Varga, József

English

Periodica Polytechnica (Budapest University of Technology and Economics)

CONVERSIONS BETWEEN GEOGRAPHICAL AND TRANSVERSE MERCATOR (UTM, GAUSS.KRUGER) GRID COORDINATES J. VARGA Institute of Geodesy, Technical University, Budapest, H-1521 (Received: March 2, 1981.) Presented by Prof. Dr. F. Sarkozy, Director To now, mutual converSIOns between Univel"sal Transverse 1Ylercato:r (UTM) grid coordinates and geographical coordinates applied computers or tables. Programmable desk and pocket calculators revolutionized calcula-tions not only in the office but also on the terrain. A single magnetic card may replace volumes of tahles. A programmable calculator, fed with correct data input provides correct results as against the multitude of sources of error, inherent in the use of tahles. Another significant advantage of programmed calculation is a lower qualified operator needed to ohtain error-free results, at a reduced running time. The mutual conversion hetween UTM and geographical coordinates hy means of a minicomputer will be presented helow. The chosen computer methods require low data and program storage capacity fit to a smaller programmable pocket calculator. The prepared pro-grams with the indicated constants provide for a mean error of ± 1 mm in the grid coordinate and : 0.0002" in the geographical coordinate, all along the el-lipsoid. Notations a = semi-major axis of the ellipsoid; b = semi-minor axis of the ellipsoid; c = a2jb, polar radius of curvature; 9 a2 - b2 f h f' .. e- - , square 0 t e ITst eccentrICIty; a2 " a2 - b2 e2 •• e'2 = = --- . square of the second eccentrICIty; p l-~' . fP = latitude of point; fPl = latitude of foot of perpendicular to central meridian from point; A = longitude of point; /10 = longitude of origin (the central meridian) of the projection; 9 J. VARGA ), = difference of longitude from the central meridian; ), = A - Ao in the Eastern Hemisphere; ), Ao - A in the Western Hemisphere; t = tg q); 'rJ2 = e'2 cos2 q); V = V 1 + 'rJ2 = 1f 1 + e'2 cos2 q) ; M = c/T/"3, radius of curvature in the meridian; M = cl V, radius of curvature in the prime vertical; S = true meridional distance on the ellipsoid from the equator; ko = central scale factor; Xo= false northing; Yo= false easting; x = grid distance from the equator; y = grid distance from the central meridian; X = x + Xo = grid northing; Y = Y Yo = grid easting. Relationships between grid and geographical coordinates in the transver-sal conformal Mercator projections of the ellipsoid are: Geographical to projection: x = (S + A z ),2 + A4 ),4) ko, Y = (AI)' + A3),3 + A5).5) ko, X= x +Xo, Y=y + Yo' Projection to geographical: X-Xo x= , ko Y- Y o y= , ko (1) (2) After substitutions and transformations, (1) and (2) take the following, efficiently programmable form: [2] x = ko {S + N }.2 sin <P cos <P[1 + ~ cos2 <P (5 _ t2 + 9'YJ2 + 41]2) + 2 12 + ~ cos4 <P(61 - 58 t2 + t4 )]}, (3) 360 GRID COORDINATES y = koi..N cos (P [1 + ~ cos2(P(1 - t2 + 7]2) + + ~ COS" (P (5 - 18 t2 + t" + 147]2 - 58 t27]2)] , 120 (P = (PI - y2 ft [1 --~ (5 + 3tt + 7]t - 9ti7]t) + 2 MINI 12Nf + y4 (61 + 90 tt + 45 ti)l ' 360 Ni i.. = y (1 - L (1 + 2 ti + 7]i) + y4 (5 + 28 ti + 24 tn] . NI COS (PI 6 Ni 120 Ni 241 (4) (5) (6) The angle values have to be substituted in radians into formulae (3) to (6) to avoid calculation with powers ofthe analytical angle unit e. S and (P~ can be determined as follows. Subscripts 1 in (5) and (6) indicate that for calculating the given quantity (PI has to be substituted into the formula. 1. Determination of S (trne meridional distance on the ellipsoid) For determining S and (PI' VINCENTY suggested a useful direct method in [7]. For each ellipsoid 7 constant values have to be calculated and stored. Constants will be derived in a similar way, some of them are, however, simple to express from the others and therefore it is needless to store them separately. LEE [3] calculates the plane coordinates by lJI using the isometric latitude to obtain symmetrical terms for x andy, easy to program exempt from S. Dis-advantage of this method is that series have to be taken into consideratian to a high order number likely to inhibit use of a minicomputer. S is given by integrals: (7) or <1> <1> ~ f dtP S=jMdtP=c (1 + e'2 cos2 tP)3/2 (8) o 0 showing the arc length S of the meridian to be proportional to the area between curves of functions w = (1 - e2 sin2tP) -312, 9* 242 J. VARGA or OJ = (1 and the coordinate axis OJ = o. It is also seen that for determining S, storage of two constants for each ellipsoid may be enough; in one case these are a and e2, in the other c and e'2. Functions OJ = f(tP) are continuous, mildly ascending in the interval 0 < tP ::s;: n/2 fit to numerical integration by dividing interval o to tP into relatively few parts to yield the area below the curves at the desired accuracy. Let us write Simpson's parabola formula for S divided to even n parts according to (8): where h = tP/n and OJi is the substitution value ofthe function at the i-th place. n = f (tP) namely with increasing tP, the interval has to be divided into ever more parts. This method of calculating S with a pocket calculator takes much time, therefore the use of a faster desk computer is recommended. For n 30, generally S is obtained with mm accuracy even near the poles. In most practical works, calculations refer to the area of a single continent or country. In this case it is expedient to calculate So once accurately for the limit parallel circle if>o outside the boundary of the area, to store it in the memo-ry as a constant and to divide only the interval tPu to tP into n parts. n can be determined empirically in each case. This latter method takes not much time even with a pocket calculator because n ::s;: 10. The great advantage of this method is that only c and e'2 have to be stored, these constants being also necessary for solving Eqs (3) to (6). For determining S, a faster method but needing an increased storage capacity will be shown below. Ellipsoid Everest Bessel Qarke/66 Clarke/80 Hayford Krassowsky IUGG/1967 Table 1 c(m) 6398547.993 6 398 786.849 6399902.551 6400057.735 6 399 936.608 6399698.902 6 399 617.430 e'! ·10-' 6.682201 989 6.719218798 6.814784832 6.850 116 152 6.768170197 6.738525415 6.739725198 6366680.291 6366742.521 6367399.690 6 367 386.645 6 367 654.500 6 367 558.497 6367471.748 GRID COORDINATES Series expanding (8) and integrating term by term: S A ' m B' . 2'" I C' . 4'" D' . 6'" = c"... - - c sm ~,- c SIn ~ - - C SIll ~ + ... 246 A' = 1 - ~e'2 + 45 e'4 _ 175 e'6.L 11025 e'8_ 4 64 256 I 16384 B'= C' = D'= I 3 '2 ,-e 4 15 '4 I 525'6 2205 '8 I -e ---e ----e - ... 16 I 512 2048 I 15'4 105'6 I 2205 '8 -e ---e ,---e 64 256 4096 35 "6 --e' 512 315 '8 I ---e, 2048 Reducing (9): where S = atP - fJsin 2 tf> + y sin 4 tf> - <5 sin 6 tf> ... ., et = A' c, B' fJ =-c, 2 C' y=-c, 4 - D' o =--c. 6 243 (9) (10) (11) a, fJ, y, <5 are constants, to be determined separately for each ellipsoid. The values for the most frequently used ellipsoids are found in Table 1. 2. Determination of tPl Fig. 1 shows that coordinate x of foot point PIon the central meridian at point P of latitude tf> equals SI' permitting (11) to he written as: P y PI::. :c-/,::: ~/::. ·10-' . 10-' ·10-' 15900.693 22 16.5466 0.021 2.497486 2.599 3.4 15 988.638 53 16.72995 0.022 2.511275 2.628 3.5 16216.94389 17.2094 0.023 2.546871 2.703 3.6 16 300.700 72 17.3876 0.023 2.560030 2.731 3.6 16107.03468 16.9762 0.022 2.529508 2.666 3.5 16036.48028 16.8281 0.022 2.518466 2.643 3.5 16039.10744 16.8338 0.022 2.518913 2.644 3.5 244 J. VARGA Fig. 1 Expressing lP 1 at a good approximation: m X I f3 • 2m r· 4m I ~ • 6m 'PI = - T - srn 'PI - - SIn 'PI T - SIn 'PI + ... (12) cc cc cc cc As lPl occurs on hoth sides of the sign of equality, it can he determined hy iteration. First a fair approximation value of lPl is determined from ($1) = ~. cc Substituting lPl into the right-hand side of (12) yields a closer approximation for lP1• The numher of re-substitutions depends on the needed accuracy hut it is never more than three. Disadvantage of the method is to need iteration, its advantage is, how-ever, not to need storage of further constants that are easy to express from con-stants involved in calculating S. In case of a computer with an increased storage capacity, also values of f3/cc, r/cc, a/cc can he stored, of them those referring to ellipsoids most frequently used are as well included in Table 1. 3. Programs The programs were developed for a Hewlett-Packard 67/97 calculator. For converting geographical to grid coordinates Eqs (3), (4) and (ll); for converting grid to geographical coordinates Eqs (5), (6) and (12) were used. The programs can he accommodated on a magnetic card, and the constants helonging to an ellipsoid are stored on one side of a card. The prepared programs are shown hy Program Listings 1 and 2, their use hy Users' Instructions 1 and 2. The memory content is identical in hoth pro-grams; this is illustrated in Table "Registers". a/cc was not separately stored GRID COORDINATES 245 but built in into the program as a constant. From Table 1 it is seen that its value may be taken for all the ellipsoids uniformly as 3.5 . 10-9• Similarly also ~ might have been built into the program with a value of 0.022. The elaborated examples refer to the Universal Transverse Mercator grid system, with the following properties to be reckoned with in our calculations. 1. Ellipsoid: Hayford. The necessary constants are found in Table l. 2. False northing, Xo = 0 m (10000000 for Southern Hemisphere). 3. False easting, Yo = 500 000 m. 4. Scale factor at the central meridian, ko = 0.9996. As in the former case, the stored matter of the memories is recorded in the Data Sheet, supposing work on the Northern Hemisphere. The programs can of course be used for any ellipsoid, even for tangential or intersecting position of the image plane. Remind that in tangential position ko = 1. Data Sheet 6.367654500 + 06 0 1.610703468 + 04 1 1.697620000 + 01 2 2.200000000 - 02 3 2.529508000 - 03 4 2.666000000 - 06 5 9.996000000 - 01 6 0.000000000 + 00 7 0.000000000 + 00 8 0.000000000 + 00 9 5.000000000 + 05 A 0.000000000 + 00 B 0.000000000 + 00 C 6.399936608 + 06 D 6.768170197 - 03 E 0.000000000 + 00 I Constants of ellipsoids missing from Table 1 are determined from Eqs (10) and (ll). The t'wo programs confined suit conversion between belts, using the fol-lowing method. Grid coordinates for one belt are converted to geographical coordinates, then, ·with the geographical longitude of the central meridian of the adjacent belt as input, grid coordinates ofthe point on the adjacent projec-tion belt are calculated with the other program. 246 J. VARGA User Instructions Geographical to projection INPUT KEYS OUTPUT STEP INSTRUCTIONS DATA/UNITS - DATA/UNITS 1 Load sides 1 and 2 of CJCJ program card CJCJ CJCJ 2 Load side 1 of d·ata cord c=Jc=J (Choose an ellipsoid " at first only CJCJ CJCJ 3 Longitude of the central meridian. Ao LDCJ degree, if needed: deg., min., sec. CJCJ CJCJ 4 Latitude, deg., min., sec. q, IENT.t1 CJ 5 Longitude -11- A CJ CJr=:J 6 Northern Hemisphere, East from Greenwich [TICJ West -u- u::J [:=J Southern Hemisphere, East -11- CDCJ West -11- C[JCJ - CJCJ 7 UTM grid coordinates CJCJ X (m) 8 -11- ~CJ Y (m) CJCJ For 0 new case go to step 2, CJCJ or 3, or 4. CJCJ EXAMPLE STEP INSTRUCTION INPUT DATA/UNITS KEYS OUTPUT OAT A fUNITS CJCJ 1 Load program CJCJ 2 ·Load data CJCJ (Hayford 1 CJCJ. 3 Longitude of 9' CD I I , the cent. meridian CJCJ 4 Latitude 47° 15' 38,4257' IENTlI CJ I 5 Longitude 60 27' 49,7791 u CJ I 6 N. Hemisphere, East. o;::J CJ 7 UTM\ X. CJCJ 5237353\489 m ! 8 UTM, Y. CE:ill CJ 308121,657 m CJ CJCJ CJI I User Instructions 1 GRID COORDINATES 247 User Instructions Projection to geographical INPUT KEYS OUTPUT STEP INSTRUCTIONS DATA/UNITS DATA/UNITS 1 load sides 1 and 2 of c:::=J c:::=J program card c:::=J c:::=J c:::=J c:::=J 2 load side 1 data card c::J c::J (Choose an ellipsoid), at first on(y c:::=J c:::=J CJ c:::=J 3 Longitude of the centraL meridian Ao CD c:::=J degree, if needed: deg" min., sec. c:::=J c=:::J ~ c:::=Ji I H- UTM g,id ccord: :-:ates X (m) lENT. t : , 0 -h- Y (m) c:::=J CJ c:::J 6 Northern Hemisp"r-,ere, =-'::15: 7rom G .. eeD, .... ic·r I I I V/est -1.- o::::J c:::J Southern Hemisphere, ::.::15: -Il- I CL] I I West -li- j L:Q:J c:::J c:::J c:::=J i 7 latitude, deg" min./ sec. I c:::J c:::J <P 8 Longitude, -11- CB:ill c:::=J A c:::=J c:::=J For a new case go to step 2, c:::J c:::=J or 3, or 4, c:::=J c:::=J EXAMPLE I STEP INSTRUCTION INPUT DATA / UNITS KEYS OUTPUT DATA/UNITS 1 load program c:::J c:::=J 2 load data (Hayford) c:::J 3 lonqitude of the 9° COCJ cent. meridian. c:::=J c:::=J 4 UTM, X 5237353,489 m lENT. t I c:::=J 5 UTM, Y 308121 I 657 m c:::=J CJ 6 N. Hemisphere, Ea sI. u:::::J CJ 7 latitude c:::=J c:::=J 47° 15' 38,4257' , 8 Longitu-ie IB:TIJ c:::=J 6° 27' 49,7791' CJ c:::=J c:::=J CJ I c:::=J c:::=J c:::=J c:::J I User Instructions 2 248 J. VARGA Program Listing I 001 *LBLB 057 6 113 , 169 + , 002 SFO 058 114 6 170 CFO 003 GTOA 059 ST09 115 1 171 CFl 004 *LBLD 060 x 116 RCL6 172 DEG 005 SFO 061 1 117 5 173 RjS 006 "LBLC 062 + 118 8 174 RCLI 007 SFl 063 5 119 x 175 RCL6 008 *LBLA 064 RCL6 120 176 x 009 RAD 065 1 121 RCL6 177 RCLA 010 P:;tS 066 8 122 X2 178 + 011 HMS--- 067 x 123 + 179 RTN 012 RCLC 068 124 RCL8 180 -LBLE 013 069 RCL6 125 3 181 HMS---014 FO? 070 X2 126 182 STOC 015 CHS 071 + 127 x 183 R~ 016 D---R 072 RCLS 128 + 184 RTN 017 STOO 073 1 129 RCL7 185 RjS 018 X:;tY 074 4 130 RCLO 019 HMS--- 075 x 131 x 020 D---R 076 + 132 RCL3 021 ST02 077 RCL5 133 x 022 SIN 078 RCL6 134 2 023 ST03 079 x 135 024 LSTX 080 5 136 x 025 COS 081 8 137 RCL2 026 ST04 082 x 138 P:;tS 027 083 139 ST07 028 X2 084 RCL8 140 RCLO 029 ST06 085 X2 141 X 030 RCL4 086 1 142 + 031 X2 087 2 143 RCL7 032 RCLE 088 0 144 2 033 x 089 145 x 034 ST05 090 ST08 146 SIN 035 1 091 x 147 RCL1 036 , 092 ..L 148 x , I 037 yX 093 RCL7 149 038 RCLD 094 x 150 RCL7 039 X:;tY 095 STOI 151 4 040 096 5 152 x 041 RCL4 097 RCL6 153 SIN 042 x 098 154 RCL2 043 RCLO 099 RCL5 155 x 044 x 100 9 156 + 045 ST07 101 x 157 RCL7 046 1 102 + 158 6 047 RCL6 103 RCL5 159 x 048 104 X2 160 SIN 049 RCL5 105 4 161 RCL3 050 + 106 x 162 x 051 RCL4 107 + 163 052 X2 108 RCL9 164 RCL6 053 RCLO 109 2 165 x 054 X2 110 166 F1? 055 x 111 x 167 CHS 056 ST08 112 1 168 RCLB GRID COORDINATES 249 Program Listing 2 001 *LBLB 057 ST09 113 9 169 -L I 002 SFO 058 DSZI 114 x 170 1 003 GTOA 059 GTOO 115 171 -L I 004 *LBLD 060 STOI 116 RCL4 172 RCL4 005 SFO 061 RCL7 117 x 173 x 006 *LBLC 062 p~S 118 1 174 6 007 SFl 063 STOO 119 2 175 008 *LBLA 064 RCLD 120 176 009 RAD 065 RCLI 121 177 1 010 RCLA 066 COS 122 1 178 I T 011 067 X2 123 + 179 RCLO 012 RCL6 068 RCLE 124 RCL5 180 x 013 069 x 125 y:x 181 RCL3 014 FO? 070 STOl 126 x 182 015 CHS 071 1 127 RCLO 183 RCLI 016 ST07 072 + 128 X2 184 COS 017 X=y 073 yX 129 x 185 018 RCLB 074 ST02 130 RCL2 186 R ..... D 019 075 131 X2 187 RCLC 020 RCL6 076 ST03 132 X2 188 + 021 077 133 x 189 ..... HMS 022 ABS 078 X2 134 RCLD 190 DEG 023 RCLO 079 ST04 135 X2 191 P=S 024 080 X2 136 2 192 RTN 025 ST08 081 ST06 137 x 193 *LBLE 026 ST09 082 3 138 194 HMS ..... 027 3 083 6 139 CHS 195 STOC 028 STOI 084 0 140 RCLI 196 Rj. 029 *LBLO 085 141 I 197 RTN T 030 RCL8 086 RCLI 142 R ..... D 198 RIS 031 RCL9 087 TA1~ 143 ..... ai\IS 032 2 088 X2 144 CFO 033 x 089 ST05 145 CFl 034 SIN 090 9 146 RIS 035 RCL4 091 0 147 RCL5 036 X 092 >< 148 2 037 + 093 6 149 8 038 RCL9 094 I 150 x 039 4 095 + 151 5 040 x 096 RCL5 152 , , 041 SIN 097 X2 153 RCL5 042 RCL5 098 4 154 X2 043 X 099 5 155 2 044 100 x 156 4 045 RCL9 101 + 157 x 046 6 102 x 158 , , 047 >< 103 RCL5 159 RCL6 048 SIN 104 3 160 x 049 3 105 x 161 1 050 106 5 162 2 051 5 107 ...L 163 0 , 052 EEX 108 RCLl 164 053 9 109 + 165 RCL5 054 CHS 110 RCLl 166 2 055 x III RCL5 167 x 056 + 112 x 168 RCLl 250 J. VARGA. Registers 0 d. , ;3 12 P er 4 a- (3/« 5 6!c£. 6 ka 7 a 9 so 51 152 153 $4 SS 56 57 sa 59 A Ye la le 0 lE Z I' x 0 .../,!..IJ c e 4. Transformation of the programs for lower accuracy requirements The enclosed programs can also be used for the highest accuracy require-ments on the entire ellipsoid; therefore they exceed that needed for lower ac-curacy requirements. If a mean error of =1 cm is made up with, steps 72 to 83 and 99 to 107 in program 1 and steps 108 to 115 in program 2 may be omitted. Working ,·tithin a domain of 10° of the geographical latitude and having So calculated up to a boundary parallel circle q>o, (3) to (5) may be replaced by simpler expressions: y = ko)' N cos q> [1 + A; cos2 q> (1 where: - Xoo 3x60, xg 90 < <PI = <Po + --Vo - ---'1}0 to -j- --'1}0 (to --I)! ... , No 2 N5 2N~ or },2. },'" . 2 < <PI = <P + - V2 sm <P cos <P + - sm <P cos3 <P (5 - t ) ! ... 2 24 These formulae can be composed to a program uniting the two programs above so that a single input on magnetic card before start suffices and there-after also conversion between the projectional belts to and from can be carried out. Similar simple formulae are those by FIELD [12]. Also nomograms al'e made for non-programmable pocket calculators facilitating conversions such as those by REITHOFER [10] for the domain 45° q> < 57°. GRID COORDINATES 251 Summary :Most of difficulties in mutual computer conversions between Universal Transverse :Mercator (UTM) grid and geographical coordinates arise in determining S (the meridian arc length from the equator) and cfJ1 (the ellipsoidal geographical latitude of the foot point which is on the central meridian). Effective methods for such conversions are suggested, suiting pocket calculators \\ith less storage capacities, just as are programs for calculators type HP 67/97 suitable for con-versions on the entire surface of any ellipsoid, as indicated in the title. References 1. HAZAY L-T.-iRczY-H. A.: A Gauss-Kriiger koordimitak szamltasa. (Akademiai Kiad6 Budapest 1951.). 2. JORDAN-EGGERT-KN;EISSL: Handbuch der Vermessungskunde, IV. (J. B. ::\Ietzlersche Verlagsbuchhandlung, Stuttgart, 1958.). 3. LEE, L. P.: The Transverse Mercator Projection of the Entire Spheroid. Empire Survey Review, XVI (123), 1962. 4. LEE, L. P.: The length of a meridian arc, and the Transverse :Mercator Projection of the entire spheroid. Survey Review, XVII (133), 1964. pp. 343. 5. HAZAY 1.: Vetiilettan. (Tankonyvkiad6, Budapest, 1964.). 6. GDOWSKI, B.: The Transverse Mercator Projection of the entire spheroid. Survey Review, XVIII (141). 1966. pp. 339-341. 7. VINCENTY. T.: The meridional distance problem for desk computers. Survey Review, XXI (161), 1971. 136-140. ' 8. GERSTBACH, G.: Zur Losung ellipsoidischer Aufgaben mit elektronischen Taschenrechnern. Zeitschrift fUr Vermessnngswesen. 5. 1974. pp. 205. 9. HUBENY, K.: Isotherme Koordinatensysteme und konforme Abbildungen des Rotations-ellipsoides. :Mitteilungen der geodatischen Institute der Technischen Universitat. 27. Graz. 1977. 10. REITHOFER, A.: Koeffiziententafeln und Rechenprogramme fUr die Gauss-Kriiger-(UTM)-Koordinaten der Ellipsoide von Bessel. Hayford, Krassowsky und des Refe-renzellipsoids 1967. Mitteilungen der geodatischen Institute der Technischen Uuiver-sitat, 27, Graz. 1977. 11. JACKSOl'\', J. E.: Transverse ::\Iercator Projection. Survey Review, XXIY (188). 1978. pp. 278-285. 12. FIELD, N. J.: Conversions Between Geographical and Transverse :Mercator Co-ordinates. Survey Review. XXV (195), 1980. pp. 228-230. Dr. J6zsefYARGA, H-1521 Budapest 

CONVERSIONS BETWEEN GEOGRAPHICAL AND TRANSVERSE MERCATOR (UTM, GAUSS-KRÜGER) GRID COORDINATES

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CONVERSIONS BETWEEN GEOGRAPHICAL AND TRANSVERSE MERCATOR (UTM, GAUSS-KRÜGER) GRID COORDINATES

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Periodica Polytechnica (Budapest University of Technology and Economics)