majhmatikoð upologismoð. To biblðo mporeð na qwristeð jematikĺ se treic enìthtec. Thn prÿth enìthta apoteloôn

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Transcript:

Prìlogoc To parìn sôggramma apeujônetai se proptuqiakoôc foithtèc TmhmĹtwn Poluteqnikÿn Sqolÿn kai Teqnologikÿn Ekpaideutikÿn IdrumĹtwn sta opoða didĺskontai eisagwgikĺ topografikĺ majămata. Epiplèon apeujônetai se epaggelmatðec MhqanikoÔc ìlwn twn eidikotătwn pou epijumoôn mia parousðash twn arqÿn leitourgðac twn basikÿn topografikÿn orgĺnwn mètrhshc kai mejìdwn upologismoô, kajÿc kai se epistămonec suggenÿn gnwstikÿn perioqÿn pou anazhtoôn mia prÿth eisagwgă sthn epistămh kai to antikeðmeno thc TopografÐac. Proerqìmenoc apì mia diaforetikă ereunhtikă perioqă, ìtan prin apì tèssera qrìnia anèlaba didaktikĺ kajăkonta sto PoluteqneÐo tou BerolÐnou me antikeðmeno thn TopografÐa, ărja antimètwpoc, katĺ th diadikasða proetoimasðac tou didaktikoô ulikoô twn sqetikÿn proptuqiakÿn majhmĺtwn, me kĺpoia basikĺ erwtămata. Orismèna apì autĺ, pou apotèlesan kai shmeðo ekkðnhshc tou parìntoc biblðou, ătan ta exăc: Poia eðnai h majhmatikă diatôpwsh twn basikÿn topografikÿn problhmĺtwn; Poia h sqèsh twn topografikÿn orgĺnwn mètrhshc me to fusikì peribĺllon twn metrăsewn kai poio rìlo paðzoun basikĺ kataskeuastikĺ qarakthristikĺ twn orgĺnwn sth sqèsh aută; Pÿc sundèontai oi topografikèc metrăseic pedðou me to telikì proðìn miac topografikăc ergasðac, dhladă th sôntaxh enìc topografikoô qĺrth ă enìc topografikoô diagrĺmmatoc; Prospajÿntac na diatupÿsw kĺpoiec apantăseic sta parapĺnw erwtămata kai èqontac wc odhgì to ulikì pou eðqe sugkentrwjeð apì tic sqetikèc paradìseic sto BerolÐno kai sth JessalonÐkh xekðnhsa th suggrafă tou parìntoc. H filosofða pou dièpei to grĺyimo twn kefalaðwn kai ton trìpo parousðashc eðnai na dojeð èmfash stic arqèc leitourgðac twn topografikÿn orgĺnwn mètrhshc kai sth sôndesă touc me to fusikì peribĺllon twn metrăsewn. Gia thn katanìhsh tou eðdouc thc plhroforðac pou dðnetai apì èna topografikì diĺgramma parousiĺzontai epðshc kĺpoioi basikoð orismoð susthmĺtwn anaforĺc kai susthmĺtwn suntetagmènwn kajÿc kai oi sqèseic metasqhmatismoô pou sundèoun diaforetikĺ sustămata sto epðpedo. Basikă ènnoia pou apoteleð to sundetikì krðko metaxô twn epð mèrouc kefalaðwn eðnai aută tou orizontðou epipèdou. To orizìntio epðpedo apoteleð to epðpedo anaforĺc gia ìlec tic metrăseic pou lambĺnontai apì ta epð mèrouc metrhtikĺ sustămata sto pedðo, enÿ tautìqrona eðnai kai h epifĺneia sthn opoða anafèrontai ìloi oi

vi Prìlogoc majhmatikoð upologismoð. To biblðo mporeð na qwristeð jematikĺ se treic enìthtec. Thn prÿth enìthta apoteloôn ta prÿta tèssera kefĺlaia, sta opoða sugkentrÿnontai ìlec ekeðnec oi jewrhtikèc ènnoiec kai oi orismoð pou dièpoun to sônolo twn praktikÿn topografikÿn ergasiÿn. H parousðash gðnetai men sunoptikĺ, èqei ìmwc katablhjeð prospĺjeia na gðnei me ìso to dunatìn pio autìnomo trìpo. Sth deôterh enìthta, sthn opoða anăkoun to 5o kai to 6o kefĺlaio, parousiĺzontai ta kôria ìrgana kai oi mèjodoi mètrhshc twn basikÿn topografikÿn parathrăsewn, dhladă twn gwniÿn kai twn apostĺsewn. Sto shmeðo autì èqei dojeð idiaðterh èmfash sthn arqă leitourgðac kai ta qarakthristikĺ enìc hlektromagnhtikoô orgĺnou mètrhshc apostĺsewn, pou apoteleð plèon anapìspasto mèroc twn sôgqronwn gewdaitikÿn stajmÿn. Tèloc, sthn trðth jematikă enìthta, pou apartðzoun ta kefĺlaia 7 kai 8, dðnontai oi basikèc mèjodoi upologismoô thc jèshc qarakthristikÿn shmeðwn tou edĺfouc sto epðpedo enìc topografikoô diagrĺmmatoc. O orizontiografikìc prosdiorismìc exetĺzetai sto 7o kefĺlaio kai o prosdiorismìc uyomètrwn kai uyometrikÿn diaforÿn sto 8o kefĺlaio. Apì ta trða parartămata pou paratðjentai sto tèloc tou keimènou axðzei edÿ na anaferjeð to prÿto apì autĺ, sto opoðo èqoun sugkentrwjeð ìlec oi basikèc monĺdec mètrhshc me idiaðtero endiafèron gia thn TopografÐa. To sugkekrimèno parĺrthma mporeð na qrhsimopoihjeð autìnoma wc ènac qrăsimoc odhgìc (pocket guide) gia tupikoôc arijmhtikoôc upologismoôc, ìpwc p.q. gia thn allagă tou sustămatoc monĺdwn mètrhshc miac gwnðac. Sto shmeðo autì mou dðnetai h eukairða na ekfrĺsw tic jermèc mou euqaristðec se ìlouc ìsouc sunèbalan me èmmeso ă Ĺmeso trìpo sthn oloklărwsh autoô tou biblðou. Sthn prÿth kathgorða anăkoun sðgoura ìloi oi prÿhn kajhghtèc mou kai nun sunĺdelfoi sto Tmăma Agronìmwn kai TopogrĹfwn Mhqanikÿn tou AristoteleÐou PanepisthmÐou JessalonÐkhc. Oi empneusmènec dialèxeic touc kai o enjousiasmìc touc katĺ th diĺrkeia twn proptuqiakÿn mou spoudÿn mou dhmioôrghsan gia prÿth forĺ thn idèa miac akadhmaðkăc karièrac. KatĹ th diĺrkeia thc metèpeita ergasðac mou sto Mìnaqo kai sto BerolÐno thc GermanÐac eðqa epðshc thn tôqh na sunergastÿ me Ĺristouc sunadèlfouc kajhghtèc kai episthmonikoôc sunergĺtec. IdiaÐterec euqaristðec anăkoun ston prÿhn sunĺdelfo sto PoluteqneÐo tou BerolÐnou Dr. Svetozar Petrović gia tic apolaustikèc kai sunăjwc ateleðwtec suzhtăseic mac gôrw apì diĺfora gewdaitikĺ zhtămata. Pèra ìmwc apì ìlouc touc parapĺnw stic selðdec autoô tou biblðou katagrĺfontai tìso stì keðmeno ìso kai se arketĺ apì ta sqămata pollèc eôstoqec parathrăseic foithtÿn apì to BerolÐno kai th JessalonÐkh, pou proèkuyan kurðwc katĺ thn pragmatopoðhsh twn antðstoiqwn ergasthrðwn

Prìlogoc vii kai thn prÿth touc epafă me ta topografikĺ ìrgana metrăsewn. To aðnigma thc qrăshc ellhnikÿn qaraktărwn apì to sôsthma tupografðac L A TEX, me to opoðo grĺfthke autì to biblðo, mou èlusan me ton plèon didaktikì trìpo oi kôrioi BasÐleioc AndritsĹnoc, Dhmătrioc Arampèloc kai Apìstoloc Surìpouloc. Gia thn upomonă touc katĺ thn exăghsh diĺforwn teqnikÿn zhthmĺtwn me tètoion trìpo, ÿste akìma kai egÿ wc mh eidikìc na mporÿ na katalĺbw, touc euqaristÿ polô. H kurða IoulÐa KarrÐnth diĺbase to keðmeno kai prìteine arketèc suntaktikèc kai glwssikèc diorjÿseic. Thc axðzoun idiaðterec euqaristðec, tonðzontac ìti oi ìpoiec asĺfeiec èqoun apomeðnei barônoun apokleistikĺ kai mìnon to suggrafèa. Ja ăjela tèloc na euqaristăsw jermĺ ton kôrio Arh SÔrmo kai tic Ekdìseic Zăth gia th boăjeia pou mou prosèferan sto telikì stĺdio thc proetoimasðac tou biblðou, kajÿc kai gia thn èmprakth sumbolă touc sth beltðwsh thc poiìthtac orismènwn sqhmĺtwn. JessalonÐkh, AÔgoustoc 2005 Dhmătrioc TsoÔlhc

Contents 1 Introduction 1 2 Basic definitions of reference frames and coordinate systems 11 2.1 Introduction............................... 11 2.2 Local coordinate systems and fundamental transformations on the plane................................... 14 2.3 Projection on the map plane...................... 33 3 Basic definitions of the gravity field 43 3.1 Introduction............................... 43 3.2 Potential, gravity and acceleration................... 45 3.3 Equipotential surfaces and geoid.................... 56 3.4 Global coordinate systems....................... 68 3.5 Gravity field approximations...................... 71 3.5.1 The ellipsoidal approximation................. 72 3.5.2 The spherical approximation.................. 73 3.5.3 The planar approximation................... 74 4 Definition of height and height difference 81 4.1 Introduction............................... 81 4.2 Definition of the different height systems............... 85 5 Instruments and methods for angle measurements 99

386 Introduction to Surveying 5.1 Introduction............................... 99 5.2 Optical Prisms............................. 105 5.3 Theodolite................................ 111 5.3.1 Connection of the Theodolite with the field of measurements 123 5.3.2 Levelling and centering..................... 131 5.3.3 Measurement of horizontal and vertical angles........ 133 6 Instruments and measurement methods of distances 137 6.1 Introduction............................... 137 6.2 Measuring tape............................. 138 6.3 Basis of known length......................... 142 6.4 The electromagnetic method...................... 146 6.4.1 Carrier and measuring signal.................. 153 6.4.2 Pulse method.......................... 160 6.4.3 Phase comparison method................... 163 6.4.4 Range and accuracy of EDMs................. 169 6.4.5 Internal error sources...................... 173 6.4.6 External error sources..................... 177 6.5 Calibration of distance measurement instruments.......... 188 6.5.1 Measuring tape calibration................... 189 6.5.2 EDM calibration........................ 191 7 Methods of computing the horizontal position of field points 205 7.1 Introduction............................... 205 7.2 The cartesian coordinates method................... 206 7.3 The polar coordinates method..................... 210 7.3.1 Polygonal traverse....................... 215 7.3.2 The intersection problem.................... 229 7.3.3 The three-point resection problem............... 236

Contents 387 7.3.4 The remote base problem................... 240 7.3.5 Line intersection........................ 242 7.3.6 Circle center computation................... 245 7.3.7 Eccentric station........................ 248 7.4 Indirect computation of inclined distances.............. 254 8 Instruments and methods for the computation of heights and height differences 257 8.1 Introduction............................... 257 8.2 Geometric levelling........................... 259 8.2.1 Line of sight error........................ 278 8.2.2 Accuracy of geometric levelling................ 295 8.3 Trigonometric levelling......................... 302 8.3.1 Effect of the gravity field.................... 306 8.3.2 Effect of the atmospheric refraction.............. 311 8.3.3 Computation of the refraction coefficient from the mutual and simultaneous observation of zenith angles........ 318 8.4 Barometric levelling........................... 327 8.5 Gravimetric levelling.......................... 335 8.6 Algorithm for the computation of heights and height differences.. 337 Appendix A: Measurement units 343 Appendix B: Basic definitions in adjustment theory 355 Appendix C: Similarity transform on the plane 363 Literature 375 Index 377

KefĹlaio 1 Eisagwgă Parìlo pou h prÿth istorikă anaforĺ sth GewdaisÐa wc anexĺrththc episthmonikăc perioqăc gðnetai apì ton Aristotèlh sta MetafusikĹ ton 5o aiÿna p.q., h sôgqronh antðlhyh gia thn epistămh thc GewdaisÐac akoloujeð ton orismì pou thc èdwse o germanìc gewdaðthc Friedrich Robert Helmert (1843-1917) to 1880 sth monografða tou Die mathematischen und physikalischen Theorien der Höheren Geodäsie. Sthn prÿth parĺgrafo tou istorikoô autoô èrgou anafèretai wc GewdaisÐa... h epistămh thc mètrhshc kai apeikìnishc thc găinhc epifĺneiac, orismìc pou stic mèrec mac ja prèpei gia lìgouc plhrìthtac na epektajeð me thn prosjăkh... kai twn qronikÿn metabolÿn thc. O sôntomoc kai periektikìc autìc orismìc perilambĺnei to sônolo twn jewrhtikÿn kai praktikÿn asqoliÿn tou sôgqronou gewdaðth, tou sôgqronou Agronìmou kai TopogrĹfou MhqanikoÔ. EÐte prìkeitai gia thn anĺlush doruforikÿn eikìnwn me mejìdouc Thlepiskìphshc, eðte gia th sullogă twn aparaðthtwn metrăsewn pedðou kai to metèpeita sqediasmì enìc topografikoô diagrĺmmatoc sta plaðsia miac tupikăc topografikăc apotôpwshc, o orismìc tou Helmert apodeiknôetai tìso petuqhmènoc ìso kai diaqronikìc. H epistămh thc GewdaisÐac, ìpwc aută orðzetai apì ton Helmert, kalôptei ìlec tic dunatèc klðmakec apeikìnishc perioqÿn thc ghc kai fainomènwn pou lambĺnoun qÿra sthn epifĺneiĺ thc. EĹn periorðsoume th sunolikă èktash thc perioqăc melèthc tìso, ÿste gia thn anaparĺstash tou kampulìmorfou sqămatoc thc găinhc epifĺneiac na mporeð na qrhsimopoihjeð èna epðpedo, katalăgoume se mða exeidikeumènh gnwstikă perioqă thc GewdaisÐac, thn TopografÐa. Enac genikĺ apodektìc orismìc thc epistămhc thc TopografÐac dðnetai apì to Dhmătrio BlĹqo (1987) wc exăc: H TopografÐa eðnai h epistămh pou didĺskei tic mejìdouc me th boăjeia twn opoðwn apeikonðzetai upì klðmaka h epifĺneia tou edĺfouc epĺnw se èna epðpedo. O diaqwrismìc eðnai safăc. Enÿ ston orismì tou Helmert sumperilambĺnontai fainìmena kai parathrăseic

2 Eisagwgă pou perigrĺfoun th gh sto sônolì thc, h epistămh thc TopografÐac periorðzetai se polô mikrìterec ektĺseic, afoô to telikì proðìn miac topografikăc melèthc - to topografikì diĺgramma ă o topografikìc qĺrthc - apoteleð mia apeikìnish tou upì melèth kommatioô thc găinhc epifĺneiac sto epðpedo tou qĺrth me morfă sqedðou upì klðmaka. H TopografÐa periorðzetai loipìn se melètec mikrăc èktashc, ìpou h ènnoia tou epipèdou eðnai kurðarqh allĺ kai epitreptă. H poluplokìthta thc pragmatikăc topografikăc epifĺneiac kai o qaraktărac twn orgĺnwn kai twn mejìdwn mètrhshc pou qrhsimopoioôntai gia tic topografikèc efarmogèc eðnai tètoia, pou den epitrèpoun th genðkeush thc qrăshc tou epipèdou parĺ mìnon kĺtw apì kĺpoiec proôpojèseic, pou upeisèrqontai mĺlista kai sto montèlo paratărhshc wc anagkaðec paradoqèc kai proseggðseic pou ja analôsoume me leptomèreia sta epìmena kefĺlaia. To basikì majhmatikì antikeðmeno thc TopografÐac eðnai o prosdiorismìc qarakthristikÿn shmeðwn tou qÿrou me th morfă suntetagmènwn se kĺpoio topikì sôsthma anaforĺc. H gnÿsh autÿn twn suntetagmènwn epitrèpei sth sunèqeia thn apeikìnish twn shmeðwn sto orizìntio epðpedo tou topografikoô diagrĺmmatoc perigrĺfontac me autìn ton trìpo kataskeuèc, ktðsmata, ìria idiokthsiÿn kai genikĺ to sônolo twn anjrÿpinwn parembĺsewn sthn perioqă pou èqei apotupwjeð. O zhtoômenoc prosdiorismìc èqei apì majhmatikă Ĺpoyh kajarĺ gewmetrikĺ qarakthristikĺ kai ja mporoôse na antimetwpisteð plărwc me ta upologistikĺ ergaleða thc klasikăc analutikăc gewmetrðac. Wstìso, kajÿc pragmatopoieðtai mèsa sto fusikì peribĺllon twn metrăsewn me th boăjeia kĺpoiou topografikoô metrhtikoô sustămatoc, adunateð na parèqei èna monosămanto apotèlesma. Autì shmaðnei ìti ta qarakthristikĺ tou peribĺllontoc twn metrăsewn, ìpwc h epðdrash thc diĺjlashc thc atmìsfairac kai h qronikă metabolă thc katĺ th diĺrkeia thc hmèrac, h diaforĺ sth dieôjunsh thc katakorôfou anĺmesa se dôo opoiadăpote shmeða tou qÿrou, oi metaballìmenec atmosfairikèc sunjăkec katĺ th diĺrkeia thc hmèrac pou epidroôn se diĺforec metroômenec paramètrouc, ephreĺzoun thn akrðbeia tou prosdiorismoô thc jèshc twn epilegmènwn shmeðwn tou pedðou. KurÐwc ìmwc h akrðbeia tou prosdiorismoô twn en lìgw shmeðwn eðnai Ĺmesa exarthmènh apì thn kataskeuastikă teleiìthta twn topografikÿn orgĺnwn mètrhshc pou qrhsimopoioôntai ă pou eðnai diajèsima gia to skopì autì. Ac exetĺsoume gia parĺdeigma, to jemeliÿdec prìblhma tou upologismoô thc apìstashc metaxô dôo shmeðwn A kai B pou anăkoun se èna prokajorismèno epðpedo. Gia th majhmatikă epðlush tou problămatoc eðnai aparaðthtoc o orismìc kĺpoiou koinoô sustămatoc suntetagmènwn sto Ðdio epðpedo kai o prosdiorismìc twn sunte-

Eisagwgă 3 s AB. B A. Sqăma 1.1: Gewmetrikă perigrafă tou problămatoc upologismoô thc apìstashc metaxô dôo shmeðwn A kai B pou anăkoun se èna prokajorismèno epðpedo tagmènwn (x A, y A ) kai (x B, y B ) twn dôo shmeðwn A kai B antðstoiqa wc proc autì to sôsthma. GnwrÐzontac thn arijmhtikă timă twn suntetagmènwn o upologismìc thc zhtoômenhc apìstashc apoteleð mða aplă Ĺskhsh thc eukleðdeiac gewmetrðac me to monosămanto apotèlesma s AB = (x B x A ) 2 + (y B y A ) 2. Wstìso, autìc o upologismìc mporeð na qarakthristeð wc monosămantoc me th basikă proôpìjesh ìti oi timèc x A, x B, y A kai y B apoteloôn katĺ apìluto trìpo gnwstèc arijmhtikèc timèc, den eðnai me Ĺlla lìgia ephreasmènec apì kanenìc eðdouc abebaiìthtec ă sfĺlmata. Sta plaðsia afhrhmènwn problhmĺtwn thc analutikăc gewmetrðac mða tètoia paradoqă eðnai epitreptă. OrÐzetai èna trðto shmeðo sto en lìgw epðpedo wc h afethrða tou koinoô sustămatoc suntetagmènwn kai dôo kĺjetoi metaxô touc Ĺxonec pou tèmnontai sthn afethrða. H afethrða tou sustămatoc apoktĺ ex orismoô tic suntetagmènec (0, 0), enÿ oi suntetagmènec x A, x B, y A kai y B prokôptoun metrÿntac thn apìstash twn dôo shmeðwn A kai B apì touc antðstoiqouc Ĺxonec. Kajÿc h apìstash enìc shmeðou apì eujeða orðzetai sthn analutikă gewmetrða wc to mètro tou kĺjetou eujôgrammou tmămatoc apì to shmeðo proc thn eujeða, mia isodônamh èkfrash gia tic parapĺnw suntetagmènec prokôptei wc to mètro thc parĺllhlhc metatìpishc twn axìnwn apì thn afethrða tou sustămatoc mèqri ta shmeða A kai B. Gia ton trìpo orismoô twn susthmĺtwn anaforĺc kai twn susthmĺtwn suntetagmènwn ja anaferjoôme analutikĺ sto epìmeno kefĺlaio. EÐnai ìmwc profanèc ìti sth diamìrfwsh twn telikÿn arijmhtikÿn timÿn twn suntetagmènwn shmantikì rìlo paðzei o trìpoc orismoô twn axìnwn kajÿc kai o trìpoc upodiaðresăc touc sta stoiqeiÿdh tmămata Ðsou măkouc, ta opoða apoteloôn kai th bĺsh gia ton praktikì upologismì twn x A, x B, y A kai y B. EĹn gia parĺdeigma exetĺsoume to sugkekrimèno prìblhma sto epðpedo tou qartioô, tìte ta stoiqeiÿdh tmămata pou mìlic anafèrjhkan tautðzontai me tic mikrìterec upodiairèseic tou qĺraka pou qrhsimopoioôme gia th mètrhsh thc apìstashc twn shmeðwn apì touc antðstoiqouc Ĺxonec. EĹn

4 Eisagwgă y y B B y A A (0,0) x x x Sqăma 1.2: O upologismìc thc orizìntiac apìstashc apaiteð ton orismì suntetagmènwn gia ta dôo sunepðpeda shmeða A B upojèsoume ìti (1) h kajetìthta twn axìnwn twn suntetagmènwn èqei epiteuqjeð me apìluto trìpo, (2) ta tmămata pou perigrĺfoun thn apìstash twn shmeðwn A kai B apì touc Ĺxonec eðnai apìluta kĺjeta se autoôc, (3) h mètrhsh me to qĺraka tou măkouc twn tmhmĺtwn de sunodeôetai apì kamða phgă sfĺlmatoc (p.q. sfĺlma anĺgnwshc thc alhjoôc timăc upodiaðreshc tou qĺraka ă adunamða taôtishc thc sugkekrimènhc upodiaðreshc me to antðstoiqo shmeðo sto qartð) kai (4) o qĺrakac pou qrhsimopoioôme de fèrei kĺpoio susthmatikì sfĺlma, ìpwc sfĺlma upodiaðreshc ă sfĺlma klðmakac, tìte mporoôme na poôme ìti o upologismìc thc apìstashc s AB me to majhmatikì tôpo pou anafèrjhke parapĺnw eðnai kai monosămantoc kai apallagmènoc apì sfĺlmata tìso tou montèlou perigrafăc tou problămatoc ìso kai tou metrhtikoô sustămatoc (qĺrakac) pou qrhsimopoiăjhke gia th diexagwgă twn metrăsewn. EĹn ìmwc to prohgoômeno jewrhtikì prìblhma èqei monosămanth lôsh mìno kĺtw apì sugkekrimènec proôpojèseic ta prĺgmata gðnontai akìmh pio polôploka tìso

Eisagwgă 5 apì pleurĺc montèlou ìso kai apì pleurĺc apaităsewn gia akrðbeia twn qrhsimopoioômenwn metrhtikÿn susthmĺtwn, ìtan anaferìmaste ston prosdiorismì thc jèshc qarakthristikÿn shmeðwn tou qÿrou katĺ th diexagwgă tupikÿn topografikÿn ergasiÿn. Se autăn thn perðptwsh oi suntetagmènec enìc shmeðou den eðnai potè posìthtec gnwstèc katĺ mða apìluth ènnoia, allĺ eðnai to apotèlesma thc epexergasðac kĺpoiwn prwtogenÿn dedomènwn pou èqoun sulleqjeð sto pedðo me kĺpoio apì ta diajèsima topografikĺ ìrgana metrăsewn. AnĹloga me thn orjìthta twn gewmetrikÿn sqèsewn kai twn qarakthristikÿn twn mhqanikÿn merÿn pou sunjètoun to eswterikì twn topografikÿn orgĺnwn pou ja qrhsimopoihjoôn gia tic anagkaðec metrăseic pedðou kai anĺloga me to katĺ pìso ta qarakthristikĺ autĺ paramènoun stajerĺ me to pèrasma tou qrìnou ă tic metaballìmenec periballontikèc sunjăkec emfanðzontai sugkekrimèna sfĺlmata stic metrăseic. Etsi parathreðtai to gegonìc oi arijmhtikèc timèc tou Ðdiou gewmetrikoô megèjouc sto pedðo me to Ðdio metrhtikì sôsthma na diafèroun metaxô touc ìtan h mètrhsh èqei epanalhfjeð se dôo diaforetikèc qronikèc stigmèc. Sthn ousða h alhjină timă opoioudăpote gewmetrikoô ă fusikoô megèjouc pou parathreðtai sta plaðsia miac topografikăc efarmogăc apoteleð pĺntote mða Ĺgnwsth parĺmetro. To gegonìc autì mĺlista den allĺzei akìma kai me thn epanalambanìmenh mètrhsh tou Ðdiou megèjouc me to Ðdio metrhtikì sôsthma. Autì pou epiqeiroôme kĺje forĺ eðnai, ekmetalleuìmenoi thn teqnologikă exèlixh pou mac prosfèrei gia ìlo to fĺsma twn efarmosmènwn episthmÿn kataskeuastikĺ ìlo kai teleiìtera basikĺ ìrgana mètrhshc kai bohjhtikèc metrhtikèc diatĺxeic, na metrăsoume èna sugkekrimèno gewmetrikì (ă fusikì se orismènec periptÿseic) mègejoc ston trisdiĺstato qÿro me ìlo kai megalôterh akrðbeia. H diaforĺ metaxô tou jewrhtikĺ agnÿstou parathroômenou megèjouc kai mðac memonwmènhc paratărhshc ekfrĺzei to sfĺlma thc mètrhshc, gia to opoðo eidikĺ sthn TopografÐa diakrðnoume sunolikĺ treic sunistÿsec, to susthmatikì, to tuqaðo kai to qondroeidèc sfĺlma. H jewrða sfalmĺtwn kai anĺlushc twn dedomènwn, pou apoteleð mða Ĺllh basikă sunistÿsa thc ekpaðdeushc tou TopogrĹfou MhqanikoÔ, analôei se leptomèreia tic ènnoiec autèc kai tic sundèei sthn perðptwsh thc TopografÐac me tic praktikèc topografikèc metrăseic (dieujônseic, gwnðec kai apostĺseic) pou sullègontai apì ta antðstoiqa ìrgana metrăsewn sto pedðo. Pèra ìmwc apì ton kataskeuastikì parĺgonta pou upeisèrqetai sthn poiìthta twn topografikÿn metrăsewn (mða ènnoia pou sunduĺzei thn akrðbeia kai thn axiopistða), exðsou shmantikìc eðnai kai o orismìc tou majhmatikoô montèlou pou perigrĺfei to sugkekrimèno prìblhma ìtan anaferìmaste se pragmatikèc topografikèc efarmogèc. Pìso orjă eðnai h majhmatikă perigrafă tou montèlou kai pìso realistikĺ perigrĺfei to zhtoômeno prìblhma sto plaðsio tou fusikoô peribĺllontoc twn me-

6 Eisagwgă. B s AB A. Sqăma 1.3: To prìblhma tou prosdiorismoô thc keklimènhc apìstashc s AB metaxô dôo shmeðwn A kai B sto qÿro trăsewn; Qarakthristikă eðnai kai pĺli h perðptwsh tou upologismoô thc apìstashc pou orðzoun dôo shmeða A kai B ìtan autĺ anafèrontai ston pragmatikì trisdiĺstato qÿro. Se antðjesh me thn teleutaða perðptwsh ìpou to epðpedo sto opoðo orðzontai ta dôo shmeða tautðzetai me to epðpedo upologismoô ìlwn twn sqetikÿn gewmetrikÿn posotătwn (epðpedo tou qartioô) sto nèo prìblhma apaiteðtai prosektikìc orismìc enìc epipèdou anaforĺc epĺnw sto opoðo ja efarmostoôn oi basikèc arqèc thc epðpedhc analutikăc gewmetrðac pou anafèrjhkan parapĺnw. O lìgoc eðnai ìti h eujeða AB tèmnetai sto qÿro me ènan Ĺpeiro arijmì diaforetikÿn epipèdwn. JewrhtikĹ kĺje èna apì autĺ ta epðpeda ja mporoôse na qrhsimopoihjeð wc to epðpedo upologismoô thc apìstashc s AB me ton trìpo pou perigrĺyame prohgoumènwc. Kajÿc o upologismìc thc sugkekrimènhc apìstashc entĺssetai sthn topografikă apotôpwsh miac eurôterhc perioqăc, eisĺgetai h ènnoia tou orizontðou epipèdou. Prìkeitai gia mða gewmetrikă posìthta me xeqwristă fusikă ermhneða, pou perigrĺfei me eniaðo trìpo thn perioqă melèthc kai ja exetasteð analutikĺ sth sunèqeia. Etsi, to zhtoômeno tou upologismoô thc orizìntiac apìstashc metaxô dôo shmeðwn A kai B sto qÿro anĺgetai se èna prìblhma pou apaiteð dôo xeqwristĺ bămata. Sto prÿto băma pragmatopoieðtai h mètrhsh thc keklimènhc apìstashc pou orðzoun ta dôo fusikĺ shmeða A kai B sto qÿro qrhsimopoiÿntac kĺpoio apì ta diajèsima gia to skopì autì ìrgana mètrhshc apostĺsewn, enÿ sto deôtero băma pragmatopoieðtai h anagwgă thc keklimènhc apìstashc sto orizìntio epðpedo. Se kĺpoiec efarmogèc, ìpwc gia parĺdeigma ston upologismì uyometrikÿn diaforÿn mèsw miac

Eisagwgă 7 trigwnometrikăc qwrostĺjmhshc, zhtoômeno eðnai to mètro thc keklimènhc apìstashc s AB. Se kĺje perðptwsh ìmwc apaitoôntai kĺpoia diakritĺ bămata sto pedðo pou epitrèpoun thn praktikă mètrhsh thc apìstashc metaxô dôo shmeðwn. ArqikĹ apaiteðtai h ulopoðhsh twn shmeðwn me kĺpoion eudiĺkrito trìpo epĺnw sthn topografikă epifĺneia. Gia to skopì autì qrhsimopoioôntai diĺfora bohjhtikĺ exartămata ìpwc karfiĺ ă pĺssaloi, pou epitrèpoun, anĺloga me thn efarmogă, mða proswrină ă pio makroprìjesmh sămansh tou shmeðou sto pedðo. To eujôgrammo tmăma pou nohtĺ sundèei ta dôo shmeða (korufă twn karfiÿn ă passĺlwn pou èqoun paktwjeð katĺllhla sto èdafoc) mac dðnei thn keklimènh touc apìstash stic treic diastĺseic. Me thn emplokă miac seirĺc metrhtikÿn orgĺnwn kai bohjhtikÿn kataskeuÿn pou ja perigrĺyoume sth sunèqeia, prospajoôme na upologðsoume thn kalôterh dunată ektðmhsh gia to mètro thc keklimènhc autăc apìstashc kajÿc kai thc probolăc thc sto orizìntio epðpedo pou èqei oristeð eniaða gia thn perioqă. H akrðbeia twn topografikÿn orgĺnwn, h pistìthta tou majhmatikoô montèlou kai to gegonìc ìti ìlec oi metrăseic pragmatopoioôntai sto idiaðtera polôploko peribĺllon twn katÿterwn strwmĺtwn thc atmìsfairac dðnoun èna mètro thc poluplokìthtac tou aploô autoô egqeirămatoc enìc praktikoô upologismoô thc apìstashc metaxô dôo shmeðwn sto qÿro. EĹn sto shmeðo autì sunupologðsoume to gegonìc ìti upĺrqei genikĺ mia apaðthsh twn apodektÿn miac topografikăc ergasðac gia auxhmènh akrðbeia kai axiopistða tou telikoô proðìntoc, tìte katadeiknôetai me ton kalôtero trìpo h idiaðtera auxhmènh apaðthsh thc TopografÐac gia akrðbeia se ìla ta epðpeda: (a) sth majhmatikă diatôpwsh tou montèlou paratărhshc, (b) sth sullogă twn dedomènwn apì ta topografikĺ ìrgana mètrhshc, (g) sth diadikasða anĺlushc kai epexergasðac twn metrăsewn pedðou gia ton upologismì twn zhtoômenwn parĺgwgwn gewmetrikÿn posotătwn kai tèloc (d) ston upologismì thc akrðbeiac twn telikÿn apotelesmĺtwn. EĹn kĺpoioc epijumoôse na perigrĺyei to rìlo thc TopografÐac, allĺ kai thc GewdaisÐac genikìtera, ja mporoôse na pei ìti prìkeitai gia ekeðnec tic epistămec pou epiqeiroôn na ulopoiăsoun sthn prĺxh afhrhmènec majhmatikèc kai fusikèc ènnoiec, ìpwc autèc twn suntetagmènwn, twn susthmĺtwn anaforĺc kai twn susthmĺtwn suntetagmènwn ă tou uyomètrou kai twn uyometrikÿn diaforÿn. Gia parĺdeigma, epistămec ìpwc h Fusikă ă ta MajhmatikĹ qrhsimopoioôn gia tic dikèc touc efarmogèc katĺ kìron gewdaitikèc ènnoiec, ìpwc to Ôyoc, th gewdaisiakă grammă ă to qĺrth, qwrðc sunăjwc na anafèrontai stic sugkekrimènec mejìdouc ulopoðhshc ă mètrhshc autÿn twn megejÿn. Apì ta ìsa anafèrjhkan parapĺnw mporeð kaneðc na fantasteð tic idiaðterec apaităseic kai ton idiaðtero qaraktăra thc TopografÐac se sqèsh me Ĺllec epistămec, ă to giatð èqei apoktăsei ton epijetikì prosdiorismì

8 Eisagwgă thc epistămhc twn pollÿn dekadikÿn. Akribÿc lìgw thc fôshc tou antikeimènou thc h TopografÐa den mporeð na isquristeð thn apìluth orjìthta twn apotelesmĺtwn thc, pĺnta apì mða austhră majhmatikă skopiĺ. MporeÐ ìmwc na isquristeð ìti, me bĺsh to majhmatikì montèlo pou uiojetăjhke kai kĺtw apì tic dedomènec exwterikèc sunjăkec pou epikratoôsan katĺ th qronikă stigmă twn metrăsewn, oi a- rijmhtikèc timèc twn telikÿn apotelesmĺtwn brðskontai mèsa sta ìria akrðbeiac kai axiopistðac pou kajorðzontai apì ta antðstoiqa statistikĺ ergaleða twn mejìdwn anĺlushc twn dedomènwn kai sunìrjwshc twn parathrăsewn pou èqoun sulleqjeð sto pedðo. Etsi, enÿ den filodoxoôn na parĺgoun apotelèsmata pou na eðnai a- pallagmèna apì kĺje Ðqnoc sfĺlmatoc ă abebaiìthtac, oi teqnikèc mètrhshc kai oi mèjodoi upologismoô thc TopografÐac eðnai se jèsh na mac parèqoun thn kalôterh dunată ektðmhsh gia to mètro sugkekrimènwn gewmetrikÿn megejÿn pou perigrĺfoun th sqetikă jèsh qarakthristikÿn shmeðwn epĺnw sthn epifĺneia thc ghc kajÿc kai na ekfrĺsoun posotikĺ thn akrðbeia thc telikăc apìdoshc autăc thc plhroforðac epĺnw sto epðpedo tou topografikoô diagrĺmmatoc. H ektðmhsh thc akrðbeiac twn telikÿn apotelesmĺtwn proèrqetai apì thn efarmogă analutikÿn statistikÿn kanìnwn kai antikatoptrðzei afenìc tic metrhtikèc ikanìthtec ă isodônama ta epðpeda akrðbeiac twn topografikÿn orgĺnwn mètrhshc pou qrhsimopoiăjhkan gia tic metrăseic pedðou, afetèrou de thn poiìthta twn metrăsewn kai thn empeirða thc omĺdac pou tic pragmatopoðhse. O skopìc twn arqÿn kai mejìdwn pou perigrĺfontai sta epìmena kefĺlaia eðnai prwtðstwc na analujoôn oi arqèc leitourgðac twn orgĺnwn mètrhshc kai oi mèjodoi prosdiorismoô thc jèshc shmeðwn thc găinhc epifĺneiac epĺnw sto epðpedo enìc topografikoô diagrĺmmatoc. Wstìso, to telikì proðìn autăc thc diadikasðac, pou eðnai ènac katĺlogoc suntetagmènwn qarakthristikÿn shmeðwn tou edĺfouc wc proc kĺpoio katĺllhla orismèno topikì ă uperkeðmeno sôsthma suntetagmènwn, prèpei na exetĺzetai sto plaðsio thc mellontikăc axiopoðhshc autăc thc plhroforðac apì mða eurôterh omĺda qrhstÿn. Kajÿc ta shmeða tou qÿrou pou prosdiorðzontai mèsa apì mða klasikă topografikă apotôpwsh perigrĺfoun thn katĺstash enìc ktðsmatoc ă miac kataskeuăc th dedomènh qronikă stigmă, ja prèpei, sta plaðsia e- nìc swstĺ organwmènou topografikoô sqediasmoô, tìso oi topografikèc metrăseic ìso kai oi antðstoiqoi upologismoð na apojhkeôontai katĺllhla se èna Gewgrafikì SÔsthma Plhroforiÿn kai na eðnai prosbĺsima gia mellontikèc efarmogèc. Se autăn thn katĺllhla diamorfwmènh hlektronikă bĺsh dedomènwn ja prèpei na kataqwroôntai ìla ekeðna ta stoiqeða, pou ja epitrèpoun thn parakoloôjhsh kai thn tekmhrðwsh twn opoiwndăpote metabolÿn thc fusikăc epifĺneiac thc ghc, eðte lìgw fusikÿn fainomènwn (seismoð, plhmmôrec, fusikèc katastrofèc), eðte exaitðac thc

Eisagwgă 9 anĺgkhc efarmogăc diafìrwn dikastikÿn apofĺsewn (allagă orðwn idiokthsiÿn, prĺxeic efarmogăc k.lp.). AxÐzei edÿ na shmeiwjeð ìti, enÿ h ènnoia tou gewgrafikoô sustămatoc plhroforiÿn prèpei na apoteleð èna anapìspasto tmăma enìc swstĺ organwmènou EjnikoÔ KthmatologÐou, h efarmogă hlektronikÿn bĺsewn dedomènwn gia thn katagrafă metrăsewn kai upologismÿn se diaforetikèc qronikèc periìdouc apoteleð basikă arqă ulopoðhshc miac megĺlhc kathgorðac efarmogÿn me uyhlèc apaităseic se akrðbeia ìpwc biomhqanikèc efarmogèc, dianoðxeic shrĺggwn kai toônel, parakoloôjhsh paramorfÿsewn teqnikÿn èrgwn k.lp.