THE SOLDNER PROJECTION OF THE WHOLE ELLIPSOID

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1 THE SOLDNER PROJECTION OF THE WHOLE ELLIPSOID Pawù Pêdzich Warsaw Univrsity of Tchnology, Institut of Photogrammtry and Cartography, Politchnii Sq. 1, -661 Warsaw, Abstract Th papr prsnts th rsults of rsarch on utilisation of lliptic intgrals in th prparation of th Soldnr projction. Th purpos of rsarch was to dvlop a mthod of construction of Soldnr projctions of th ntir llipsoid by applying lliptic functions. Th papr consists of two main parts; a thortical part which contains xtractions of formulas for coordinats and distortion, algorithms, and a practical part which contains: maps prsnting distortions. Th dvlopd mthod allows to calculat coordinats in th Soldnr projction of th ntir llipsoid with high prcision. Using this mthod w can also calculat th lngth of mridian and godtic lin on an llipsoid. 1. THE APPLICATION OF ELLIPTIC INTEGRALS IN DETERMINING COORDINATES IN SOLDNER PROJECTION Lt us suppos that th coordinats of th rfrnc llipsoid ar dfind in th systm of rcd godtic coordinats u,l as r x a cos u cos L, y a cos u sin L, z bsin u, whr tan u 1 tan B. Soldnr coordinats ç and î of th llipsoid ara ar dfind as î th lngth of th arc of th orthogonal godtic lin to th cntral mridian L=L that lins points P (u,l ) and P(u,L), whr P is its point on mridian L=L, ç th lngth of th arc of mridian L=L that lins points (u=,l=l ) and P (u=u,l=l ). Dtrmining Soldnr coordinats dfind in this way thrfor boils down to solving two problms: dtrmining th godtic lin s turning point P (u,l ) and dtrmining th rlvant lliptic intgrals. Th rctangular coordinats x,y in r ' x, y. Soldnr projction ar dfind dirctly as 1.1. Dtrmining turning point P(u,L ) of godtic lin Th proposd mthod for th dtrmination of Soldnr coordinats rquirs th dfinition of a godtic lin quation which is dtrmind by th turning point P (u,l ). Th godtic lin quation on th surfac of th flattnd sphroid in rcd godtic coordinats u,l dtrmind by th turning point P (u,l ) (Panasiu 1995) can b prsntd as L L 1 dv (1) 1 tan u cos v whr cos v cot u tanu. Equation (1) can b prsntd in th form of 1 L v 1 J 1 L () whr

2 1 r 1 r 1 tan u J r1 r r J, (3) 1 w r 1 J r 1 J r r rp w p rp, (4) 1 a b and also p and tan v w,. cos u b Th trm J r in (4) for r=1 rcivs th form of 1 cos u arctan wcos J u. (5) Dtrmining th rcd godtic latitud u of th turning point of th godtic lin which crosss th point P(u,L), can b don with th itrativ mthod according to formulas (-5). Th initial approximation of u has th sphric shap in th form of u ( ) arccot cos L L cot u (6) 1.. Th application of Jacoby s lliptic intgrals in dtrmining Soldnr coordinats Th sarch for Soldnr coordinats (,) boils down to solving (Panasiu 1995) th following intgrals u a tdt 1 cos (7) u a 1 cos t tdt cos (8) dt u whr dt cos u cos t 1 cos t cos t cos u (9) Having substitutd t s th intgral (7) rcivs th form of

3 a 1 sin s ds. (1) u Using th dfinition of th total lliptic intgral E (Byrd 1954) E s, sin t dt, (11) s 1 th intgral (1) can b prsntd as a E, E u,. (1) This is Lgndr s lliptical intgral of th scond ind in its normal form with paramtr. But w transform th intgral (8), aftr substituting (9) and ' 1 to th form of u cos t 1 ' sin t a 1 dt (13) sin u u sin t 1 sin u Lt us considr (Achizr 197) a nw variabl sin t cn w. (14) sin u Aftr bilatral diffrntiation (14) w obtain cos t sn wdn w dt, (15) sin u whr sn w sin am w,, (16) is an xprssion of Jacoby s lliptic sin of variabl w with paramtr, whras

4 cn w cos am w, (17) dnots Jacoby s lliptic cosin of variabl w with paramtr, and also am(w,) dnots th amplitud of variabl w with paramtr, and dn w 1 sn w, (18) is th dlta of th amplitud of variabl w with paramtr (Byrd 1954). Having insrtd (14) and (15) to (13) and having don simpl convrsions, w obtain w ' sin u a 1 1 ' sin u 1 sn w dn w. (19) 1 ' sin u Th formula (19) can b also prsntd as ' sin u dn w, a 1 1 ' sin u 1 ' u sin. () Th right sid of quation () can b prsntd as lliptic intgrals a 1 1 ' sin u E am w,,, (1) whr sin u 'sin u, sin u 1 ' sin u cn w,. () In (1) E[am(w,),] stands for Lgndr s lliptic intgral of th scond ind in its normal form with paramtr. 1.. Dtrmining flat rctangular coordinats x,y in Soldnr projction basd on llipsoidal coordinats B,L In this tas, point Ps(,L ) is givn and it dfins th origin x=, y= of coordinats in Soldnr projction and any point P(B,L). W nd to dtrmin th coordinats of point P (x=ç, y=î) which is an imag of point P(B,L) in Soldnr projction. Th tas can b solvd in th following stags: Dtrmining th rcd latitud on th llipsoid s surfac. Itrativ dtrmination of th turning point P (u,l ) of th godtic lin that crosss point P(u,L) from th formulas (6). Calculation of coordinats of point P (x=ç, y=î) in Soldnr projction according to formulas (11), (1), (). Function E in formulas (11), (1), () can b dtrmind with Fourir sris (Byrd 1954) in th form of

5 1 m E, t m, (3) m m whr t 1 t sin cos 1 t 4 3 sin cos 3 sin 8 m 1 1 m1 t m t m1 sin cos m m. Projction distortions in Soldnr projction of th whol llipsoid using Jacoby s lliptic functions Th dtrmination of projction distortions in Soldnr projction rquirs th counting up th drivativs d d d d,,,. Ths drivativs ar composit functions. d.1. Dtrmining drivativ d Drivativ, has th form of d d. (4) Th drivativ d prsnt on th right sid of th quation (4) has th form of d 1 cos u, (5) whil drivativ cos u 1. (6) cos B Th drivativ prsnt in (4) must b dtrmind numrically.

6 d.. Dtrmining drivativ, W prform bilatral diffrntiation of quation (1) in rlation to variabl B. Th drivativ on th lft sid (1) acquirs th form of d a 1 1 ' sin u a 1 a 1 1 ' sin u ' sin u 1 ' sin u. (7) Th drivativ on th right sid (1) has th following form de d am de d am d de d. (8) d am d am d d Th drivativ de according to (Byrd 1954) is xprssd by th dpndnc d am de dam 1 sin am( w, ), (9) whil drivativ d am dam( w) dnw. (3) Th drivativ is obtaind as a rsult of diffrntiating th first of th formulas () cos u sin u sinu cos u sin u dcn dcn d d, (31) whr dcn snwdnw, (3)

7 dcn snwdnw ' w Ew snwc, (33) d ' cnw c, (34) dnw ' 1, (35) w F am w,,. (36) d W dtrmin th drivativ according to th scond of th formulas () as d 'cos u 3 ' sin u sin u 1 ' sin u 1 ' sin u 1 ' sin u. (37) de d am Drivativs and d d (Byrd 1954) rciv th form of amw,, Eam( w, ), Famw, de, (38) d dam dnw Ew ' w snwc. (39) d ' d Th drivativ is dtrmind dpnding on (7) aftr taing into considration (8-39). d.3. Dtrmining drivativ d Th drivativ can b prsntd in th form of d d, (4) whr drivativ d is givn by formula (5), and drivativ is dtrmind according to (9) as

8 cos u cos u cos u. (41) cos u 1 cos u Th drivativ is dtrmind numrically. d.4. Dtrmining drivativ Aftr bilatral diffrntiation (1) th drivativ on th lft sid rcivs th form of d a 1 1 ' sin u a 1 a 1 1 ' sin u ' sin u 1 ' sin u, (4) and th drivativ on th right sid (1) th form of de d am de d am d de d (43) d am d am d d W obtain th drivativ in (43) as a rsult of diffrntiating th first of th formulas () as sin u cos u sin u dcn dcn d d (44) d Th othr drivativs prsnt in (43) ar dtrmind according to (9), (3), (339). Th sought drivativ is dtrmind from (4)..5. Dtrmining th valus of projction distortions in Soldnr projction of th whol llipsoid. With ths dpndncis dcd, rspctiv algorithms wr constructd togthr with softwar abl to dtrminat coordinats x,y in Soldnr projction, as wll as projction distortions. Figur 1 prsnts a map graticul and projction distortions in Soldnr projction of th whol llipsoid. Th figur on th lft hand sid prsnts th lin of idntical maximum lngth distortions, whil th figur on th right sid prsnts th llips of distortions.

9 Fig. 1. Map graticul in Soldnr projction of th whol llipsoid and th distribution of projction distortions. Soldnr projction is a quidistant projction, orthogonal to th imag of th axial mridian. Th axial mridian is projctd as a lin of zro distortions. Local distortions grow with th distanc from th axial mridian s imag. Th lins of constant distortions of aras, angls, and lngths in this projction ar clos to straight lins paralll with th imag of th axial mridian. Thr ar som intrsting pculiaritis in this projction, which rsult from th shap of godtic lin on llipsoid. Thy ar not shown in this papr. Th proposd mthods utilising lliptic intgrals and Jacoby s lliptic functions to dtrmin in a rlativly asy way and with high prcision th rctangular coordinats and projction distortions in Soldnr projction. Rfrncs: J.Panasiu, J.Balcrza, U. Porowsa: "Wybran zagadninia z podstaw torii ozorowañ artograficznych", Oficyna Wydawnicza PW, Warszawa P. Byrd, M. Fridman: "Handboo of lliptic intgrals for nginrs and physicists," Springr-Vrlag, Brlin N. Achizr: "Elminty tiorii lipticzsich funcji", Izd. Naua, Moswa 197.

Pawl Pdzich was born at Wgrow, Poland on August, 1971. H graatd Warsaw Univrsity of Tchnology, Faculty of Godsy and Cartography, spcialization cartography, in 1997.

10 Pawl Pdzich was born at Wgrow, Poland on August, H graatd Warsaw Univrsity of Tchnology, Faculty of Godsy and Cartography, spcialization cartography, in In this yar h startd doctoral studis at Warsaw Univrsity of Tchnology. In 1998 h did acadmic practics at Naval Acadmy in Gdynia. In h obtaind his doctor's dgr. Th thsis was on map projctions. In 3 h startd to wor, as a lcturr, at Institut of Photogrammtry and Cartography, Warsaw Univrsity of Tchnology. In th sam yar h also startd to wor as a lcturr at Institut of Civil Enginring and Godsy, Military Univrsity of Tchnology in Warsaw. H lcturs mathmatical cartography at Warsaw Univrsity of Tchnology, cartography, topography, gographic information systms at Military Univrsity of Tchnology. H is an author of svral publications on map projctions, prsntations on national confrncs. H ladd svral rsarch projcts.

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