
c2 + b 2 — a 2
cos Ci = ------------------2 be
c2 + e 2 — d 2
cos C.> = ------------------2 ce
(1)
(2)
b 2 + e2 — /2
cos C, = --------------- —
2 be
(3)
spherical excess in triangle AB C;
Zi =
£2 = spherical excess in triangle BCD ;
and
£3
= spherical excess in triangle ACD.
If the doubly-braced quadrilateral were in adjustm ent, the figure would
have closed exactly, satisfyin g the geom etrical relation :
1
Ci + —
£1
1
1
+ C3 H— — £3 — C2 H— — S2
But in general there w ill be observational errors o f accidental nature
in the measured lengths, givin g rise to corresponding angular errors in the
com puted spherical angles :
Let An, A b , ... A f be the corrections to the measured lengths a, b, ... f,
AC1; AC2, AC3 be the corresponding corrections to the computed
1
1
1
spherical angles C 1 -{-----ej, C2 + — e2, C3 -|------£3 , such that :
3
3
3
and
(C j +
—-
O
s1 +
A C j)
+
( C , H— —
O
£3
+
A C 3) =
O
( C 2 H— —
£2
+
A C ,)
or :
Cl +
c3—
1
C2 +
—
3
( e i 4-
£3
—
£2 )
=
ACo —
ACt —
AC3 =
AC
(4 )
D ifferen tiatin g relations (1 ), (2) and (3), we have after sim plification,
1
ACi = ------------ (A a — A b cos A j — A c cos B ^
c sin B x
(5)
AC2 = -------------(A d — A c cos B 2 — Ae cos D 2)
c sin B2
(6)
AC3 = ------------ (A/ — A b cos A g — Ae cos D 3)
e sin D 3
(7)
and :
F ro m the above we have, on expressing AC in seconds o f arc, the
geom etrical condition equation in the fo llo w in g fo rm :
AC" =
206300
-----------------(A a — A b cos A t — Ac cos B ^
c sin Bj
206300
-----------------(A / — A b cos A 3 — Ae cos D 3)
e sin D 3
206300
H---------------( A d — Ac cos B 2 — A e cos D 2)
c sin B2
or, after sim plification,
AC" = —
K aAa + K SA b — K cA c + K dA d — K eAe — K fA f
(8)
where
206300
206300 sin A,
c sin B i
b
sin A-l sin A a
206300 sin B4
K„ = -----■---- ------ ----- : K rf =
c sin B t sin B2
206300
(9)
c sin B 2
206300 sin D4
206300
K b = -----— -------— - ; K, =
e sin D 2 sin D3
e sin D 3
N ow , if the w eights o f linear m easurem ents are assumed equal
irrespective o f the length measured, the consequent adjustm ents becom e
d irectly p roportion al to their respective coefficients in the con dition equa­
tion (8), reducing the on ly norm al equation to the fo rm :
[ K K ] X — AC = 0
whence the X-correlate becomes :
AC
X = -------------------------------
(10)
Ka
2 + K? + ... + K f
and the linear adj vistments w o rk out as :
Aa = — K aX ; A b =
K b\
;
Ac = — K CX ; Ad =
K dX ;
Ae = — K^. ; A f = — K t\ ;
'j
L
J
(11)
Thus, excluding the diagonals, the lon ger sides w ill obviou sly receive
the sm aller adjustm ents and the shorter sides the greater w hen actually
evaluated, which appears unnatural and disquieting. It w ou ld th erefore
“ appear preferable, i f not essential, to adopt some other system o f w eigh tin g
w hich w ill give adjustm ents o f amounts m ore in keeping w ith exp erien ce” .
Hence i f in the T ellu rom eter determ ination, the probable error o f a
measurement is taken to v a ry d irectly as lr, w here I is the length m easured
and r any constant, the corresponding w eigh ts o f measured lengths w ill
v a ry as 1/ l 2r and the consequent adjustm ents are as l2r tim es their respective
coefficients in the condition equation (8), reducing the o n ly norm al equation
to the fo rm :
[(/’-K ) ( l r K ) ] X — AC = 0
w
h e n c e
t h e
^ - c o r r e l a t e
b e c o m
e s
:
AC
---------------------------------------------a-"-K* + b 2 ’-K? + ... + p K f
( 12)
and the linear adjustm ents w o rk out as :
A a =
—
Ac =
— C2'K f2 X ;
Ad =
— e SrK ?
Af ~
Ae
T ) ,,4 J J U l
^
^
3 J I1 C C
U i
,. 11
<Xl 1
4- V-* «
«
l l i c a c
=
„-2rK
» ,
♦ o
v> U l g i i t J ,
2 X
.
4- V i n
L iiv
A b
r . t->
o
w n w j
=
6 2 rK
2
X
.
^
;
J
I-
d~rK ' j \ ;
\
»>
jv i u v
i n rt
i u m g
n
1 il p t m
a u j u o i m
n n f o
^ m o
t V» /"*
m v
(13)
m
A c t
A ü vjo t.
in keeping w ith practical experience are either 1// or l//2, the systems o f
w eigh tin g actually considered in the present article are as follow s.
(1) w eights all equal for r = 0, givin g adjustm ents as :
A a = — K„X ; A b =
K b\ ; AC = — K..X ;
Ad =
K dX ; A e = — K eJ. ; A f = — K,X ;
y
(u )
1
(2) w eights v a ry in g as 1// fo r r — — , givin g adjustm ents as :
A« = —
Ad =
d K dl
;
Ab =
;
Ae
Z>K(JX ; A c = — cK,X ;
= — e K f\ ;
A f — — fK,\
;
(3) weights v a ry in g as 1//2 for r = 1, givin g adjustm ents as :
A a = - - « 2K„X ; A b —
Ad
=
d 2’ K dX
;
b - K b\
;
Ac —
Ac = — e 2K eX ;
— c2K (X ;
A f — — f 2 K ,\
;
}
'
(16)
On substituting the above linear corrections in relations (5), (6) and (7),
it becomes possible to evaluate the angular corrections also, with the help
o f trigonom etrical tables.
N o w from relations (9) we have, after sim plification ,
Ku _
^
Ky
sin A*
sin A 4
sin A,
or :
K„
K„
K,
---- _f__ = ----------- ± --------= ------ Z__
sin A 3
sin (180“— A 4)
sin A,
(17)
w hich shows that the values o f K a, K„ and K f are such as can be represented
in m agnitudes by the three sides o f a triangle taken in this order, and
having its angles equal to A 3, 180“— A 4 and A, opposite to their respective
sides.
K,
K,
Kd
sin D 4
sin D 2
sin D 3
or :
K„
K,
Kd
sin I>4
sin D.,
sin (1 8 0 "— D rt)
which shows that the values o f K e, K , and K d are such as can be represented
in m agnitudes by the three sides o f a trian gle taken in this order and
having its angles equal to D 4, D 2 and 180°— D 3 opposite to th eir respective
sides.
K tt
Kc
Kd
sin B2
sin B4
sin Bj
K0
Kc
Kd
sin B2
sin B 4
sin (1 8 0 °— Bj)
or :
(19 )
w hich shows that the values o f K a, K c and K d are such as can be represented
in m agnitudes by three sides o f a triangle taken in this order and having
its angles equal to B2, B4 and 180°— Bj^ opposite to their resp ective sides.
K fc
K„
sin C2
sin C3
K„
sin
or :
sin (1 8 0 ° — C2)
K„
K„
sin C3
sin Cj
( 20)
which also shows that the values o f K;,, K c and K e are such as can be
represented in magnitudes by three sides o f a triangle taken in this order
and having its angles equal to 180°— C2, C3 and Cj opposite to th eir respective
sides.
Hence from relations (17), (18), (19) and (20) it easily fo llo w s that all
the six values o f K„, K 6, ... K f can be represented in m agnitudes b y the fou r
sid es and two diagonals o f a doubly-braced qu adrilateral o f fo u r triangles,
each having its angles m ade up o f the supplem ent o f the fu ll angle at a
corner o f the given qu adrilateral, plotted to a convenient scale, (F ig . A ) and
the other tw o angles constituting the same full angle at that corner. Th e
second doubly-braced qu adrilateral, representing the six values K a, K 6, ... K,
in magnitudes as described above, can be plotted with advantage side by
side w ith the given qu adrilateral (F ig. A ), m akin g use o f the same scale
and starting from the corner h avin g the largest angle w h ich in the present
case is C<*>. Now , on produ cing one o f the flan kin g sides, say c, and
d raw in g a straight line tow ards A ' parallel to the diagonal d b y means o f
a suitable parallel ruler, it is possible to obtain the angles B2 and D2 as
parts o f 180° — C2 . T o fix the scale o f the figure, the value o f K d
corresponding to the d iagon al opposite the fu ll angle C2 can be computed
fro m the relation :
206300
Kd=
c sin B->
or :
206300
(21)
K d = ----- ------“a
(* ) T h e s ta tio n C should a ls o be one at o r t o w h ic h a z i m u t h has e ith e r been
ob serv ed o r a lr e a d y k n o w n so t h a t once the c o m p u te d sp herical a n gles C „ C 2 an d C;, are
ad ju s te d and the c o o rd in a te s at a n y on e station b e c o m e known, t h e r e w i l l not be a n y
d if fic u lt y in r ed ucin g the c o o rd in a t e s o f the r e m a i n i n g stations a n d t h e r e w i l l no
necessity to com p u te o r to a d ju s t th e r e m a i n i n g a n gle s f o r th is purpo se.
where h d is the distance fro m C o f the point o f intersection o f BD and the
arc o f the circle in F ig. A drawn w ith BC as diam eter, and plotted at
a convenient scale as C'A'. Then on draw ing A 'B ' parallel to AB and A 'D '
p arallel to DA, D 'B ' becomes parallel AC, thus com pleting the required
figu re (F ig. B) representing in magnitudes all the values o f K a, K 6, ... K^.
T h e value o f K d being known, the rem aining values o f K a, K,,, K c, K e and K ,
can be easily scaled to three significant figures w ith the help o f figures B L,
B 2 or B 3 .
A lternately, as revealed in a recent investigation, fig. B 1( B L, or B 3 can
easily be draw n m ore precisely and con fiden tly in the fo llo w in g manner :
F irst plot BD on a convenient scale (viz. half, double etc. or at the
same scale as that o f Fig. A ) along the diagonal BD subtending the full
206300
antrlp C„ at C,. so that R l l ' = K . = --------- . where h j is the perpendicular
hd
distance o f C from BD.
Then draw D'C' parallel to AB, intersecting BC produced at C' and
satisfying the relation :
206300
D 'C' = K „ = --------^a
where h a is the perpendicular distance o f C from AB.
F in a lly draw D 'A ', B A ' and C 'A ' parallel respectively to DA, BA, and
CA and intersecting one another at a com mon point A ' sim ultaneously
satisfying the relation :
206300
D 'A ' = K , = --------hf
where h f is the perpendicular distance o f C from DA, thus com pleting the
required diagram , w ith Fig. Bj drawn over Fig. A itself, as shown in the
diagram at the end o f this article.
R eferrin g back to Fig. Bj, B^, or B3, instead o f to Fig. A, the relations
(5), ( 6 ) and (7) can be reduced to the follo w in g form :
AC" =
=
AC," =
and :
AC ,'
K „ A « — K „A j, cos A , — K „A c cos (180°— ^ — A 5)
K „ A « — KaAa cos A
K / A / — K/A 6
=
KfAf
=
K d& d —
cos
1
— K „A c cos (C j+ A j)
A 3 — K rAe cos (180°— A :i— C3)
— K/Af, cos A 3 — K f Ae cos (A s+ C 3)
K dA c cos B 2
—
K d\ e
cos D,
H ow ever on substituting the values o f Art, A b, ... A f given in relation
(13), we have fro m the above,
AC," = — A [a 2 rK ttK a — b 2 rK aK b cos A , — C2 rK aK, cos (Cj + A , ) ]
(22)
AC," =
— X [a2rK /K r +
b 2 rK fK b
cos A 3 + e2 rK f K e cos (C 3 + A 3) ]
(23)
X [ d 2 rK aK d +
c 2 rK dK c
cos B 2 + e 2 rK dK,. cos D 2
(24)
and :
AC !' =
N o w con siderin g a 2 rK„, b 2 rK h, c 2 rK r ; f 2 rK ;, b 2 rK,„ e 2 rK r ; c 2 rK c, d 2 rK d, e 2 rK r;
a 2 rK„, d 2 rK d, f 2 rK ; as fou r sets o f vector elements having their m oduli the
same as the corresponding scalar quantities a2rK„, b 2 rK,„ ... f 2 rK f and their
directions as denoted b y the arrows along the straight lines at the four
corners o f fig. B lf B2 or B 3, the relations (22), (23) and (24) can be expressed
in vector form as follo w s :
I C r = — X ( K „ a ^ + K, . 5 ¾
+ K a^ K c)
= — XK a (^ K a + b * K b + 0 ¾ )
= — X K,,(a2r + £>2r + c2") B'O^
w here (a2,"+fc2r+ c 2r) B 'O j denotes the sum o f the vectors a 2rK tt+fc2rK (J+ c2rK 0
and the modulus j(a2r+ 6 2r+ c 2r) B'Oj|, or ( a 2r + b 2r+ c 2r) B 'O l5 is the distance
from B' o f the point O a d ividin g the straight line D 'O ^ in the ratio
(a 2r+ c 2r) / b 2r, 0 { being again the point divid in g the straight line C 'A ' in the
ratio a 2 r/ c 2r, or again :
A C " = — X ( a2r + b 2r + c2r) K œB'Oj
or :
A C " = — X (a2r + b 2r + c2-) K aL a = — \ K ' a
(25)
where L a is the resolute o f the vector B'O i in the direction o f the vector K a
o f the first set, and is equivalent to the distance from B' o f the point of
intersection o f B 'A ' and the arc of the circle drawn with B 'O j as a diameter.
S im ilarly :
AC3' = — X ( f 2r + b 2r + e 2r) K f D '0 3
or :
AC£' = — X ( P + &2r + e2n K f L f = — X K ',
(26)
where ( f 2 r+ b 2 r-j - e 2r) D 'Q 8 denotes the sum o f the vectors f 2 rK f + b 2 rK b+ e 2 rK e,
the modulus |(f2r+ 62r+ e r2) D '0 3| or ( f 2 r+ b 2 r+ e 2r) D 'O a is the distance from D'
o f the point 0 3 d ivid in g the straight line B '0 3 in the ratio ( f 2 r+ e 2 r) / b 2r,
0 3 being again the point divid in g the straight line C 'A' in the ratio f 2 r/ e 2r,
and L f is the resolute o f the vector D '0 3 in the direction o f the vector
o f the second set and is equivalent to the distance from D ' o f the point of
intersection o f D 'A ' and the arc of the circle drawn w ith D '0 3 as diameter,
and :
AC 2 ' = X (ef2r + c2r + e 2r) K d C'Oa
or again :
AC*' = X ( d 2r + c2r + e2-) K Æ
L d = \ K 'd
(27)
where ( d 2 r+ C 2r-f e2r) C '0 2 denotes the sum o f the vectors d 2 r + c 2 rK c+ e 2 rK e,
the modulus |(cf2r+ c 2r+ e 2r) C '0 2| or (d2' + c2r+ e 2r) G '0 2 is the distance fro m D'
o f the point 0 2 d ivid in g the straight line A '0 '2 in the ratio ( c 2 r+ e 2 r) / d 2r,
0 2 being again the point d ividin g the straight line B 'D ' in the ratio e 2 r/ c 2r
and L æ is the resolute o f the vector C '0 2 in the direction o f the vector K d
o f the third set and is equivalent to the distance from C' o f the point of
intersection of C 'A' and the arc of the circle drawn w ith C '0 2 as diameter.
Ca s e
( / ) : H a v i n g w e i g h t s a ll e q u a l f o r r = 0.
F or r — 0, we have fro m relations (25), (26) and (27) :
ACJ' = — Xlfa-4 B '0
w h ere 4 B'O denotes the sum o f the vectors K a+ K 6+ K e and the modulus
jB'OJ, or B'O, is the distance from B' o f the m iddle point o f the straight line
jo in in g the m iddle points o f the diagonals b and d in fig. Bi.
Or again :
A C i'
= — 4 X K l-ïÿ O
or :
ACJ' = — 4 X K œ•L a = — X K '
(28)
w here L a is the resolute o f the vector B'O in the direction o f the vector K n
o f the first set and is equivalent to the distance from B ' o f the point o f
intersection o f B 'A ' and the arc of the circle drawn w ith A 'O , as diameter,
and K ' is equal to 4 K 0 L 0 obtainable by the direct approxim ate method
o f m ultiplication, or by using in fig. Bx an ordinary slide rule retaining
three significant figures only.
S im ilarly :
A C " = — 4 X K , ■L , = — À K;
(29)
where
is the resolute o f the vector D O in thp direction o f the vector K,
o f the second set and is equivalent to the distance fro m D ' o f the point
o f intersection o f D 'A ' and the arc o f the same circle as in the case of
relation (28), and Wf is equal to 4 K j l ^ p obtainable by the direct approxim ate
m ethod o f m u ltiplication or b y using in fig. Bi an ord in a ry slide rule
retaining three significant figures only.
And :
A G ^ = 4XKi -L d = XK^
(30)
where L d is the resolute o f the vector C'O in the direction o f the vector K d
o f the third set and is equivalent to the distance from C' o f the point o f
intersection o f C ' A ' and the arc o f the same circle as in the case o f
relation (28), and K'd is equal to 4 K d L.d obtainable by the direct approxim ate
method o f m ultiplication or b y using in Fig.
an ord in a ry slide rule
retaining three significant figures only.
1
Case (2 ) : H a v in g w eights
v a r y i n g as
1// f o r r = — .
1
F o r r = — , we have from relations (25), (26) and (27) :
2
AC'/ = — X ( a + b + c ) K a L a = — X K '
(31)
A C * '= — l ( f + b + e) K r L , = — XK/
(32)
and :
A C !/=
X ( d + c + e ) Krf-L* =
X K d'
(33)
where values o f K ', KJ and K'd can be obtained by the direct approxim ate
m ethod o f m ultiplication or b y using in Fig. B 2 an o rd in ary slide rule
retaining three significant figures only.
Case ( 3 ) : H a v in g
w eights
v a r y i n g as
1//3 f o r r = 1 .
F o r r = 1, w e have fro m relations (25), (26) and (27) :
A C " = — X(a2 + &*+c2) K a L a = — X K '
(34)
AC 3' = — X ( f 2 + b 2 + e 2) K f ~Lf =
— XK/
(35)
XK^
(36)
and
AC2" =
X(e?2 + c 2 +e2) K d-L„ =
where values o f K ', K; and
can be determ ined first by m aking use o f the
values o f the squares o f the lengths obtained from com putations in the
cosine form ulae (1 ), (2) and (3) and then by the direct approxim ate method
o f m ultiplication or by using in Fig. B2 an ordinary slide rule retaining
three significant figures only. Hence com bining the values fo r AC^ A C " and
AC3,' and rem em bering that AC2 — ACt — AC3 = AC, we obtain fo r all three
above cases,
AC” =
X
(IVj "1“ ky + 1¾ )
or
X =
AC"
/ (K£ + K; + 1¾ )
(37)
N ow substituting the above value for X in relations (14), (15) and (16),
the linear corrections A a, A b, ... A f can be easily obtained by the direct
approxim ate method o f m ultiplication or hy using an o rd in ary slide rule
retaining three significant figures only. O bviously the signs o f the linear
corrections fo r the sides are opposite to those fo r the diagonals.
S im ilarly substituting the above values fo r X in relations (28) and (29)
fo r case (1), in (31) and (32) for case (2) and in (34) and (35) for case (3),
the angular corrections AC! and AC3 can also be easily obtained by the
direct approxim ate method o f m u ltiplication or by using an ord in ary slide
rule retaining three significant figures only. On obtaining the values of
ACi and AC3, the corresponding value o f AC2 for the w hole angle can
im m ediately be deduced from the relation :
AC2 = ACj + AC3 + A C
In short, the entire com putational drill can be summarised as under :
(a )
P lo t fig. A w ith measured lengths.
( b ) Compute AC from relation (4), an operation entailing a single
addition and a single subtraction which can easily be carried out mentally.
(c) Scale h d from fig. A.
(d) Compute K d from relation (21), an operation entailin g a single
division in volvin g on ly three significant figures, which can be easily carried
out by the approxim ate method o f division or by using an ordinary slide
rule.
(e) P lo t fig. Bj, B2 or B3 according to instructions on page 122 w ith
C 'A ' = K d .
(/) Scale K a, K 6, ... K r and L„, L , and L d from fig. B1; B2 or B3.
( g ) Compute K ' , K/ and K'd from relations (28), (29) and (30), or (31),
(32) and (33), or (34), (35) and (36), as the case m ay be, an operation of
simple m ultiplications in volvin g three significant figures in each case, which
can be carried out by the approxim ate method o f m ultiplication or sim ply
by using an ordinary slide rule.
( h ) Sum up K '-fK J + K^ — an operation entailing a sim ple addition
in volvin g three significant figures only w h ich can easily be carried out
mentally.
( i ) Compute X fro m relation (37) — an operation entailing a single
division which can easily be carried out as in (d ) above.
( j ) Compute A a, A b, ... A f fro m relations (14), (15) and (16), as the
case m ay be, an operation entailing simple m ultiplications in volvin g in
each case three or less significant figures, w hich can be carried out b y the
approxim ate method o f m u ltiplication or sim ply by using an ord in ary slide
rule.
( k ) Compute AC i and AC3 fro m relations (28) and (29), or (31) and (32),
or (34) and (35), as the case m ay be, an operation entailing sim ple m u ltip lica­
tions in vo lvin g in each case three or less significant figures, w h ich can be
carried out as in ( j ) above.
(/) Compute AC2 fro m relation (4 ), an operation sim ilar to (ft) above.
A n exam ple is w o rk ed out b elow in order to make the practical routine
o f com putations clearer. T h e num erical exam ple quoted is taken fro m the
articles by B . T . M u r p h y and G.T. T h o r n t o n - S m i t h (E.S.R., X IV , 106,
O ctober 1957 and E.S.R., X V I, 124, A p ril 1962) so that direct com parison
w ith the results com puted b y the suggested method m ay be possible.
Given data
M easured lengths :
a =
69 847.62 feet
b
=
83 587.77
c
=
44
679.24
d
=
102 017.34
e
=
65 824.23
f
=
94 277.10
Computed plane angles :
Cx =
56° 3 9 ' 5 9 "1 3
C2 = 133° 53'55'.'97
C3 =
77° 14' 02'/85
Computed spherical angles :
1
Cx H----- £l =
3
56° 39' 59'/38
1
C2 + — e2 = 133° 53' 56'.'13
3
C3 + y
e3 =
77° 14' 03"27
Adjustment
(? ) : H a v i n g w e i g h t s all eq u a l.
Case
P lo t fig. A w ith measured lengths.
F ro m relation (4 ) d erive : A C " = +6752 .
F ro m fig. A scale : h d = 20800.
206300
K j = --------- = 9.92 .
Æ 20800
F ro m relation (21) compute :
F
P lo t fig. Bj .
F ro m fig. B j scale :
K 0 = 4.62 ;
Ke
K „ = 6.56 ;
= 7.60 ;
K c = 8.88
K ; = 3.63
and also :
La =
2.58 ;
L , = 1.16 ;
Ld
= 6.25
F ro m relations (28), (29) and (30) obtain :
K ' = 4 L a •K„ =
47.7
K ; = 4 L / -K , =
K'd = 4 L d K d =
248.0
Sum
F ro m relation (37) compute :
16.8
= 312.5
6.52
X -- ------ - 0.0208
313
F ro m relation (14) com pute :
Aa = — 0.096 ; \ b =
Ad
=
+ 0.207 ;
+ 0.137 ;
Ae = — 0.158 ;
AC = — 0.186 ;
Af
—
— 0.076 .
F ro m relations (28) and (29) com pute :
A C i' = — 0799
F ro m relation (4) com pute :
Case
and
AC3
" = -0-/34
AC2' = + 5719
( 2 ) : H a v i n g w e i g h t s v a r y i n g as 1/1.
P lo t fig. A w ith measured lengths.
F ro m relation (4) derive :
F ro m fig. A scale :
hd
A C " = +6752
= 20800.
F ro m relation (21) com pute :
K d = 9.92 .
P lot fig. B2.
F ro m fig. B2 scale :
K a = 4.62 ;
K t = 6.56 ;
K e = 7.60 ;
K , = 3.63 ;
L „ = 3.97 ;
L ; = 1.97 ;
K 0 = 8.88 ;
and also :
L d=
8.67 .
F ro m relations (31), (32) and (33) obtain :
K'a
=
K; =
K t’
(er + &+ c )L a-K 0 =
36.3 x 10s
=
17.4 x 105
(f + b + e )~ L r K ,
=
(d + c + e )L
d- Ktf = 184.3 x 10s
Sum = 238.0 x 105
F ro m relation (37) com pute :
6.52
1 0 -5 = 0.0273 x 1 0 -5
X = ------------ x
238.0
F ro m relation (15) com pute :
Aa =
— 0.09 ;
Ab =
Ad =
+ 0 .2 8 ;
A e = — 0.14 ;
+ 0 .1 5 ;
Ac = — 0.11
Af
= — 0.09
F ro m relations (31) and (32) com pute :
AC/' =
— 0799
and
AC3" = — 0748
From relation (4) com pute : AC2' = +5705.
Case
(.?) : H a v i n g w e i g h t s v a r y i n g as 1/12.
P lo t fig. A w ith m easured lengths.
F ro m relation (4) derive :
F ro m fig. A scale :
hd
A C " = + 6752 .
= 20800.
F ro m relation (21) com pute :
K d = 9.92 .
P lo t fig. B 3 .
F ro m fig. B3 scale :
Ka =
4.62 ;
K„ =
6.56 ;
Ke =
7.60 ;
K, =
3.63 ;
L „ = 4.42 ;
L, =
2.43 ;
Kc =
8.88 ;
=
8.99 .
and also :
L d
F ro m relations (34), (35) and (36) obtain :
K; =
( a2+&2 + c2) L a K a =
28.3 x 105
K'f =
(/ 2+ & 2+ e 2) L r K, =
17.8 x 105
KJ =
(cf2+ c2+ e2) L d K d = 149.3 x 10s
Sum = 195.4 x 10®
F ro m relation (37) com pute :
X
6.52
= --------- X 10-5 = 0.0334 x 10-5
195.4
F ro m relation (16) com pute :
A a = — 0.08 ;
A& =
+ 0.15 ;
Ac = — 0.06 ;
Ad =
Ae =
— 0.11 ;
Af
+ 0 .3 4 ;
= — 0.11
F ro m relations (34) and (35) com pute :
ACT =
— 0795
F ro m relation (4 ) com pute :
and
A C ."=
AC:;' =
- 0759 .
+4798 .
ABSTRACT OF RESULTS
Adjustment c orrections at various weights
Old values
New values
1
0.10
0.14
0.19
0.21
0.16
0.08
0".99
5".19
0".34
—
+
—
+
—
—
—
+
1//
0.09
0.15
0.11
0.28
0.14
0.09
0".99
5".05
i
00
A Ct
A C2
A C,
—
+
—
+
—
—
—
+
—
©
a
b
c
d
e
f
—
+
—
+
—
—
—
+
—
/
p
0.08
0.15
0.06
0.34
0.11
0.11
0".95
4".98
0".59
—
+
—
+
—
—
—
+
—
1
0.10
0.14
0.19
0.21
0.16
0.08
0".99
5". 18
0".35
—
+
—
+
—
—
—
+
—
1//
0.09
0.15
0.11
0.28
0.14
0.09
1".01
5".03
0".48
1/P
— 0.08
+ 0.15
— 0.06
+ 0.35
— 0.11
— 0.11
—
—
—
T h e results o f the above com parison are found to be h igh ly satisfactory
— the disagreem ent being o n ly o f the order o f ± 0.01 in the case o f linear
adjustm ents and o f ± 0 .0 2 in the case o f angular adjustm ents.
CONCLUSION
In v ie w o f the sim p licity o f the practical routines fo r the com putations
enum erated in the exam ple quoted and o f the ease and speed w ith which
the results o f adjustm ents can be w ork ed out to w ith in the geodetic standard
o f accuracy, the graphical method suggested should p rove v ery suitable for
adjustm ents o f trilatération figures considered in the present article.