.S. Department of Commerce 

Frederick H . Mueller, Secretary 

Coast and Geodetic Survey 

H . Arnold Karo, Director 

Publication 65- 2 

PLANE COORDINATE PROJECTION TABLES 

PUERTO RICO 

and 

VIRGIN ISLANDS 

WASHINCTON 1 1961 
Foreword 

The plane coordinate system used in Puerto Rico, the 

Virgin Islands, and St. Croix Island is based on a Lambert 

conformal conic projection with two standard parallels. 

These areas are all included in the same basic projection. 

However, in order to avoid ne gative y coordinates for 

points on st. Croix, the y coordinates there have been increased by 100 1 000.00 feet. This is taken care of automatically in computing coordinates by using the proper 

constants from the tabulation immediately preceding Table I. 

The tables in this publication are used for the conversion 

of geographic positions to plane coordinates or plane coordinates ~o geographic positions. 

The methods of computation have been designed for 

machine calculation, using tables of natural trigonometric 

functions. A table of these functions has been published 

by the Coast and Geodetic Survey to ten decimal places 

with ten-second intervals for o0 to 6° in Special Publication No. 246 and is sold for a nominal sum by the Superintendent of Documents, u.s. Government Printing Office, 

Washington 25, D.c. 

The formulas and sample computations which follow 

show the general methods for computing either type of coordinates. 

Plane Coordinates from Geographic Positions 

x = R sin 9 + c 

y = Rb - R cos g 

Grid azimuth = geodetic azimuth - 9 + second term 

where 

R is the radius for the latitude of the station, 

Rb is a constant for a zone, 

Q is the mapping angle for the longitude of the 

station, 

and 

C is the value of x assigned to the Central 

Meridian for a zone. 

3 
4 

The second term for the reduction of geodetic to 

grid azimuths may be neglected for most work. However, 

for lines five mile s or more in length, if the same degree of accuracy is desired as is obtained by geogi·aphic 

computations, this term should be evaluated and used. 

Second term = y + 

0 

2 f 0 

2 sin l" 

Geographic Positions from Plane Coordinates 

The formulas show the method of computing R and Q 

from which the latitude and longitude may be obtained. 

where 

and 

x' = x - c 

tan e = x' ":" (~ - y) 

R = (~ - y) . 

":'" cos e 

L\). = e . 

- 1 

A. = Central Meridian - ~A, 

R, Rb, 9, and C are the same as previously 

defined 

J is a constant for a zone. 
l"on 72:\d 

( 1-22-57) 

U.S. DEPARTMENT DP COMMERCE COAST AND GEODETIC SURVEY· 

PLANE COORDINA ru ON UMBERT PROJECTION 

5 

(Cood.-fbrm tor calculalu>c·mothlno eompulallon) 

X=Rsin8+C 

C= soo ooo oo ' ' ...., Stat•-Zon• Puerfo Rico 

Grid Az =- Gt1od Az - 8 

Station Lafifud• R sin 8 x 

Longifud• 8 cos8 y 

I I 8 29 079S'I C,3 '-150 (,55./9 -0.00321 01338 296 3/L.f. 9 I 

Paco , 1901 ' fo7 01 /~. 215 - 0 I I 02..1337(,2 0 . 99999 '/!£/ /S 237. 151.18 . I II 

Grid arimutfl to orimuffl mark 

2 18 1 s 3o.9'f7 '13 51/f Cf07.S2 +0.00.307 (,0292 '195 373. 71 Frio (U.S.G.s), foS 52 12.191 +O I03'/.'f77S(,,S 0.'19999S2&90 172 872.// /93'-I I 

5 

4 

Grid ozimufh fo ozltn11fh mark 

Grid ozimufh 

Grid azimuth 

U.S. DEPARTMENT O F COMMERCE 

CO AST AN O GEODETIC SURVEY 

FORM J42b 

AEV . APRIL ll!U 

fo ozitn11th mark 

fo azimuffl mar* 

I 

v I 

. 

. I II 

I 

GEODETIC POSITIONS FROM LAMBERT COORDINATES 

(CALCULATING MACKINE COMPUTATION) 

STATE-Z NE -~B_.u~c,~rfi~_L _l o ______ _ l = 0.31288 62.28/ 

Station Paco 1901 ' 

c soolooo.09 Rb &3 087 If 79. tf'I .r 29fo . .3 Llf. <JJ y 23i JS /. !8 -- - ) 

x' = x-C - Z.03. (;;_</S 09 Rb-y 03 J./50328.2" J 

tan fJ = x ' -;. (Rb-y) - 0 .0032 1 Q/503 f) - t,02'.' 1338 

f) I - 0 0 I I ' 02.'/..38~ ti>.. = 0 7 l - 2 I I (o '.' 21 SI I 

lo.qq9qq L/8'f7S -0 0 3S I 

cos f) ti>.. /fo. '21s1_ 

R = (R~-y) 7 COS f) r.o3.'ISO toSS.19 Central Meridian t.& 0 i.&' oo.'oooo 

"' 18 ° 29' o 7. cfs J./o ,\ -= C. M.-.\A (,, 7 0 Of I l b.2/SI 

Station r ·o (U.S.G.5.) ) I 934 

C 00 000. 00 ~b'---+-...=....o~'-'-'-+----'-_,_"'--'--~ 

~r 3 . 71 Y 

. .-'- .,.-c - + 195, 373. 7 I Rb- y 

tan fJ -= :r'-;. (Rh -y) I + 0 . 00 3 0 7 ~ 0 'f 3_7 __ ~ ___ fJ ---f------'-""----'---'---'-----1 

0 I II 

f) + 0 0 . 7 " ti.\ :- o_-;._l _-+---o,,--~--=----==--1 

cos f) I o. C/'1999 5zc;,~ ____ .\_---+- +---=o---="""-----'-...<...-=~'--t ' 

R = (Rb-y ) 7cos fJ I &3.,5 I~ 0 . 8 2 Central Meridian '' 

0 

(,--~--i 

I <P i 18 ° I 8 '_jg " o > = C. M_._-_ti._\ __.__~"--5_0 _S."'"""C. ___.__~~ 
6 

Cons tants 

r- ~~~~~~~~~·~ -Z_ ~~I ~--Z;ne 2 

Puerto Rico St. Cr o ix 

& 

Constant 

c 

Central Meridian 

1 

2fo 2 sin l" 

1 

log 2fo 2 sin l" 

log i 

log K 

Virgin Islands I 

( Axc e pt S t . Cr o i x) I 

Island 

I 

500, 000.00 ft. 

I 

500, 000 . 00 f t. 

I 

66° 26 1 00 000 

63 ,687, h79. 44 f t . 63, 787, 479. 44 f t. 

145, 257.78 f t . 245, 257 .78 ft. 

0. 31288 82281 0. 31288 82281 

-10 

2.368 x 10 

0. 374 4035 - 10 

- 10 2. 3D8 x 10 

0. 374 4035 - 10 

4 I 

I 

9. 49538 92238 - 10 9.1 ~9538 92238 - 10 

7.84676 66905 
7 

Puerto Rico and Vir gin Islands (except St . Croix ) 

Tabl e I 

Lat. R Y' a er ca e ca e 

(feet ) (y value on difference tmi t s of expr essed 

central f or 1 sec. 7th place as a 

meridian) of lat. of logs ratio 

feet f eet 

17 o • 3, 7, 79. 0 100. 0 +7 • . o 0 7 

51 63 , 681,427 . 69 6, 051. 75 100. 86233 +70. 3 1. 00001619 

52 63, 675, 375. 95 12,103 . 49 100. 86250 +62.1 1. 00001430 

53 63 , 669 , 324. 20 ,155 2~ l 00. 86233 +54. 3 1.000012so 

54 63 , 663 ,272.46 24, 206. 9 100 . 86250 +46 . 8 1.00001078 

55 63 , 657, 220.71 30, 258. 73 100. 86233 +39. 7 1. 00000914 

17 56 63 , 651,168. 97 36 ~7 l 00 . 86250 +32. 9 1.00000758 

57 63 , 645,117. 22 42, 362. 22 100. 86250 +26 . 5 1. 00000610 

58 63 , 639 ,065. 47 48, 413. 97 l 00. 86250 +20 . 5 1.00000472 

59 63 , 633 ,013. 72 54,465. 72 l 00 . 86250 +14. 8 1.00000341 

18 00 63 , 626, 961.97 60 , 517. 47 100. 86267 + 9. 5 1.00000219 

18 01 63 , 620, 910. 21 66 , 569 . 23 100. 86267 + 4. 6 1.00000106 

02 63 858 45 72, 620. 99 100. 86283 o.o 1.00000000 

03 63 , 60 , 806. 68 78, 672. 76 100.86283 - 2 0.99999903 

04 63, 602,754. 91 84, 12i.s3 100. 86300 - • 0 0. 99999816 

05 63 , 596 , 703 .13 90, 77 . 31 100. 86300 -11 .s 0. 9999973.5 

18 06 63, 590,651.35 96, 828. 09 100. 86317 -14.6 0. 99999664 

07 63 .58~ .599 56 102, 879. 88 100. 86333 -17. 3 0. 99999602 

08 63 , 57 , 547. 76 108, 931.68 100. 863.50 -19. 7 0. 99999546 

09 63, 572,495.95 114, 983 .49 100. 86350 -21.7 0. 99999500 

10 63 , 566,444.14 121 , 03.5 . 30 l00 . 86367 - 23 . 4 0. 99999462 

18 11 63 , 560, 392. 32 127 , 087 . 12 100. 86383 -24. 7 0. 99999431 

12 63 55~ 3~0. 49 133 ,138. 95 100 . 86400 - 25. 6 0. 99999411 

13 6 3 I 54. I 2 8 • 6 5 139,190. 79 100 . 861~ 7 - 26.1 0. 99999399 

14 63 , 542, 236 . 80 145, 242. 64 100. 86433 -26. 3 0. 99999394 

15 63, 536,184. 94 151, 294. so 100. 86450 - 26.1 0. 99999399 

18 16 63, 530,133. 07 157 , 346. 37 100. 86467 - 2.5. 6 0. 99999411 

17 63 , 524, 081.19 163 , 398. 25 100. 86500 - 24. 7 0. 99999431 

18 63 , 518, 029 . 29 169,450 .15 100 . 86517 - 23 . 4 0. 99999462 

19 63, 511, 977. 38 175, 502. 06 100. 86533 -21.7 0.99999500 

20 63 , 505, 925. 46 181, 553 . 98 100. 86550 -19 . 7 0. 99999546 

18 21 63 ,499 I 873. 53 187, 605. 91 100. 86583 -17.3 0. 99999602 

22 63 , 493 , 821.58 193, 657.86 100. 86600 -14. 6 0. 99999664 

23 63 , 487 ,769 . 62 199, 709.82 100. 86633 - 11.s 99999 ~5 

24 63 ,481, 717. 64 205, 761. 80 l 00. 86667 - 8. o 0. 99999816 

25 63 , 47.5, Q65. 64 211,813 . 80 100. 86683 - 4. 2 0. 99999903 
8 

Puer to Rico and Vir gin Islands (except St. Croix) 

Tabl e I (Concluded) 

Lat. Y' a ul ar ca e n c e 

(feet) (y value on difference unit s of express ed 

cen tr al f or 1 sec. 7th pl ac e as a 

meridian ) of l a t. of logs ratio 

fe et) feet) 

1 3, 9, 13. 3 17, • 1 100 . 717 o. o 1. 00000000 

63 ,463 , 561. 60 223 17. 8~ 100. 86733 + 4.6 1.00000106 

63, 457 , 509. 56 229 , 969 . 8 l 00 . 86767 + 9 . 5 1. UU000219 

63,451,457 . 50 236 , 021. 94 l 00. 86817 +1 ~ 1. 00000341 

63 ,445,405. 41 242, 074. 03 100. 86833 +20. l. 00 001 ~ 71 

18 31 6 3 , 43 9, 3 5 3 • 31 248, 126.13 100. 86867 +26 . 5 1. 00000610 

32 63 , 433 ,301.19 254,178. 25 100 . 96883 +32. 9 1.00000758 

33 63,427,249.06 260 , 230. 3t3 100. 86933 +39 . 6 1. 00000912 

34 63,421,196. 90 266 , 282. 54 100. 86967 +46. 7 1.00001075 

35 63 ,415, 144. 72 272 , 334.72 100. 87000 +54 . 2 1.00001248 

18 36 63, 409, 092. 52 278, 386 . 92 100. 87050 +62.1 1. 00001430 

37 63 0~0.29 284, L.39 . 15 100 . 37083 +70 . 3 1.00001619 

38 63 , 396, 9 8. 04 290 , 491.40 100 . 67117 +78. 9 1. 00001817 

39 63 , 390 , 935.77 296, 543 . 67 100. 87150 +87. 13 1. 00002022 

40 63, 384, 883 . 48 302 , 595 . 96 +97 . l 1. 00002236 
9 

St. Croix Island 

Table I 

Lat. R Y' Tabular Scale in Sea e 

(feet) ( y value on difference uni ts of expressed central for 1 sec. 7th place as a 

meridian) of lat. of logs ratio 

feet) (feet) 

7 3,77 I .1 9, 3. 0 100. 3 3 + 1. 1.0000_ 7 3 

36 63,772,204.31 15,275.13 l00. 86367 +237 .1 1.00005460 

37 63,766,152.49 21,326. 95 100. 86350 +223 . 4 1.00005145 

38 63,760,100.68 27,378.76 l00.86333 +210.1 1.00004839 

39 63,754,048.88 33,430.56 100.86317 +197.2 1.00004541 

40 63,747,997.09 39,482.35 100.86317 +184.6 1.00004251 

17 41 63,741,945.30 45,534.14 l00.86300 +172.4 1.00003970 

42 63,735,893.52 51, 585. 92 100.86300 +160.5 1.00003697 

43 63,729,8~.74 57,637.70 100.86283 +149.0 1.00003432 

44 6 3 I 7 23 I 7 9 • 97 63,689.47 100 . 86283 +137 .9 1.00003176 

45 63,717,738.20 69, 741.24 100. 86267 +127. 2 1.00002928 

17 46 63,711,686.~ 75,793.00 100.86250 +116.8 1.00002689 

47 63,705,634. 9 81, 844. 7 s 100.86267 +106.8 1.00002458 

48 63,699,582.93 87, 896. 51 l00.86250 + 97.1 1.00002235 

49 63,693,531.18 93,948.26 100. 86233 + 87.8 1.00002021 

50 63,687,479.44 100,000.00 + 78.9 1.00001817 
10 

Puerto Rico and Vir gin Islands , includinR St . Cr oix 

Table II 

l" of Long. = 8823 of Q 

Long. Q Long. Q Long . Q 

64°01 1 +0°45' 22 ~1 585 64°36 1 +0 °34 25~062)06 65°11 1 + Q 0 2J I 27~'997 27 

02 +0 45 03 . 354291 37 +O 34 06 . 289012 12 +O 23 09 . 223733 

OJ +0 44 44. 580998 38 +O 33 47.515719 13 +O 22 50.450439 

04 +O 44 25. 807704 39 +0 33 28.742425 14 +0 22 Jl. 677146 

05 +0 44 07.034410 40 +O 33 09. 969131 15 +O 22 12. 903852 

64 06 +0 43 48. 26111 7 64 41 I +O 32 51.195837 

I 

65 16 +O 21 54.130558 

07 +0 43 29.487823 42 +O 32 32.422544 17 +O 21 35. 357265 

08 +O 43 10.714529 43 +O 32 13. 649250 18 +O 21 16. 583971 

09 +O 42 51. 941236 

I 

44 +0 31 54.875956 I 19 +0 20 57. 810677 

10 +O 42 33 .167942 45 +O Jl 36 .102663 20 +0 20 39 .037384 

I 64 11 +0 42 14. 394648 I 64 46 +O 31 17. 329369 65 21 +0 20 20.264090 

12 +0 41 55. 621354 47 +O 30 58. 556075 22 +O 20 01. 490796 

13 +0 41 36.848061 48 +O 30 39.782782 

I 

2J +O 19 42. 717502 

14 +0 41 18. 074767 49 +0 JO 21.009488 24 +O 19 23.944209 

15 +0 40 59.301473 50 +0 JO 02. 236194 25 +O 19 05.170915 

64 16 +O 40 40.528180 64 51 +0 29 4J. 462901 65 26 +O 18 46.397621 

17 +0 40 21. 754886 52 +0 29 24.689607 27 +O 18 27. 624328 

18 +0 40 02.981592 53 +0 29 05. 916313 28 +0 18 08 . 85103 4 

19 +0 39 44.208299 54 +0 28 47.143019 I 29 +O 17 50 . 077740 

20 +0 J9 25.435005 55 +O 28 28.369726 30 +O 17 31.304447 

64 21 +O 39 06.661711 64 56 +O 28 09.596432 65 31 +O 17 12. 531153 

22 +O 38 47.888418 57 +O 27 50. 823138 32 +0 16 53 . 757859 

23 +O 38 29.115124 58 +O 27 32.049845 33 +O 16 34. 984566 

24 +O 38 10.Ji18JO 59 +0 27 l J .276551 34 +0 16 16. 211272 

25 +O 37 51.5 8536 65 00 +O 26 54.503257 35 +0 15 57.437978 

64 26 +0 37 32. 795243 65 01 +0 26 35.729964 65 36 +o 15 38. 664684 

27 +0 37 14.021949 02 +O 26 16. 956670 37 +O 15 19. 891391 

28 +0 36 55.248655 03 +o 25 58.183376 38 +o 15 01.118097 

29 +0 36 36.475362 04 +O 25 39. lμ0083 39 +O 14 42. 344803 

30 +O 36 17.702068 05 +0 25 20 . 636789 40 +0 14 23.571510 

64 31 +O 35 58.928774 65 06 +O 25 01. 863495 65 41 +O 14 oi.798216 

32 +O 35 40.155481 07 +0 24 43.090202 42 +O 13 4 .024922 

33 +0 35 21. 382187 08 +0 24 24. 316908 43 +o 13 27 . 251629 

34 +0 35 02.608893 09 +0 24 05.543614 44 +0 13 08. 478335 

35 +O 34 43. 835600 10 +O 23 46. 770320 45 +0 12 49.705041 
Long. 

65°46 1 

47 

48 

49 

50 

65 51 

52 

53 

54 

55 

65 56 

57 

58 

59 

66 00 

66 01 

02 

03 

04 

05 

66 06 

07 

08 

09 

10 

66 11 

12 

13 

14 

15 

66 16 

i~ 19 

20 

11 

Puerto Rico and Virgin Islands, including St. Cr oix 

Table II (Concluded) 

l" of Long. = 0~' 31288823 of Q 

Q Long. Q Long. 

+0 30~93 1748 66°211 +0 01 33~866468 66°56 1 

+0 12 12.158454 22 +O 01 15.093175 57 

+0 11 53. 385160 23 +0 00 56. 319881 58 

+O 11 34. 611867 24 +0 00 37 . 546587 59 

+O 11 15.838573 25 +0 00 18. 773294 67 00 

+O 10 57.065279 66 26 0 00 00.000000 67 01 

+o 10 38.291985 27 -0 00 18.773294 02 

+0 10 19.518692 28 -0 00 37 . 546587 03 

+0 10 00.745398 29 - 0 00 56. 319881 04 

+o 09 41. 972104 30 - 0 01 15.093175 05 

+0 09 23 .198811 66 31 -o 01 33 . 866468 67 06 

+0 09 04.425517 32 -0 01 52.639762 07 

+o 08 45. 652223 33 -o 02 11. 413056 08 

+o 08 26.878930 34 -o 02 30.186350 09 

+0 08 08.105636 35 -o 02 48. 959643 10 

+0 07 49 . 332342 66 36 -0 03 07.732937 67 11 

+O 07 30 . 559049 31 -0 03 26 . 506231 12 

+o 07 11.785755 38 -0 03 45. 952~ 13 

+o 06 53.012461 39 -o 04 04. 05281 14 

+O 06 34. 239167 40 -o 04 22. 826112 15 

·>-0 06 15.465874 66 41 -0 04 41. 599405 67 16 

+o 05 56. 692580 42 -0 05 00. 372699 

+0 05 37.919286 43 -0 05 19.145993 i~ +o 05 19.145993 44 -0 05 37 . 919286 19 

+o 05 00. 372699 45 - 0 05 56.692580 20 

+0 04 41 . 599405 66 46 -o 06 15. 465874 

+o 04 22.826112 47 - 0 06 34. 239167 

+o 04 04.052818 48 -o 06 53. 012461 

+o 03 45.279524 49 -0 07 11.785755 

+0 03 26 . 506231 50 -o 07 30.559049 

+o 03 07. 732937 66 51 -0 07 49. 332342 

+O 02 48.959643 52 -o 08 08.105636 

+O 02 30.186350 53 -0 08 26.878930 

+0 02 11.L.13056 54 -0 08 45.652223 

+O 01 52. 639762 55 -0 09 04. 425517 

Q 

-0 °09 I 23 9881 1 - -o 09 41.97210 

-o 10 00.7~539 

-o 10 19. 51869 

-o 10 38.29198 

-0 10 57 . 06527 

-0 11 15. 8J857 

~ 2 

5 

9 

3 

-o 11 34. 6ll867 

-0 11 53 . 38516 0 

-o i 2 i2.158u5 4 

-0 12 30.93174 8 

-0 12 49.705041 

-0 13 08.47833 5 

-0 13 27.251629 

-o 13 46.02492 2 

-o 14 04.79821 6 

-o 14 23. 571510 

-0 14 42 3~803 -o 15 01.11 097 

-o 15 19. 89139 

-0 15 38. 66468 

-0 15 57. 43797 

-0 16 16. 21127 

1 

i 

2 

-0 16 34.98456 6 

-o 16 53 . 757859 
12 

CORRECTIONS TO NATURAL SCALE RATIOS* 

(in units of the 7th decimal place) 

For Lambert Projection 

Af' a s a r g u m e n t 

1 

2 

~ 5 

6 

~ 9 

10 

11 

12 

i~ 15 

16 

i~ 19 

20 

21 

22 

~~ 25 

26 

~~ 29 

30 

Corr•n 

(Plus) 

0 

0 

0 

1 

1 

1 

2 

2 

~ 

l+ 

5. 

6 

7 

8 

9 

10 

11 

13 

l'+ 

15 

17 

19 

20 

22 

21+ 

26 

27 

29 

32 

A~• 

-

31 

32 

M 

35 

36 

~~ 

~ 

l+l 

1+2 

~ 45 

1+6 

1+7 

1+8 

49 

50 

51 

52 

~ 55 

56 

~~ 59 

60 

Corr•n 

(Plus) 

31+ 

36 

~ 43 

1+5 

1+8 

51 

~i 

59 62 

65 

68 

71 

71+ 

~f Bl+ 

88 

91 

§§ 

102 

106 

110 

114 

118 

122 

126 

~~· is the difference 1n 

latitude in minutes 

of the ends of the line. 

For Lambert or 

transverse Mercator 

Projection 

io,ooo 

20,000 

30,000 

I+o,ooo 

50,000 

60,000 

70,000 

80,000 

90,000 

100,000 

110,000 

120,000 

130,000 

140,000 

150,000 

160,000 

170,000 

180,000 

190,000 

200,000 

210,000 

220,000 

230,000 2l+o,ooo 

250,000 

260,000 

270,000 

280,000 

290,000 

300,000 

310,000 

320,000 

330,000 3G-O,ooo 

350,000 

Corr•n 

(Plus) 

0 

0 

1 

2 

2 

3 

l 

8 

10 

11 

11+ 

16 

19 

21 

21+-

27 

31 

34 

38 

1+2 

l+6 

50 

55 

59 

64 

69 

ra 

86 

91 

97 

103 

110 

116 

•scale ratio interpolated for mean latitude or mean x• of 

the ends of a line and corrected by the above table is a 

true mean value accurate to within one in the seventh 

decimal place. 

t'. ~. GA)\'LH!\\H !'T PHll'<Tt?\G OFI- Kt_ 1'972 0 - 464-17 1