Appendix 5 contains the precise formulas fo r the interconversion between geodetic and rectangular coordinates.
The f i r s t group o f formulas are the basic form ulas, the terms o f the formulas have been tabulated fo r the UTM p ro je c tio n . Table 14 may be used fo r the values o f R and N. These formulas are reproduced from
Thomas (1952). /
The second pa rt contains formulas th a t are more re a d ily adapt
able to computer programming (Umbach, 1967, has a FORTRAN IV program using th is method). Table A has the constants needed fo r the Transverse Mercator computations; Table B has the constants needed fo r the Lambert
computations. The formulas and tables are reproduced from C la ire (1968).
When one gets a considerable distance from the Central Meridian in the Lambert p ro je ctio n the arc-to-chord correction given on page 51 produces an unnecessary e rro r th a t can be remedied by using the formula put fo rth by Schmid (1962):
X2 - *1 + 0 ^ 2 - Y-j) (Y2 - Y1 - O ^ X g - X-j)
2 arc 1„ 3
Y! - Y0 - 01(X1 - C) 2
where O-j is the mapping angle fo r the i n i t i a l s ta tio n ( in minutes) to the nearest minute times 0.0002909 and Y^, Yq and C are the given con
stants o f the p ro je c tio n .
123 f
TRANSVERSE MERCATOR PROJECTION
Transverse Mercator coordinates, scale, and convergence from geographic coordinates.
( 5 -1 8 ,'+ (« ),
+ B‘n * 008 * + 2 4 -p« f8' n * cos’ *) (5 —0 .
* « , ( l + - , ' ) + A -X- ^8- ^ ( 5 - 4 0 .
7 - A X a in * [ l + A X : ^ * ( 1 + 3 , 1 ( 2 - P ) ]
Geographic coordinates, scale, and convergence from rectangular coordinates.
Avw * - * , = p(, [ - , 2 ^ + 2 ^ - ^ ; (5 + 3 0 ] '
A X = X — X0= p m c 4>: g (■ + 2 i ! + a ! ) + j ^ g ( 5 + 2 8 ( ; + 2 4 ( ! ) J .
t= 1 + 5 (j9i) (1 + , ' 3 + 2 !4 (lO( 1 + 6 ’, i ) '
T==''‘,[i;""l(s;) 0+'!" "0+15.(70 12 <-5«<-3‘{)}
The ( l — T ) corrections.
tf-Tt ~(l / : —!/,)(/:i 2z,),
t j - T 7 - (.Vi - y i ) ( z i l - 2 z j ) .
( ’oordinates from hearing and distance.
For a first approx
imation assume the projected geodesic and" its rectilinear chord to be coincident.
We have then z, — x, + S' sin a, i/3 —y,d S’ cos a where S is the spheroidal geodesic dis
tance. W ith approximate values of z3 and y, computed from these formulas we then compute 8 and /9 from the formulas
0, , 2 jr '>.
Then ' " •
Z i- z , 4 8 sin <3, Vj - Vi 1-8 cos j9 and
a ' --- P ;t 1 80° + P(3/l 'r2/6,)/^r:| + -2z- -
Bearing and distance from coordinates.
0- tftn-1 ^ s=- (z, —z,) esc 0 = (y ,—yd sec /
^ t e r ’y+K^r-')]’
a=0+»(y--i'V^>t2zj.)
The bearings in the above formulas are taken from true north.
125
LAM BERT PRO JECTION
Conversion of geographic to Lambert rectangular coordinates.
r o = r ( * » ) = # c c o t A r = S + - ~ j j ± S ( 5 i ^ 4 ^
S 6(5 + 3 tan1 4>o) , S '6(7 + 4 tan1 «^) tan 4>*
^ 120 RoNl ± 240 So AT*
A X = X — X 0, t= A X s i n 0o, x = ( r 0 + A r ) s i n 7, y = A r + x t a n ^ 7.
A X is p o s i t i v e e a s t o f t h e c e n t r a l m e r i d i a n , n e g a t i v e w e s t o f t h e c e n t r a l m e r i d i a n . S is t h e m e r i d i a n d i s t a n c e b e t w e e n l a t i t u d e s a n d <f>. A r is t a b u l a t e d f o r s u i t a b l e i n t e r v a l s o f <t>.
Conversion of Lambert rectangular coordinates to geographic coordinates.
tan 7 = -- 2. ' Ar = y —x tan - 7, AX= 7 cosec <#>q.
Knowing A r , the corresponding latitude can be obtained by interpolation from the table giving A r for different values of 4>. Alternatively, Scan be obtained by successive approximations from the formula above for A r and <f>then found from a table of merid
ional distances.
Computation of Lambert rectangular coordinates from bearing and distance.
0 ^ yxS*sin1 a S cos a-S1sin2 a y l — y] . S yl cos a sec3 5 cos 3 5 tan fo y , - V , = S cos o --- g-gr--- + - 61 T + 6 % ---, S*cos 4 a tan 4>0 , S 2y? sec1 5 cos 2 ( a + 6 ) tan ^ , S 3yisec Scoe ( 3 a + 5) tan 4*
^ 2 4 R I 4 R I + 6 R I
o . yZS s i n a S 2 c o s 2 a - S ' s i n a , Sy? s i n a s e c 3 5 c o s 3 5 t a n s m o + — 2 > _ --- g % ---+ - ; ---
---, S 'y ] sec1 5 sin 2 (a + 5 ) tan ^ , S ’ t/i sec 5 sin (3 a + 5 ) tan <&, , S'* sin 4a tan ^
+ T i? i + 6 % + 2 4 R I ;
where tan &= ~> <f>o=latitude of the origin, R l,= R 0N l.
When a line is only a few miles in length and no t more than about 150 miles from the origin, coordinates may be computed from:
o F i , , m 8 tan , m* (5 + 3 tan1 fo)-]
T 1 +2RJ7'+-6R-Nf + 24Rjn j
! , = *,+ « sin
a<=y±180"+(52=^l+yi), y i = y i + 8 c o a f l ,
where m is the true m eridian distance of the m id p o in t of the line and 0 is the grid bearing.
When the lines are long or the x’s and y ’s are large we compute s fro m the above form ula, using the scale factor fo r the m id p o in t of the line and the angle 0 computed
S sin a ( y i + i s ' cos a) fr° m ^ “ + 2 % s in I" , •
Then x3= X i+ 8 sin 0, i/2= y i + s cos 0
For lines about 30 miles in length the latte r formulas w ill give accuracy to about 1/100,000. For mucii longer lines where highest degree of accuracy is required no really satisfactory formulas have been derived for point-to-point working directly in terms of Lambert conformal coordinates. In such cases one may compute geographic coordinates by Puissant’s or Clarke's formulas and then transform these geographic coordinates into Lambert conformal coordinates.
Distances and bearings from Lambert rectangular coordinates.
One may use successive approximations in the formulas above for computing Lambert conformal coordinates from bearing and distance.
A first approximation from these gives
tan S = ( y t — yi) sec a = ( z , ^ - z 1) cosec a.
These values are used in the second and succeeding terms to get new values for S' sin a and S cos a.
For short lines not too far f ro m the origin c o le u l a to 0and a from tan f} ~ —— —,
yi—y\
s ~ {y i — yt) sec /9 = (z3 -x ,) cosec 0.
Calculate S and a from
s- 0 - 2a - T i $ f > •-•S x f.rr*
--Scale and scale error.
For long lines m2 = m, + iS cos A, where m2 is the approximate meridional distance of the end point of the line, m, that of the beginning, and
+s’ T.'iw un *"}
<&, is the latitude of the origin, tf> that of the initial point of the line and A the azimuth at that point. For appropriate length lines, any of the above formulas connecting 8 and S may be uoed.
TRANSVERSE M ERCATOR PROJECTION
The formulas used for the transverse Mercator projections for the state plane coordinate systems, excepting those tor Alaska, are not strictly exact. However, they are sufficiently exact for 'he purpose intended. The absolute values of the coordinates of a point may not be quite correct, but the differences between stations in any particular area w ill be sufficiently correct for all practical purposes. The computation of plane coordinates of all points computed before the appearance of the electronic computer was by a logarithm ic method and later a machine method based on the logarithm ic method.
Therefore, the following equations are based on the logarithmic method of computation, so that the original plane coordinate computations can be reproduced as closely as possible.
F O R W A R D C O M P U T A T IO N
F o r m u la s .—The following formulas are used for computing plane coordinates from geographic positions.
<£ = 1atitude of station X = longitude of station
30.92241 724
S, = 24 cos <t> \ T
5580sin2<*>),/4L (1 -0 .0 0 6 7 6 86580$
— X - 3.9174
^wk) j
(X in seconds) . S „ = S , + 4 . 0 8 3 l ( ^ yx= T , + 3.28083 333 Smn + (— 8— ^ SmTiJr«
(Txand xw ill be in the m illio n s in New Jersey; xmay be in the m illio n s in ce rta in o th e r.sta te s.)
<£, = d>+ (1 — 0.00676 86580 s i n 2 d>)2 t a n <£ { ( f )i n s e c o n d s )
<t>2 = <t>+ 25,502f 1 % (1 -0 .0 0 6 7 6 86580 s in ! la n
y = 101 27940 65 A {6(X»' - T,) + <t>" ~ Tt - [1.052.89388 2
— (4.48334 4 — 0.02352 0 cos2 c/hz) cos2 ^>2] sin <f>2 cos </>2}
f < f> ' is degrees and m inutes o f d>2 in w hole m inu te s 1 { 0 " is the re m a in d e r o f 0 Z in seconds J
A a = (7 2- X ) j^sin :
, 1.9587 10'
k —
{T-z — X)2 sin cos2 (X in seconds)
_ f (1 + 0.00681 47849 cos2 0 ) 2 / % - T A 2l 881.74916 2 7,2 \ l ^ / J
The constants (T’s) are listed for the various states and zones at the end of this publication. The rectifying latitude of the latitude of the origin is divided into two parts, T3 and T4. The degrees and minutes in whole minutes are represented by 73. The remainder in seconds is represented by Tv This separation is made because the equations were designed for an electronic computer which had a fixed word capacity of only 10 figures and the rectifying latitude expressed in seconds would be 11 figures.
128
IN V E R S E C O M P U T A T IO N
F o r m u la s .—T ito fo rm u la s fo r vu m |)u tin g geographic p o sitions from plane coordinates, fo llo w :
0.30480 0 h (W r „ / S,„ \ :ll
x n l-'_ 7, “ HT(pj.
a » '= (degrees and m in u te s o f tu in w hole m in ute s)
„ ^ , O A m S l 367SS 3 3 . ,
(t) = / j H--- j, y (re m a in d e r of cu in seconds)
(JL) — ( l) + O j "
ld ) T = 7’:i ldegree;- and m in u te s o f 4> and (f)' in w hole m in ute s) (</,')" = o )"+ [1.(07.347)71 0 + (6.19276 0
-e 0.031W1 2 eos- to) cos- w | sin oi.eos to (re m a in d e r o f <f>' in seconds)
4>'=(<!>')'+ (<t>')"
(d > r= ( < /> ')" - 25.52381 (1 - 0 .0 0 6 7 6 86580 s in2 </>')2 ^ ) ’ tan <£'
(f>= (<&')'+ (<&)"
S . = 5 * - 4 . 0 8 3 1
.Si = 5 ™ -4 .0 8 3 1
S,( 1 - 0.00676 86580 sin2 <Q" 2
' 30.92241 724 cos </>
AX„ = AX, + 3.9174
X"= T2 — AX, — 3.9174
TABLE A. Constants fo r the Transverse Mercator Projection
!:
185216.62358 185216.62358 227130.53702 1*5326.09287?:
.99990 00000::
**26.09287 179225.53386 179225.53386 1 1 2 *39.5271*i:
.36175.99990 9300000 .99990.38175 9300000i: 2*9118.35156 2*9118.35156 2*18.35156 219137.0*639
:: .99997 5 0 0 0 0
:: 219177.0*639 22*132.8*965 22*132.8*965 262115.15187
:: .9999* 11765
.36113 32 .99996 66667
.361106* .99996 66667
.36110 6* #8?
130
L A M B E R T P R O J E C T I O N
T h e L a m b e rt p ro je c tio n s used in the plane c o o rd in a te system s o f the U n ite d States are L a m b e rt co n fo rm a l co n ic p ro je ctio n s w ith tw o sta n d a rd p a ra lle ls. T h e p ro je c tio n fo r A m e ric a n Samoa is a L a m b e rt c o n fo rm a l co n ic p ro je c tio n w ith one sta n d a rd p a ra lle l. T h e fo rm u la s given are a d a p ta tio n s o f those used o rig in a lly in c o m p u tin g the p ro je c tio n ta b le s by w h ic h a ll o f the plane co o rd in a te s were co m p u te d by desk c a lc u la to r before 1957.
These e q u atio n s are not the o nly ones th a t can be used in c o m p u tin g L a m b e rt co o rd in a te s. T h e y may not even be the best. H o w e ve r, they y ie ld co o rd in a te s w h ich ch e ck values c o m p u te d using the p ro je c tio n ta b le s w ith in a h u n d re d th o f a foot (or at m ost tw o h u n d re d th s).
F O R W A R D . C O M P U T A T I O N
F o r m u la s . — T h e fo rm u la s fo r c o m p u tin g co o rd in a te s fro m g e o graphic p o sitio n s fo llo w :
</> = la titu d e o f sta tio n X — lo n g itu d e o f sta tio n
5 = 101.27940 65{ 6( XL7- </ >' ) - f 4 > " + (1,052.89388 2
— (4.48334 4 — 0.02352 0 cos- 0 ) cos- 0 ] sin cos <£}
(</)' is the degrees and m in u te s o f expressed in w hole m in ute s) (</)" is the re m a in d e r o f (f> in seconds)
R = U + s / ^ j l + U - ( i ^ ) /-io + ( y jp ) /-i
0 — / #;( X)
[0 and X are in seconds)
x = L \ + R sin 0 y = U - R + 2R s in - 1
C o n ve rg e n ce = # S cale fa c to r = /c
= U R { \ - 0 .0 0 6 7 6 86580 s in’2 <fr)^
20,925,832.16 cos <f>
T h e value o f 0 w ill not be g re a te r th a n 5° 07' e xce p t in A la s k a , zone 10, w h e re it w ill n o t be g re a te r than 9° 3 4 '.
T h e c o n s ta n ts (Z/s) are lis te d fo r th e va rio u s states a n d zones at th e e n d o f th is p u b lic a tio n . T h e r e c tify in g la titu d e o f th e c e n tra l p a ra lle l is d iv id e d in to tw o p a rts, L 7 a n d L H. T h e degrees a n d m in u te s in w h o le m in u te s are re p re s e n te d by L 7. T h e re m a in d e r in seco n ds is re p re s e n te d by L h. T h is s e p a ra tio n is m ade because th e e q u a tio n s were d e s ig n e d fo r an e le c tro n ic c o m p u te r w h ic h had a fixe d w o rd c a p a c ity o f o n ly 1 0 fig u re s a n d th e re c tify in g la titu d e e xp re sse d in se co n ds w o u ld be 1 1 fig u re s.
132
I N V E R S E C O M M U T A T IO N
F o r m u l iis . — F orm ulas fo r c o m p u tin g geo gra p h ic p o sitio n s from plane co o rd in a te s fo llo w :
0 — are tan 7— —
U - y
X
/? =
and X are in seconds) U - y
cos 0
L A — L^ — y + 2R s in'2 ^
u Si
l + { w ) ^ - { w ) L^ + { w ) ' L"
1 + { w ) u ~ { w ) L l t + { w ) Ll1
o)' = Z, 7 — 600 (degrees and m in u te s o f oj in w hole m in ute s) a / '= 3 6 ,0 0 0 + /„„ - 0.00987 3675S S3 s (re m a in d e r o f co in seconds)
o; = oj' + cu"
(}>' — L 7 — 6(X). (degrees and m in u te s o f (/> in w hole ih in u te s )
4>', = o j" - Jr | 1 ,()47.S4f)71 0 I (f>.l<)27h 0
I O.OSOV 1 2 cos* cu) cos* (u ] sin w cos co (re m a in d e r o f </> in seconds)
</> • </>' -t </>"
TABLE B. Constants fo r the Lambert Projection
w A m » e M A U x c H u s m s m a s s a c w s t t t s m*c h»g a n
H f t a n v A N U FB*aYlVAf*A SOUTH G U O X A SOUTH CAIOUNA
REFERENCES CITED
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Brown9 C urtis M. and W infield H. E ldrid g e , Evidence and Procedures fo r Boundary Location. W iley, New York: lW 2 . ’ C la ire 8 Charles N ., State Plane Coordinates by Automatic Data Processir
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C lark, David$ Plane and Geodetic Surveying (Vol. I I ) . Constable9 London: 5th ed.', 1963.
Colvocoressesg Alden P ., A U nified Plane Coordinate Reference System.
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M itc h e ll, Hugh C .» D e fin itio n s o f Terms Used in Geodetic and Other Surveys. USUSSTTpecWTuFlTcItTon ”2427' U, S. Government P rin tin g O ffic e , Washington: 1948.
M itc h e ll, Hugh C. and Lansing G. Simmons, State Coordinate Systems (A Manual fo r Surveyors). USC&GS Special P ublication 235. U. S.
Government P rin tin g O ffic e 9 Washington: 1945.
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Constable,
Rayner, W illiam H. and M ilton 0. Schmidt, New York: 1957.
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Reynolds„ Walter F .* Relation Between Plane Rectangular Coordinates and Geographic Coordinates. USC&GS SpecTaTTuBlication 71. U. S.
Government P r in t i ng (iT fi ce, Washington: 1932.
Richardus, P ., P ro je ct Surveying. W iley, New York: 1966.
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246 - Sines, Cosines, and Tangents 0° to 6 °, 1949.
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Periodi cals
Col cord, J. E ., "Geodetic Sections," Surveying and Mapping, Vol. XXVI, No. 3, p. 455. Washington: 1966,
Gale, P ling M.„ "Control Surveys fo r the C ity o f Houston," paper presented to The American Congress on Surveying and Mapping Convention, 1968,
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1 3 8
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