4.6 Transformation Between Geographic and UTM Coordinates

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1 . ransfrmatin Btwn Ggraphic and UM rdinats.. nvrsin frm Ggraphic t UM rdinats Usd fr cnvrting and n an llipsid f knwn f and a, t UM crdinats. Ngativ valus ar usd fr wstrn lngituds. hs quatins ar accurat t abut a cntimtr at 7 f lngitud frm th cntral mridian Whr = 0 (latitud f th cntral mridian at th rigin f th x, y crdinats) M = ru distanc alng cntral mridian frm th quatr t acrss frm th pint M = 0 (M at ) = lngitud f cntral mridian (fr UM zn) k = (scal factr at th cntral mridian) 0

2 N. ransfrmatin Btwn Ggraphic and UM rdinats FOM EQUAION SHEE adius f urvatur in th plan f th mridian m a N sin adius f curvatur n th plan f th prim vrtical a sin m adius f curvatur at a givn azimuth mn ( ) sin N cs m N

3 . ransfrmatin Btwn Ggraphic and UM rdinats.. nvrsin frm Ggraphic t UM rdinats tan ' A cs ( ) cs whr and ar in radians M 5 5 sin a 5 5 sin 5 sin whr is in radians

4 Nrthing and Easting 70 8 ' 8 5 A A A k k 5 N 0 ' A A A k x N cs ' k k k x N 70 ' tan A A A M M k y UM Scal Factr Or in trms f Latitud and Lngitud. ransfrmatin Btwn Ggraphic and UM rdinats.. nvrsin frm Ggraphic t UM rdinats

5 . ransfrmatin Btwn Ggraphic and UM rdinats.. nvrsin frm UM t Ggraphic rdinats Usd fr cnvrting UM crdinats n an llipsid f knwn f and a, t and. Ngativ valus ar usd fr wstrn lngituds. hs quatins ar nt as accurat as th ggraphic t UM cnvrsin Whr = ftprint latitud which is th latitud at th cntral mridian which has th sam y crdinat f th pint = th rctifying latitud

6 . ransfrmatin Btwn Ggraphic and UM rdinats.. nvrsin frm UM t Ggraphic rdinats M M 7 sin 55 sin,, N, and ar,, N, and m calculatd at th ftprint latitud ( ) D M a sin 097 sin8 9 5 whr is in radians x y k N k 5

7 cs 5 0 D ' D D tan N 70 D D D ' ' ransfrmatin Btwn Ggraphic and UM rdinats.. nvrsin frm UM t Ggraphic rdinats

8 (h UM scal factr can als b apprximatd using th avrag radius f th arth.) x Apprximat UM Scal factr k k Grid Factr (scal factr) X (Elvatin factr) Elvatin factr (Scal factr t sa lvl). ransfrmatin Btwn Ggraphic and UM rdinats.. UM Map Scal Factr h lvatin factr can b apprximatd using th avrag radius f th arth (=,7,7m) and lvatin abv th gid rathr than th lvatin abv th llipsid. his is dn bcaus f th rlativly small valu f N in cmparisn t H, and bcaus th gid hight is usually usd fr lvatin. Elvatin factr h Apprx. Elvatin factr h grid scal factr fr UM maps can thn b cmputd using ithr th apprximat r tru valus E E H Grund Grid distanc Distanc Grid factr 7

9 . ransfrmatin Btwn Ggraphic and UM rdinats.. UM Map Scal Factr [viw] h H N ntral Mridian Grund surfac Man sa lvl Ellipsid surfac Prjctin surfac k =

10 . ransfrmatin Btwn Ggraphic and UM rdinats.. EXAMPLE A GIVEN: Pints n map frm gdtic bnch marks Map: NAD7, :50,000 NS map f 7H (Willw Bunch Lak) = 9 5 N = 0 0 W Apprx. Elvatin h = 0 ft = 70.m FIND: a) UM crdinats fr pint A, whr: a =,78,0. m /f = = 9 5 N = 9.5 = radians = 0 0 W = -0. = radians (UM zn ) = 05 W k = = 0 9

11 . ransfrmatin Btwn Ggraphic and UM rdinats.. EXAMPLE A Exampl 0

12 . ransfrmatin Btwn Ggraphic and UM rdinats.. EXAMPLE A f f M N N ' ,7,7. 8 m M a sin tan.8985 ' cs A cs a sin,90,0. 87 m N usd in this quatin is nt t b cnfusd with gidal hight. 5 5 sin M a sin 5 sin

13 . ransfrmatin Btwn Ggraphic and UM rdinats.. EXAMPLE A x k NA y k M M A A add a fals asting f 500,000m E = 58,58.5 m A N tan ' 5 0 A A ,58.59 m 00 0' 70 N = 5,55,.5 m O m Nrth Zn x Y UM rdinats Mridian East f ntrl Mridian ntral Mridian 500,000 m East Equatr Mridian Wst f ntrl Mridian

14 .7 Applicatin f UM rdinats GSUG - Gdtic Survy utin: UM and Ggraphic his prgram will cmput th cnvrsin btwn Ggraphic crdinats, latitud and lngitud and ransvrs Mrcatr Grid crdinats. h usr may chs this standard prjctin r may chs a dgr as dfind fr anada. h paramtrs f scal, cntral mridian, fals asting and fals nrthing may dfin any M prjctin and ar alrady dfind within th prgram fr tw standard prjctins, UM and dgr. Ggraphic t UM cmputatin utput Input Ggraphic rdinats LAIUDE: 9 dgrs 5 minuts 0 scnds NOH LONGIUDE: 0 dgrs 0 minuts 0 scnds WES ELLIPSOID: LAKE 8 ZONE WIDH: Dgr UM GSUG UM crdinats: UM Zn: Easting: mtrs EAS Nrthing: mtrs NOH Output- alculatd UM crdinats: UM Zn: Easting: mtrs EAS Nrthing: mtrs NOH

15 .7 Applicatin f UM rdinats.7. EXAMPLE B FIND: b) Latitud, lngitud and hight f pint A with rspct t NAD 8 llipsid a = 787m /f = GIVEN dx = m dy = 59m dz = 88m fr Saskatchwan Nt: dx = x

16 .7 Applicatin f UM rdinats a a' a m f f f 5 ' a N m sin M a m sin N xsincs ysinsin zcs a M h rad.0 0 dg. = 0.55 x sin y cs N hcs a h x cs cs y cs sin z sin a h = m ' 95' 0.55" rad ' 00' (.75") h ' h h 70.m ( 5.70m) =.75 N f = N = W = 7.95m EXAMPLE B sincs f a f N sin m - f M N - f sincs 5

17 .7 Applicatin f UM rdinats EXAMPLE B cn t. Natinal ransfrmatin: NAD7 - NAD8 (Nv), Nv mputatin utput Input rdinats LAIUDE: 9 dgrs 5 minuts scnds NOH LONGIUDE: 0 dgrs 0 minuts scnds WES ransfrmatin: NAD7 -> NAD8 NAD 8 Output data: LAIUDE: N Shift: 0.0 scnds Standard dviatin: 0.078m LONGIUDE: W Shift:.8797 scnds Standard Dviatin: 0.08m alculatd Output data: LAIUDE: N LONGIUDE: W

18 .7 Applicatin f UM rdinats.7. EXAMPLE B cn t. Wrks up t 50 N In Wstrn anada Natinal Gdtic Survy 7

19 .7 Applicatin f UM rdinats.7. EXAMPLE FIND: c) Latitud and lngitud f pint B with rspct t NAD 7 E = 50,000m N = 5,70,000m a =,78,0. /f = = 05 W k = M = 0 8

20 rad ' a M f f k y M M.7 Applicatin f UM rdinats.7. EXAMPLE rad sin sin 9 5 sin 55 sin sin sin a a N

21 .7 Applicatin f UM rdinats.7. EXAMPLE x D tan ' x N k cs X (fals asting) 0, rad D D 5 0 9' tan N rad ' 5.8 " N D D ' 70 D 5 8 8' cs rad D '." W 0

22 .7 Applicatin f UM rdinats.7. EXAMPLE GSUG - Gdtic Survy utin: UM and Ggraphic UM t Ggraphic cmputatin utput Input Ggraphic rdinats UM Zn: Nrthing: mtrs Easting: mtrs ELLIPSOID: LAKE 8 ZONE WIDH: Dgr UM Output ggraphic crdinats: LAIUDE: N LONGIUDE: W alculatd ggraphic crdinats: LAIUDE: N LONGIUDE: 0 0. W

23 .8 Map Azimuth and Scal Factrs f Lin A Givn: A-B has a calculatd grid Azimuth = = 9 5 N =0 0 W Find: ru Azimuth f lin frm A t B (scnds) Δα θ Δα θ sin m sc ( ) 88" sin 9.58 F sin m cs m sin " F ( ) F ntral Mridian m Grid Nrth Grid Nrth = 08 = 0 58 rrctd (Astrnmic) Azimuth A t B = = =05 W A = 9 5 N =0 0 W B

24 .8 Map Azimuth and Scal Factrs f Lin A MAP AZIMUH AND SALE FAOS OF LINE A - B rrctd (Astrnmic) Azimuth A t B = = Prcis scal Factr (S.F.) UM Scal factr A k k k A 5 8' Prcis Elvatin Factr (E.F.) M sin M N ( ) cs N Frm Gdtics anada GPS - H v. sftwar : H 70.m h 7.7m N -7.95m Prcis Elvatin Factr (E.F.) h ru grid factr = S.F. X E.F. A 8 A

25 .8 Map Azimuth and Scal Factrs f Lin A MAP AZIMUH AND SALE FAOS OF LINE A - B Apprximat scal factrs basd n a sphricalarth UM Scal Factr (S.F.) M p k 8,58.5 x 0.999,7,7 Elvatin Factr (E.F.),7, ,7,7 70 Apprximat grid factr ugh nvrsin d tan 5. tan 9 5' " 00'."

26 .8 Map Azimuth and Scal Factrs f Lin A MAP AZIMUH AND SALE FAOS OF LINE A - B Mr Prcis Sphrical nvrsin θ" " sin φ θ" 88" sin A φ B rrctd (Astrnmic) Azimuth A t B = =

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