Keflaio 1

Eisagwg

Parlo pou h prth istorik anafor sth Gewdaisa wc anexrththc episthmonikc

perioqc gnetai ap ton Aristotlh sta Metafusik ton 5o aina p.Q., h sgqronh

antlhyh gia thn epistmh thc Gewdaisac akolouje ton orism pou thc dwse o

germanc gewdathc Friedrich Robert Helmert (1843-1917) to 1880 sth monogra-

fa tou Die mathematischen und physikalischen Theorien der Hoheren Geodasie.

Sthn prth pargrafo tou istoriko auto rgou anafretai wc Gewdaisa . . . h

epistmh thc mtrhshc kai apeiknishc thc ginhc epifneiac, orismc pou stic

mrec mac ja prpei gia lgouc plhrthtac na epektaje me thn prosjkh . . . kai

twn qronikn metaboln thc. O sntomoc kai periektikc autc orismc perilam-

bnei to snolo twn jewrhtikn kai praktikn asqolin tou sgqronou gewdath,

tou sgqronou Agronmou kai Topogrfou Mhqaniko. Ete prkeitai gia thn

anlush doruforikn eiknwn me mejdouc Thlepiskphshc, ete gia th sullog

twn aparathtwn metrsewn pedou kai to metpeita sqediasm enc topografiko

diagrmmatoc sta plasia miac tupikc topografikc apotpwshc, o orismc tou

Helmert apodeiknetai tso petuqhmnoc so kai diaqronikc. H epistmh thc Ge-

wdaisac, pwc aut orzetai ap ton Helmert, kalptei lec tic dunatc klmakec

apeiknishc perioqn thc ghc kai fainomnwn pou lambnoun qra sthn epifnei

thc. En periorsoume th sunolik ktash thc perioqc melthc tso, ste gia

thn anaparstash tou kampulmorfou sqmatoc thc ginhc epifneiac na mpore

na qrhsimopoihje na eppedo, katalgoume se ma exeidikeumnh gnwstik perioq

thc Gewdaisac, thn Topografa. Enac genik apodektc orismc thc epistmhc

thc Topografac dnetai ap to Dhmtrio Blqo (1987) wc exc: H Topografa

enai h epistmh pou didskei tic mejdouc me th bojeia twn opown apeikonzetai

up klmaka h epifneia tou edfouc epnw se na eppedo. O diaqwrismc enai sa-

fc. En ston orism tou Helmert sumperilambnontai fainmena kai parathrseic

2 Eisagwg

pou perigrfoun th gh sto snol thc, h epistmh thc Topografac periorzetai

se pol mikrterec ektseic, afo to telik pron miac topografikc melthc -

to topografik digramma o topografikc qrthc - apotele mia apeiknish tou

up melth kommatio thc ginhc epifneiac sto eppedo tou qrth me morf sqe-

dou up klmaka. H Topografa periorzetai loipn se meltec mikrc ktashc,

pou h nnoia tou epipdou enai kurarqh all kai epitrept. H poluplokthta

thc pragmatikc topografikc epifneiac kai o qaraktrac twn orgnwn kai twn

mejdwn mtrhshc pou qrhsimopoiontai gia tic topografikc efarmogc enai t-

toia, pou den epitrpoun th genkeush thc qrshc tou epipdou par mnon ktw

ap kpoiec propojseic, pou upeisrqontai mlista kai sto montlo paratrhshc

wc anagkaec paradoqc kai proseggseic pou ja analsoume me leptomreia sta

epmena keflaia.

To basik majhmatik antikemeno thc Topografac enai o prosdiorismc qara-

kthristikn shmewn tou qrou me th morf suntetagmnwn se kpoio topik s-

sthma anaforc. H gnsh autn twn suntetagmnwn epitrpei sth sunqeia thn

apeiknish twn shmewn sto orizntio eppedo tou topografiko diagrmmatoc pe-

rigrfontac me autn ton trpo kataskeuc, ktsmata, ria idiokthsin kai genik

to snolo twn anjrpinwn parembsewn sthn perioq pou qei apotupwje. O

zhtomenoc prosdiorismc qei ap majhmatik poyh kajar gewmetrik qarakth-

ristik kai ja mporose na antimetwpiste plrwc me ta upologistik ergalea thc

klasikc analutikc gewmetrac. Wstso, kajc pragmatopoietai msa sto fu-

sik peribllon twn metrsewn me th bojeia kpoiou topografiko metrhtiko

sustmatoc, adunate na parqei na monosmanto apotlesma. Aut shmanei ti

ta qarakthristik tou peribllontoc twn metrsewn, pwc h epdrash thc dijla-

shc thc atmsfairac kai h qronik metabol thc kat th dirkeia thc hmrac, h

diafor sth diejunsh thc katakorfou anmesa se do opoiadpote shmea tou

qrou, oi metaballmenec atmosfairikc sunjkec kat th dirkeia thc hmrac pou

epidron se diforec metromenec paramtrouc, ephrezoun thn akrbeia tou pros-

diorismo thc jshc twn epilegmnwn shmewn tou pedou. Kurwc mwc h akrbeia

tou prosdiorismo twn en lgw shmewn enai mesa exarthmnh ap thn kataskeua-

stik teleithta twn topografikn orgnwn mtrhshc pou qrhsimopoiontai pou

enai diajsima gia to skop aut.

Ac exetsoume gia pardeigma, to jemelidec prblhma tou upologismo thc ap-

stashc metax do shmewn A kai B pou ankoun se na prokajorismno eppedo.

Gia th majhmatik eplush tou problmatoc enai aparathtoc o orismc kpoiou

koino sustmatoc suntetagmnwn sto dio eppedo kai o prosdiorismc twn sunte-

Eisagwg 3

.

.

A

BsAB

Sqma 1.1: Gewmetrik perigraf tou problmatoc upologismo thc apstashcmetax do shmewn A kai B pou ankoun se na prokajorismno eppedo

tagmnwn (xA, yA) kai (xB, yB) twn do shmewn A kai B antstoiqa wc proc aut

to ssthma. Gnwrzontac thn arijmhtik tim twn suntetagmnwn o upologismc

thc zhtomenhc apstashc apotele ma apl skhsh thc eukledeiac gewmetrac me

to monosmanto apotlesma sAB =(xB xA)2 + (yB yA)2. Wstso, autc o

upologismc mpore na qarakthriste wc monosmantoc me th basik propjesh

ti oi timc xA, xB, yA kai yB apotelon kat apluto trpo gnwstc arijmhti-

kc timc, den enai me lla lgia ephreasmnec ap kanenc edouc abebaithtec

sflmata. Sta plasia afhrhmnwn problhmtwn thc analutikc gewmetrac ma

ttoia paradoq enai epitrept. Orzetai na trto shmeo sto en lgw eppedo wc h

afethra tou koino sustmatoc suntetagmnwn kai do kjetoi metax touc xonec

pou tmnontai sthn afethra. H afethra tou sustmatoc apokt ex orismo tic

suntetagmnec (0, 0), en oi suntetagmnec xA, xB, yA kai yB prokptoun metr-

ntac thn apstash twn do shmewn A kai B ap touc antstoiqouc xonec. Kajc

h apstash enc shmeou ap eujea orzetai sthn analutik gewmetra wc to mtro

tou kjetou eujgrammou tmmatoc ap to shmeo proc thn eujea, mia isodnamh

kfrash gia tic parapnw suntetagmnec prokptei wc to mtro thc parllhlhc

metatpishc twn axnwn ap thn afethra tou sustmatoc mqri ta shmea A kai B.

Gia ton trpo orismo twn susthmtwn anaforc kai twn susthmtwn suntetag-

mnwn ja anaferjome analutik sto epmeno keflaio. Enai mwc profanc ti

sth diamrfwsh twn telikn arijmhtikn timn twn suntetagmnwn shmantik r-

lo pazei o trpoc orismo twn axnwn kajc kai o trpoc upodiaresc touc sta

stoiqeidh tmmata sou mkouc, ta opoa apotelon kai th bsh gia ton praktik

upologism twn xA, xB, yA kai yB. En gia pardeigma exetsoume to sugkekrimno

prblhma sto eppedo tou qartio, tte ta stoiqeidh tmmata pou mlic anafr-

jhkan tautzontai me tic mikrterec upodiairseic tou qraka pou qrhsimopoiome

gia th mtrhsh thc apstashc twn shmewn ap touc antstoiqouc xonec. En

4 Eisagwg

A

A

A

B

B(0,0) x x

y

y

y

x

B

Sqma 1.2: O upologismc thc orizntiac apstashc apaite ton orism suntetag-mnwn gia ta do suneppeda shmea

upojsoume ti (1) h kajetthta twn axnwn twn suntetagmnwn qei epiteuqje

me apluto trpo, (2) ta tmmata pou perigrfoun thn apstash twn shmewn A

kai B ap touc xonec enai apluta kjeta se autoc, (3) h mtrhsh me to qraka

tou mkouc twn tmhmtwn de sunodeetai ap kama phg sflmatoc (p.q. sflma

angnwshc thc alhjoc timc upodiareshc tou qraka adunama tatishc thc

sugkekrimnhc upodiareshc me to antstoiqo shmeo sto qart) kai (4) o qrakac

pou qrhsimopoiome de frei kpoio susthmatik sflma, pwc sflma upodiare-

shc sflma klmakac, tte mporome na pome ti o upologismc thc apstashc

sAB me to majhmatik tpo pou anafrjhke parapnw enai kai monosmantoc kai

apallagmnoc ap sflmata tso tou montlou perigrafc tou problmatoc so

kai tou metrhtiko sustmatoc (qrakac) pou qrhsimopoijhke gia th diexagwg

twn metrsewn.

En mwc to prohgomeno jewrhtik prblhma qei monosmanth lsh mno ktw

ap sugkekrimnec propojseic ta prgmata gnontai akmh pio polploka tso

Eisagwg 5

ap pleurc montlou so kai ap pleurc apaitsewn gia akrbeia twn qrhsi-

mopoiomenwn metrhtikn susthmtwn, tan anafermaste ston prosdiorism thc

jshc qarakthristikn shmewn tou qrou kat th diexagwg tupikn topografi-

kn ergasin. Se autn thn perptwsh oi suntetagmnec enc shmeou den enai pot

posthtec gnwstc kat ma apluth nnoia, all enai to apotlesma thc epexer-

gasac kpoiwn prwtogenn dedomnwn pou qoun sulleqje sto pedo me kpoio

ap ta diajsima topografik rgana metrsewn. Anloga me thn orjthta twn ge-

wmetrikn sqsewn kai twn qarakthristikn twn mhqanikn mern pou sunjtoun

to eswterik twn topografikn orgnwn pou ja qrhsimopoihjon gia tic anagkaec

metrseic pedou kai anloga me to kat pso ta qarakthristik aut paramnoun

stajer me to prasma tou qrnou tic metaballmenec periballontikc sunjkec

emfanzontai sugkekrimna sflmata stic metrseic. Etsi parathretai to gegonc

oi arijmhtikc timc tou diou gewmetriko megjouc sto pedo me to dio metrhtik

ssthma na diafroun metax touc tan h mtrhsh qei epanalhfje se do diafore-

tikc qronikc stigmc. Sthn ousa h alhjin tim opoioudpote gewmetriko fu-

siko megjouc pou parathretai sta plasia miac topografikc efarmogc apotele

pntote ma gnwsth parmetro. To gegonc aut mlista den allzei akma kai

me thn epanalambanmenh mtrhsh tou diou megjouc me to dio metrhtik ssthma.

Aut pou epiqeirome kje for enai, ekmetalleumenoi thn teqnologik exlixh

pou mac prosfrei gia lo to fsma twn efarmosmnwn episthmn kataskeuastik

lo kai teleitera basik rgana mtrhshc kai bohjhtikc metrhtikc diatxeic, na

metrsoume na sugkekrimno gewmetrik ( fusik se orismnec periptseic) mge-

joc ston trisdistato qro me lo kai megalterh akrbeia. H diafor metax tou

jewrhtik agnstou parathromenou megjouc kai mac memonwmnhc paratrhshc

ekfrzei to sflma thc mtrhshc, gia to opoo eidik sthn Topografa diakrnoume

sunolik treic sunistsec, to susthmatik, to tuqao kai to qondroeidc sflma.

H jewra sfalmtwn kai anlushc twn dedomnwn, pou apotele ma llh basik

sunistsa thc ekpadeushc tou Topogrfou Mhqaniko, analei se leptomreia

tic nnoiec autc kai tic sundei sthn perptwsh thc Topografac me tic praktikc

topografikc metrseic (dieujnseic, gwnec kai apostseic) pou sullgontai ap

ta antstoiqa rgana metrsewn sto pedo.

Pra mwc ap ton kataskeuastik pargonta pou upeisrqetai sthn poithta twn

topografikn metrsewn (ma nnoia pou sunduzei thn akrbeia kai thn axiopista),

exsou shmantikc enai kai o orismc tou majhmatiko montlou pou perigrfei to

sugkekrimno prblhma tan anafermaste se pragmatikc topografikc efarmo-

gc. Pso orj enai h majhmatik perigraf tou montlou kai pso realistik

perigrfei to zhtomeno prblhma sto plasio tou fusiko peribllontoc twn me-

6 Eisagwg

.

.

A

B

sAB

Sqma 1.3: To prblhma tou prosdiorismo thc keklimnhc apstashc sAB metaxdo shmewn A kai B sto qro

trsewn; Qarakthristik enai kai pli h perptwsh tou upologismo thc apstashc

pou orzoun do shmea A kai B tan aut anafrontai ston pragmatik trisdi-

stato qro. Se antjesh me thn teleutaa perptwsh pou to eppedo sto opoo

orzontai ta do shmea tautzetai me to eppedo upologismo lwn twn sqetikn

gewmetrikn posottwn (eppedo tou qartio) sto no prblhma apaitetai prose-

ktikc orismc enc epipdou anaforc epnw sto opoo ja efarmoston oi basikc

arqc thc eppedhc analutikc gewmetrac pou anafrjhkan parapnw. O lgoc

enai ti h eujea AB tmnetai sto qro me nan peiro arijm diaforetikn epi-

pdwn. Jewrhtik kje na ap aut ta eppeda ja mporose na qrhsimopoihje

wc to eppedo upologismo thc apstashc sAB me ton trpo pou perigryame pro-

hgoumnwc. Kajc o upologismc thc sugkekrimnhc apstashc entssetai sthn

topografik apotpwsh miac eurterhc perioqc, eisgetai h nnoia tou orizontou

epipdou. Prkeitai gia ma gewmetrik posthta me xeqwrist fusik ermhnea,

pou perigrfei me eniao trpo thn perioq melthc kai ja exetaste analutik sth

sunqeia.

Etsi, to zhtomeno tou upologismo thc orizntiac apstashc metax do shmewn

A kai B sto qro angetai se na prblhma pou apaite do xeqwrist bmata.

Sto prto bma pragmatopoietai h mtrhsh thc keklimnhc apstashc pou orzoun

ta do fusik shmea A kai B sto qro qrhsimopointac kpoio ap ta diajsima

gia to skop aut rgana mtrhshc apostsewn, en sto detero bma pragma-

topoietai h anagwg thc keklimnhc apstashc sto orizntio eppedo. Se kpoiec

efarmogc, pwc gia pardeigma ston upologism uyometrikn diaforn msw miac

Eisagwg 7

trigwnometrikc qwrostjmhshc, zhtomeno enai to mtro thc keklimnhc apsta-

shc sAB. Se kje perptwsh mwc apaitontai kpoia diakrit bmata sto pedo pou

epitrpoun thn praktik mtrhsh thc apstashc metax do shmewn. Arqik apai-

tetai h ulopohsh twn shmewn me kpoion eudikrito trpo epnw sthn topogra-

fik epifneia. Gia to skop aut qrhsimopoiontai difora bohjhtik exartmata

pwc karfi pssaloi, pou epitrpoun, anloga me thn efarmog, ma proswrin

pio makroprjesmh smansh tou shmeou sto pedo. To eujgrammo tmma pou

noht sundei ta do shmea (koruf twn karfin passlwn pou qoun paktwje

katllhla sto dafoc) mac dnei thn keklimnh touc apstash stic treic diast-

seic. Me thn emplok miac seirc metrhtikn orgnwn kai bohjhtikn kataskeun

pou ja perigryoume sth sunqeia, prospajome na upologsoume thn kalterh

dunat ektmhsh gia to mtro thc keklimnhc autc apstashc kajc kai thc pro-

bolc thc sto orizntio eppedo pou qei oriste eniaa gia thn perioq. H akrbeia

twn topografikn orgnwn, h pistthta tou majhmatiko montlou kai to gegonc

ti lec oi metrseic pragmatopoiontai sto idiatera polploko peribllon twn

katterwn strwmtwn thc atmsfairac dnoun na mtro thc poluplokthtac tou

aplo auto egqeirmatoc enc praktiko upologismo thc apstashc metax do

shmewn sto qro. En sto shmeo aut sunupologsoume to gegonc ti upr-

qei genik mia apathsh twn apodektn miac topografikc ergasac gia auxhmnh

akrbeia kai axiopista tou teliko prontoc, tte katadeiknetai me ton kaltero

trpo h idiatera auxhmnh apathsh thc Topografac gia akrbeia se la ta eppe-

da: (a) sth majhmatik diatpwsh tou montlou paratrhshc, (b) sth sullog twn

dedomnwn ap ta topografik rgana mtrhshc, (g) sth diadikasa anlushc kai

epexergasac twn metrsewn pedou gia ton upologism twn zhtomenwn pargwgwn

gewmetrikn posottwn kai tloc (d) ston upologism thc akrbeiac twn telikn

apotelesmtwn.

En kpoioc epijumose na perigryei to rlo thc Topografac, all kai thc Ge-

wdaisac geniktera, ja mporose na pei ti prkeitai gia ekenec tic epistmec pou

epiqeiron na ulopoisoun sthn prxh afhrhmnec majhmatikc kai fusikc nnoiec,

pwc autc twn suntetagmnwn, twn susthmtwn anaforc kai twn susthmtwn

suntetagmnwn tou uyomtrou kai twn uyometrikn diaforn. Gia pardeigma,

epistmec pwc h Fusik ta Majhmatik qrhsimopoion gia tic dikc touc efar-

mogc kat kron gewdaitikc nnoiec, pwc to yoc, th gewdaisiak gramm to

qrth, qwrc sunjwc na anafrontai stic sugkekrimnec mejdouc ulopohshc

mtrhshc autn twn megejn. Ap ta sa anafrjhkan parapnw mpore kanec

na fantaste tic idiaterec apaitseic kai ton idiatero qaraktra thc Topografac

se sqsh me llec epistmec, to giat qei apoktsei ton epijetik prosdiorism

8 Eisagwg

thc epistmhc twn polln dekadikn. Akribc lgw thc fshc tou antikeimnou

thc h Topografa den mpore na isquriste thn apluth orjthta twn apotelesm-

twn thc, pnta ap ma austhr majhmatik skopi. Mpore mwc na isquriste

ti, me bsh to majhmatik montlo pou uiojetjhke kai ktw ap tic dedomnec

exwterikc sunjkec pou epikratosan kat th qronik stigm twn metrsewn, oi a-

rijmhtikc timc twn telikn apotelesmtwn brskontai msa sta ria akrbeiac kai

axiopistac pou kajorzontai ap ta antstoiqa statistik ergalea twn mejdwn

anlushc twn dedomnwn kai sunrjwshc twn parathrsewn pou qoun sulleqje

sto pedo. Etsi, en den filodoxon na pargoun apotelsmata pou na enai a-

pallagmna ap kje qnoc sflmatoc abebaithtac, oi teqnikc mtrhshc kai oi

mjodoi upologismo thc Topografac enai se jsh na mac parqoun thn kalterh

dunat ektmhsh gia to mtro sugkekrimnwn gewmetrikn megejn pou perigrfoun

th sqetik jsh qarakthristikn shmewn epnw sthn epifneia thc ghc kajc kai

na ekfrsoun posotik thn akrbeia thc telikc apdoshc autc thc plhroforac

epnw sto eppedo tou topografiko diagrmmatoc. H ektmhsh thc akrbeiac twn

telikn apotelesmtwn prorqetai ap thn efarmog analutikn statistikn ka-

nnwn kai antikatoptrzei afenc tic metrhtikc ikanthtec isodnama ta eppeda

akrbeiac twn topografikn orgnwn mtrhshc pou qrhsimopoijhkan gia tic me-

trseic pedou, afetrou de thn poithta twn metrsewn kai thn empeira thc omdac

pou tic pragmatopohse.

O skopc twn arqn kai mejdwn pou perigrfontai sta epmena keflaia enai

prwtstwc na analujon oi arqc leitourgac twn orgnwn mtrhshc kai oi mjo-

doi prosdiorismo thc jshc shmewn thc ginhc epifneiac epnw sto eppedo enc

topografiko diagrmmatoc. Wstso, to telik pron autc thc diadikasac, pou

enai nac katlogoc suntetagmnwn qarakthristikn shmewn tou edfouc wc proc

kpoio katllhla orismno topik uperkemeno ssthma suntetagmnwn, prpei

na exetzetai sto plasio thc mellontikc axiopohshc autc thc plhroforac ap

ma eurterh omda qrhstn. Kajc ta shmea tou qrou pou prosdiorzontai

msa ap ma klasik topografik apotpwsh perigrfoun thn katstash enc

ktsmatoc miac kataskeuc th dedomnh qronik stigm, ja prpei, sta plasia e-

nc swst organwmnou topografiko sqediasmo, tso oi topografikc metrseic

so kai oi antstoiqoi upologismo na apojhkeontai katllhla se na Gewgrafik

Ssthma Plhroforin kai na enai prosbsima gia mellontikc efarmogc. Se

autn thn katllhla diamorfwmnh hlektronik bsh dedomnwn ja prpei na ka-

taqwrontai la ekena ta stoiqea, pou ja epitrpoun thn parakolojhsh kai thn

tekmhrwsh twn opoiwndpote metaboln thc fusikc epifneiac thc ghc, ete lgw

fusikn fainomnwn (seismo, plhmmrec, fusikc katastrofc), ete exaitac thc

Eisagwg 9

angkhc efarmogc diafrwn dikastikn apofsewn (allag orwn idiokthsin,

prxeic efarmogc k.lp.). Axzei ed na shmeiwje ti, en h nnoia tou gewgra-

fiko sustmatoc plhroforin prpei na apotele na anapspasto tmma enc

swst organwmnou Ejniko Kthmatologou, h efarmog hlektronikn bsewn de-

domnwn gia thn katagraf metrsewn kai upologismn se diaforetikc qronikc

peridouc apotele basik arq ulopohshc miac meglhc kathgorac efarmogn me

uyhlc apaitseic se akrbeia pwc biomhqanikc efarmogc, dianoxeic shrggwn

kai tonel, parakolojhsh paramorfsewn teqnikn rgwn k.lp.