NONZONAL EXPRESSIONS OF GAUSS- KRÜGER PROJECTION IN POLAR REGIONS

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1 NONZONAL EXPRESSIONS OF GAUSS- KRÜGER PROJECTION IN POLAR REGIONS Zhongmi Li, Shaofng Bian*, Qiang Liu, Houpu Li, Chng Chn, Yanfng Hu Dpt. of Navigation, Naval Univrsity of Enginring, Wuhan, China - lizhongmi16@16.com Commission Ⅳ, WG Ⅳ/ KEY WORDS: Gauss-krügr projction, transvrs Mrcator projction, polar survying, rfrnc fram, conformal colatitud, computr algbra systm ABSTRACT: With conformal colatitud introducd, basd on th mathmatical rlationship btwn xponntial and logarithmic functions by complx numbrs, strict quation of complx conformal colatitud is drivd, and thn thortically strict nonzonal xprssions of Gauss projction in polar rgions ar carrid out. By mans of th computr algbra systm, corrctnss of ths xprssions is vrifid, and sktchs of Gauss-krügr projction without bandwidth rstriction in polar rgions ar chartd. In th Arctic or Antarctic rgion, graticul of nonzonal Gauss projction complis with popl s rading habit and rflcts ral ground-objct distribution. Achivmnts in this papr could prfct mathmatical basis of Gauss projction and provid rfrnc fram for polar survying and photogrammtry. 1. INTRODUCTION Polar rgions hav incrasingly bn th intrnational focus in rcnt dcads. It is of grat significanc for polar navigation and scintific invstigation to slct th suitabl projction mthod. As on common conformal projction, th transvrs Mrcator (TM) or Gauss-Krügr projction is frquntly usd for charting topographic map (.g., Lauf 1983; Snydr 1987; Yang 000). Sris xpansions of mridian lngth in Krügr (191) is th basis of th most common way for calculation of Gauss coordinats. In th last cntury, scholars hav carrid on xtnsiv rsarchs about th projction. L (1976) and Dozir (1980) carrid out formula of UTM coordinats by mans of lliptic functions. Basd on complx numbrs, Bowring (1990) gav on improvd solution for TM projction. With rspct to Laplac-Bltrami and Korn-Lichtnstin quations, conformal coordinats of typ UTM or Gauss- Krügr wr carrid out dirctly in Grafarnd (006). Additionally, Brmjo (009) drivd simpl and highly accurat formulas of TM coordinats from longitud and isomtric latitud, and compard truncation rrors in diffrnt ordrs by using th program Mapl and Matlab. Karny (011) xtndd Krügr s sris to 8th ordr, constructd high-prcision tst st basd on L (1976) and discussd proprtis of th xact mapping far from th cntral mridian. Obviously, rsarchs on TM or Gauss-Krügr projction hav alrady obtaind brilliant achivmnt. In th dvlopmnt history of Gauss-krügr projction thoris, formula abov hav diffrnt faturs, for xampl, ral powr sris xpansions of longitud diffrnc l ar oftn limitd in a narrow strip (.g. l [ 3.5, ] in UTM projction; l [ 3, + 3 ] or l [ 1.5, ] in Gauss- Krügr projction). Exprssions by complx numbrs, liminating zoning rstrictions, ar difficult to b usd in polar rgions with th singularity of isomtric latitud. Karny (011) improvd L s formula, and providd an accuracy of 9 nm ovr th ntir llipsoid, bur not gav formula that can ntirly xprss th Arctic or Antarctic rgion. In attmpts on th nonzonal formula of Gauss projction in polar rgions, Bian (014) usd a nar-sphrical assumption to driv complx colatitud, which would hav an influnc on th strictnss of his formula. Givn ths, in ordr to prfct th mathmatical foundations of Gauss-krügr projction spcializd in polar rgions, by introducing th rlationship btwn conformal colatitud and isomtric latitud, an improvmnt masur will * Corrsponding author doi: /isprsannals-iii

2 b shown in this papr.. EXPRESSIONS OF GAUSS COORDINATES IN NONPOLAR REGION According to Bian (01), basd on mridian arc lngth xpansion about conformal latitud, non-itrativ xprssions of Gauss projction ar writtn as in this form whr l indicats godtic longitud, q mans isomtric latitud and is a function of godtic latitud B ( ) = arctanh ( sin ) arctanh( sin ) q B B B 1+ sin B 1 sin B (4) = ln 1 sin B 1 + sin B ( w) φ = arcsin tanh z = x + iy = a( α0φ + αsin φ + α4sin 4 φ+ α6 sin 6φ+ α8 sin8φ+ α10 sin10 φ ) (1) whr a indicats smi-major axis of th arth llipsoid. Cofficints α0, α α10 xpandd to 10 ar carrid out by computr algbra systm Mathmatica. With its strong powr in symbolic opration, cofficints xpandd to 0 or vn 40 can b asily gottn in a similar way. And finally all th cofficints could b simplifid as sris summations of arth llipsoid ccntricity. As our targt in this papr is to improv original formula of Gauss projction for polar using, w plac grat importanc on th improvmnt masurs, and do not discuss th xpansion cofficints whthr thy ar xpandd nough to a high prcision or not. Hr w tak th first ccntricity for xampl, cofficints α 0 α 10 xpandd to 10 ar listd in Eq. () α 0= α= α4= α6= α8= α10 = () Figur 1. Sktch of isomtric latitud q with godtic latitud B [0, 90 ) Taking arctanh ( sin ( B) ) arctanh ( sin B) ( ( B) ) = arctanh ( sin B) into account, q( B) = q( B) = and arctanh sin is gottn from Eq. (4), maning that isomtric latitud q is an odd function of godtic latitud B. Trnd of isomtric latitud q with godtic latitud B ranging from 0 to 90 is chartd in Figur 1. As shown in Fig. 1, isomtric latitud q incrass with godtic latitud B ranging from 0 to 90. In considration of isomtric latitud q s odvity, q bcoms an infinitly larg quantity as godtic latitud B approachs to ± 90, which brings about singularity in xprssions of complx isomtric latitud w in Eq. (3) as wll as complx conformal latitud φ in Eq. (1), and thn maks xprssions of Gauss coordinats in Eq. (1) difficult to b usd in polar zons. Additionally, in Eq. (1) φ and w indicats complx conformal latitud and complx isomtric latitud rspctivly, and w = q+ il (3) Figur. Sktch of Gauss projction in nonpolar rgion Morovr, as tanh = tanh ( q+ i ( ) = tan w in Eq. (1) contains tanh il i l, not suit for th situation whr godtic doi: /isprsannals-iii

3 longitud l approachs to 90, xprssions of Gauss coordinats in nonpolar rgion Eq. (1) only can b usd in th zon ( ) D ={ B, l : l < 90, B < 90 }. By mans of computr algbra systm Mathmatica, sktch of Gauss projction in nonpolar rgion is drawn in Figur. 3. NONSINGULAR EXPRESSIONS OF GAUSS COORDINATES IN POLAR REGIONS In ordr to carry out th xprssions of Gauss projction that can b usd in polar rgions, Eq. (1) must b transformd quivalntly. As Eq. (1) is drivd from mridian arc lngth xpansion X = a( α ϕ+ α sin ϕ+ α sin 4ϕ+ 0 4 α sin 6ϕ+ α sin8ϕ+ α sin10 ϕ+ ) whr X indicats mridian arc lngth, ϕ indicats conformal latitud, and cofficints α0 α10 ar th sam as Eq. (1). According to Xiong (1988), conformal latitud ϕ is a function of godtic latitud B π B 1 π ϕ = arctan tan + (6) sin B Whn th godtic latitud B is on th northrn hmisphr, th conformal latitud ϕ is a positiv valu. Othrwis, it is a ngativ valu. (5) Obviously, singularity of Eq. (8) dpnds on th singularity of th uniqu variabl θ. To judg th singularity of θ, insrting Eq. (6) into Eq. (7), quation of conformal colatitud θ is gottn. π B 1 θ = π arctan tan sin B = π arctan xp( q) = arccot xp ( q) = arctan xp ( q) Taking Eq. (4) into account, basd on th rlationship btwn xponntial and logarithmic functions xp( ln x) x, Eq. (10) is gottn. 1+ sin B 1 sin B xp( q) = xp ln 1 sin B 1 + sin B 1 sinb 1+ = 1+ sinb 1 sinb (9) (10) Insrting Eq. (10) into Eq. (9), w can find whn B = 90, θ = 0. Nithr θ nor Eq. (8) is singular in north pol. As nonpolar solution of Gauss projction is obtaind by dvloping th rlationship btwn mridian lngth X and isomtric latitud q from ral to complx numbr fild, xprssions usd in polar rgions can b achivd similarly. Firstly, basd on th dfinition of complx function, w = q+ il rplacs q in Eq. (9) to raliz th xtnsion of conformal colatitud θ, and thn th complx conformal colatitud θ is drivd. 3.1 Nonsingular Exprssions of Gauss Projction in Complx Form θ = arctan xp( ( q+ i ) (11) To liminat th singularity of conformal latitud ϕ whn godtic latitud B approachs to 90 in Eqs. (5) (6), conformal colatitud θ is introducd, and valus θ = π ϕ (7) Aftrwards, insrting Eq. (7) into Eq. (5), quivalnt xprssion of mridian lngth X can b writtn with conformal colatitud θ. X = a( α0θ + αsin θ α4sin 4θ + α6sin 6θ α sin8θ + α sin10 θ ) + aα π (8) Scondly, rplacing θ in Eq. (8) by θ, and turning mridian lngth X in Eq. (8) into complx coordinats z = x + iy, whr th ral part x indicats Gauss ordinat and th imaginary part y indicats abscissa, yilds to xprssions of Gauss projction. For convnint polar charting, moving zro point of th xprssions from th quator to th north pol, ordinat is rducd by 14 mridian arc lngth ( aα 0π ) and abscissa rmains unchangd. For clar prsntation, abscissa and ordinat aftr translation ar still xprssd with y and x. Omitting th drivation, nonsingular xprssion of Gauss coordinats by complx numbrs in Arctic rgion is carrid out. doi: /isprsannals-iii

4 ( z = x+ iy = a α θ+ α sin θ α sin 4 θ+ 0 4 α sin 6θ α sin8θ+ α sin10θ ) (1) Whn longitud l =0, abscissa y = 0, ordinat x quals to mridian arc lngth intgrating from th pol. Equality of th cntral mridian arc is guarantd. Bsids, transformations abov ar all lmntary oprations btwn complx functions, which ar monodrom and analytic functions in th principl valu, kp conformal in th whol transformation procsss. It is vrifid that Eq. (1) satisfy Cauchy Rimann quations, so conformality of Gauss projction is guarantd. By now, nonzonal solution of Gauss projction that can b usd in Arctic rgion hav bn finishd. 3. Nonsingular Exprssions of Gauss Projction in Ral Form As complx conformal colatitud θ is a complx variabl, to sparat θ into ral and imaginary parts θ = u+ iv, quations xp ( q+ i = xp( q)( cosl+ isin and q =arctanh( sinϕ ) ar introducd. Basd on th rlationship btwn complx function and arctangnt function, Eq. (1) is transformd quivalntly. ( ( q i ) ( )( ) ( )( ) θ = arctan xp + 1+ ixp q cosl isinl = iln 1 i xp q cos l i sin l sinh q cosl = iln + i cosh q sin l cosh q sin l = arctan csch cos arctanh sch sin = arctan tan cos arctanh sin sin ( q i ( q ( θ i ( θ (13) Taking rlationships ( ) sin u+ iv = sinucosiv+ cosusiniv, sin iv = isinh v and cos iv = cosh v into account, Eq. (11) is sparatd into ral and imaginary parts, and Eq. (15) is gottn. x = aα0u + a nu nv n= 1 y = aα0v + a nu nv n= 1 n 1 ( 1) α n sin( ) cosh( ) n 1 ( 1) α n cos( ) sinh( ) (15) By now, nonzonal xprssions of Gauss projction in ral form suit for th Arctic rgion hav bn carrid out. Actually, taking full advantag of arth s symmtry, it is no nd to driv xtra xprssions of Gauss coordinats for th Antarctic rgion. As th southrn hmisphr is symmtrical to th northrn hmisphr, viwing th south pol as nw north pol, rplacing P( B, l ) on southrn hmisphr by P( B, and insrting it into xprssions usd in Arctic rgion, sktch of th Antarctic rgion in th sam prspctiv as th Arctic rgion is gottn. Sktchs of Gauss projction in polar rgions ar drawn in Figurs 3 and 4. As shown in Figurs 3 and 4, th Arctic and Antarctic rgions ar compltly displayd basd on nonzonal xprssions abov. Through translating th origin of Gauss coordinats from th quator to th pol, Gauss ordinats ar ngativ whn longitud diffrnc l < 90, whil ordinats ar positiv whn longitud diffrnc l > 90. In polar circls, th mridians l =0, ± 90, ± 180 ar shown as straight lins aftr projctd, and ar th symmtry axs of Gauss coordinats. Aftr transformd, complx conformal colatitud θ is dvidd into ral and imaginary part, thy quals ( θ cos ( θ sin u = arctan tan v = arctanh sin (14) Obviously, whn P( B, approachs to th north pol, θ 0, rang of l rachs to [ 180,180 ], and θ has a spcific valu and not singular at crtain point P( B, l ) on th northrn hmisphr. Figur 3. Sktch of Gauss projction in Arctic rgion basd on nonzonal xprssions doi: /isprsannals-iii

5 Program (No. 01CB71990) and National Natural Scinc Foundation of China (No , ). REFERENCES Figur 4. Sktch of Gauss projction in Antarctic rgion basd on nonzonal xprssions 4. CONCLUSIONS With quations of conformal colatitud and isomtric latitud introducd, basd on rlationship btwn complx xponntial and logarithmic functions, nonzonal xprssions of Gauss projction in polar rgions ar carrid out. Conclusions ar drawn as blow. (1) With isomtric latitud singular in th pol, traditional xprssions of Gauss projction can not b usd in polar rgions. Through translating th origin of traditional Gauss projction by 14mridian lngth from th quator to th pol, thortically strict xprssions that can b usd in polar rgions ar carrid out. Ths ar of grat significanc for prfcting mathmatical systm of Gauss projction. () Compard with traditional Gauss projction, nonzonal formula drivd in this papr ar fit for th whol polar rgions without bandwidth rstriction, and could provid rfrnc fram for polar survying and photogrammtry. (3) Though paralll circls and th othr mridians ar not projctd to straight lins lik Mrcator projction, graticul of Gauss projction in th Arctic or Antarctic rgion could still comply with our rading habit, vn rflct ral ground-objct distribution bttr and mak th polar rgions absolutly clar at a glanc. Brmjo, M., Otro, J., 009. Simpl and highly accurat formulas for th computation of Transvrs Mrcator coordinats from longitud and isomtric latitud. Journal of Godsy, 83, pp Bian, S. F., Li, H. P., 01. Mathmatical analysis in cartography by mans of computr algbra systm. In: Cartography-A Tool for Spatial Analysis, InTch, Croatia, Chaptr 1, pp Bian, S. F., Li, Z. M., Li, H. P., 014. Th non-singular formula of Gauss Projction in polar rgions by complx numbrs. Acta Godatica t Cartographica Sinica, 43(4), pp Bowring, B. R., Th transvrs Mrcator projction a solution by complx numbrs. Survy Rviw, 30(37), pp Dozir, J., Improvd algorithm for calculation of UTM coordinats and godtic coordinats. NOAA Tchnical Rport NESS 81, National Ocanic and Atmosphric Administration, Washington. Grafarnd, E., Krumm, F., 006. Map Projctions. Springr, Brlin. Karny, C. F. F., 011. Transvrs Mrcator with an accuracy of a fw nanomtrs. Journal of Godsy, 85(8), pp Krügr, L., 191. Konform Abbildung ds Erdllipsoids in dr Ebn. Drucj und Vrlag von B.G. Tubnr, Lipzig. L, L. P., Conformal Projctions Basd on Elliptic Functions (No.16). BV Gutsll. Lauf, G. B., Godsy and Map Projctions. Taf Publications, Collingwood. Snydr, J. P., Map Projctions A Working Manual. US Gol Surv Prof Papr 1395, Washington D C. Yang, Q. H., Snydr, J. P., Toblr, W. R., 000. Map Projction Transformation: Principls and Applications. Taylor & Francis, London. ACKNOWLEDGES Rvisd April 016 This work is financially supportd by National 973 Plan doi: /isprsannals-iii