APPLICATION OF THE CHEBYSHEV POLYNOMIALS TO APPROXIMATION AND CONSTRUCTION OF MAP PROJECTIONS

Transcription

1 APPLICATION OF THE CHEBYSHEV POLYNOMIALS TO APPROXIMATION AND CONSTRUCTION OF MAP PROJECTIONS Pweł Pędzch Jerzy Blcerz Wrsw Uversty of Techology Fculty of Geodesy d Crtogrphy Astrct Usully to pproto of p proecto the lest squre ethod s used. Deterto of polyols coeffcets requres soluto of coplcted syste of equtos. It s possle to vod such prole usg orthogol Cheyshev polyols. It s copletely dfferet ethod of pproto, where the u dfferece etwee vlue of fucto d vlue clculted fro polyol s zed. I the pper there re preseted propertes of Cheyshev polyols, ther pplcto to p proecto pproto d coprso wth other ethods of p proecto pprotos. Moreover Cheyshev polyols y e used s ethod of zto of p proecto dstorto. The eple of such proecto s show the pper.. Ufor pproto Approto perfored wth the use of Cheyshev polyols s clled the ufor pproto. It cossts of pproto of the fucto f() y the polyol T () the tervl,, such wy, tht the devto of the hghest solute vlue ( ) T ( ) E f (), would hve the possly sllest vlue for the pproprte selecto of coeffcets of the T () polyol.. Cheyshev polyols Cheyshev polyols of the frst type d the order y e preseted the followg for (Pszows 975) T T T T T T () Cheyshev polyols of the secod type hve the followg for U U, U 4, U U U (). Propertes of Cheyshev polyols The T () polyols hs zeros the tervl, + π cos,,...- (4) d + etrees ths tervl

2 π ' cos,,,... (5) Polyols re orthogol the cotuous cse, wth the sclr product ( T, T) T ( ) T ( ) d (6) where dl π ( T, T) dl (7) π dl Ad, the dscrete cse, wth the sclr product + π ( T, T) T ( ) T ( ), cos + (8) where re zeros of the T + () polyol, d dl ( T, T) ( + ) dl (9) + dl If the t vrle s troduced, whch flls the tervl,, the the followg susttuto s ppled t.5( + ) +. 5( ) () The dervtve of Cheyshev polyol T s clculted sg o the relto T ' U () Accordg to the, so-clled, rule, the polyol T hs the sllest u or the tervl, out of ll polyols of the order wth the ledg coeffcet.. Cheyshev seres of oe d two vrles The fucto f() y e pproted y Cheyshev seres the for (Borc, Dhlqust 987) f ( ) c + ct () I the dscrete cse coeffcet of the seres re clculted fro the forule: c f ( ) T ( + ) () The F(,y) fucto y e pproted y es of Cheyshev seres the for (Leg 997) (, ) ( ) ( F y c T T y Coeffcet of the seres (4) re clculted usg the forul ε ( + ) π ( l + ) π c (, l ) cos cos ( )( ) F y + + l ( ) + ( + ) ) (4) (5) where ε4 for d, ε for d or d d ε for d.

3 Forule ()-(5) y e used the cse of uevely dstruted ucto pots. I prctce t ofte hppes tht wth the tervl uder cosderto, pots re evely dstruted d tht clculted vlues of the fucto est those pots. The, the pproch preseted y Leg (997) y e ppled. I the cse of oe vrle, f the ode s locted etwee pots,, for whch the fucto vlues f, f were clculted, the the terpolted vlue y e clculted usg the forul f ( ) wf + ( w) f (6) where w.5 / [( )(. 5( ))] (7) Ad, the cse of two vrles, the vlue of the fucto the ucto pot s clculted the followg wy: f the ucto pot (,y ) flls wth the rectgle, deleted y pots, d y c,y d, for whch vlues f, f, f c, f d were esured, the the followg forul y e used for clculto the vlue of the fucto the ucto pot f (, y ). 5( f + f + f c + f d )+ +.5( f d f c + f f )/( )(. 5( + ))+ (8) +.5 f f + f f / y y. 5 y + y ( ) ( )( ( )) d c 4. Utlsto of Cheyshev polyols for pproto of crtogrphc proectos Utlsto of Cheyshev polyols for pproto of crtogrphc proectos wll e preseted usg the sple eple of the zuthl, coforl proecto of the for π ϕ R cos + cosλ (9) 4 π ϕ y R cos + s λ 4 where the preters ϕ,λ defe the geogrphc co-ordtes o sphere. The scle of legths for ths proecto s preseted y the forul π ϕ cos + 4 (). cosϕ The fucto () ws pproted y the seres of the for p c + c T () the tervl ϕ 48, 54 whch covers the re of Pold. Coeffcets of the seres () were detered such wy tht the solute, u vlue of the error, sller th * -9 could e oted. Ths codto s et for the seres of the 6 order, of the coeffcets c c c c c c c 6.87 The scle vlues, clculted for pots, locted wth equl dstces, y es of the forul () d the seres () re lsted Tle Tle

4 ϕ p - p 48 o e- 49 o E- 5 o ,E- 5 o ,8E- 5 o ,4E- 5 o ,6E- 54 o ,9E- The proecto (9) ws pproted y es of the seres y c c' T ( ϕ) T ( λ) T ( ϕ) T ( λ) Assug the ccurcy of the order of. the followg coeffcets c of the 5 th order polyol were oted for the co-ordte Tle presets the co-ordte vlues, clculted usg the forul () d the seres () d the dffereces etwee those vlues Tle ϕ[ o ] λ[ o ] [] p [] - p [] I the cse of the y fucto pproto, the followg coeffcets c were oted Tle presets the co-ordte vlues, clculted usg the forul () d the seres () d dffereces etwee those vlues () ()

5 Tle ϕ[ o ] λ[ o ] y[] yp[] y-y p [] I the cse of pproto of the crtogrphc proecto usg Cheyshev seres (), () dstortos wll e detered y es of clculto of prtl dervtves y y ϕ λ λ c' c c' U T T ( ϕ) T ( λ) ( ϕ) U ( ) λ ( ϕ) U ( λ) ϕ c U ( ϕ) T ( ) λ Dervtves (4) re the ss for clculto of scles of dstortos of dstces, res d gles. 5. Coprso of results oted usg the ufor d e squre pproto The oted results of the ufor pproto were copred wth the results of e squre pproto. Orthogol polyols were ppled for the e squre pproto. The fucto f() ws pproted y the seres where ( ) ( ) f y p ( ), ( ) p (6) s the epoetl, orthogol polyol. Such polyols y e detered usg the recurrece relto (Rlsto 97) p + () (- + ) p () - p - (), (,,...), (7) where p (), p - (), α + p ( ) p ( ) (4) (5) (8)

6 β p ( ) p ( ) (9) () ω /d () ω ( ) fp ( ) ( ) d p p () () The ethod descred ove ws used for pproto of the scle of legth p The est pproto ws used for the 6 th order of the polyol. The coeffcets of the followg vlues were detered E E E E E E E- 6 Tle 4 presets the scle vlues clculted usg the forul (), p pproted usg the polyol () d dffereces detered for selected pots locted wth equl dstces. Tle 4 ϕ M p - p 48 o E- 49 o E- 5 o E-9 5 o E-9 5 o E- 5 o E-9 54 o E-9 Coprg the results lsted Tle 4 d Tle t y e see, tht pproto usg Cheyshev seres llows for otg etter results. For the se order of the polyol, the hgher ccurcy y e oted. The proecto (8) ws pproted y seres the for p ( ϕ) () ϕ λ (4)

7 y ' ϕ λ (5) Coeffcets of the seres (4) d (5) were detered y es of the lest squre ethod. For the seres of the 5 th order, specfed y the forul (4) the followg coeffcets were oted Tle 5 presets the vlues of the p co-ordte, clculted y es of the seres (4) d errors detered t selected pots, locted wth equl dstces. Tle 5 ϕ[ o ] λ[ o ] [] p [] - p [] For the seres (5) of the 5 th order the followg coeffcets were oted Tle 6 presets the vlues of co-ordtes, clculted usg the forul (9) d the seres (5), s well s dffereces etwee those vlues. Tle 6 ϕ[ o ] λ[ o ] y[] yp[] y-y p []

8 Coprg the results lsted Tles 5,6 d Tles,4 t y e see tht pproto usg Cheyshev seres llows for otg the hgher ccurcy. For the se order of the polyol the hgher ccurcy y e oted. 6. Utlsto of Cheyshev polyols for zg dstortos of crtogrphc proectos Bsg o wors perfored y Gdows (Gdows 969), eple of costructo of crtogrphc proecto of ellpsod to ple, for whch Cheyshev polyols wll e used for zg proecto dstortos, wll e preseted ths secto. Tht proecto wll e copred to the proecto detered sg o the ethod ppled y Tssot. The co-ordte syste s costructed o the ellpsod the followg wy. A rtrry pot O s ssued o the ellpsod d geodetc les re costructed ech drecto. The, orthogol trectores of those les re costructed, whch re the Guss geodetc crcles. Thus, the se-geodetc grd s oted o the ellpsod. The t s ssued tht les vcost re geodetc les, whch cross the pot O, provded, tht v s such gle, whch s creted y the gve geodetc le wth the erd L L O of the geodetc co-ordtes (B,L). Geodetc crcles re red s ucost. If u es the legth of the rc of gve geodetc les, the the I squre for of the ellpsod surfce y e preseted the for ds du G( u, v) dv. (6) The G s preseted the for K 4 G u u K u u +..., (7) 6 where K es the Guss curvture of the gve surfce. The we ssg () v K u. (8) Now the geerlsed zuthl proecto should e cosdered the for r( u) cosv (9) y r( u) s v where u, γ d v,π. Coforl proectos do ot est the clss of proectos (9). However, such proectos y e detered out of those proectos, whch wll hve the possly lowest dstorto of res. Assug r ( u) the re defort y e preseted the for rru K P + u u + (4) G 6 K u +... After equtg the coeffcet of the seres (4) to zero, the followg vlues re oted K,,. The coeffcet of u s equl to d t cot e equted to 4 zero, sce 4 s the costt uer, d (v). π Assug 4, where 5 π dv oe ots u (4)

9 ( )... p u + (4) The ove preseted ethod ws ppled the Tssot theory. Now, order to ze the dstorto (4) Cheyshev polyols wll e ppled. K Assug A the dstorto (4) y e preseted the followg for K u + u + u +... (4) A A A A Ltg cosdertos to four ters of the seres (4), the ssue of zto of tht epresso s the ssue of deterto of such polyol of the rd order of the coeffcet t the hghest power equl to, whch hs the sllest devto fro the tervl u,γ. Followg the rule, those codtos re et y Cheyshev polyol. For the tervl, t hs the for Ad T, (44) 4, 4 4. (45) I order to detere Cheyshev polyol of the thrd order for the tervl u, γ the vrle should e odfed usg the relto u γ (46) γ The, Cheyshev polyol of the thrd order, for the tervl u, γ s oted the for 9 T ) u γu + γ u γ (47) 6 where ) γ T (48) u, γ After equtg coeffcets (47) d (4) d solvg the syste of equtos the followg reltos re oted ( ) γ ( )... γ + (49) 6 K ( ) γ Thus, the proecto, for whch dstorto of res s pprotely equl p ( ) Tˆ (5) ws detered, wth the u devto fro zero, equl γ ( ). (5)

10 The u dstorto (5) s pprotely tes sller th the dstorto (4). Cocludg rers The ethod of the ufor pproto, descred ths pper, s the ltertve for the cooly used ethod of the e squre pproto. Usg seres sed o orthogol Cheyshev polyols the ecessty to solve lrge systes of equtos y e voded, wht llows to vod y uercl proles. As t y e see o the preseted eple, the descred ethod gves etter results wth respect to the ccurcy, th typcl ethods ppled the cse of the e squre pproto. The sple costructo of lgorths used for clcultos, s the ddtol dvtge, whch covces to the selecto of ths ethod. The rule llows for utlsto of Cheyshev polyols for costructo of crtogrphc proectos, d llows for zg the proecto dstortos. Refereces Borc A., Dhlqust G., Metody uerycze, Nuercl ethods, Pństwowe Wydwctwo Nuowe, Wrszw 987 Gdows B., Mlzc zesztłceń w odwzorowch powerzch, Mzg dstortos proectos of surfces, Prce Nuowe Geodez Nr 6, Wydwctw poltech wrszwse, Wrszw 969 Leg G., Copresso of rcrft erodyc dtse usg ultvrle Cheyshev polyols, Advces Egeerg Softwre 8 (997) -4 Pszows S., Zstosow uerycze weloów szeregów Czeyszew, Nuercl utlsto of Cheyshev polyols d seres, Pństwowe Wydwctwo Nuowe, Wrszw 975 Rlsto A., Wstęp do lzy uerycze, Itroducto to uercl lyss, Pństwowe Wydwctwo Nuowe, Wrszw 97