CLFCTON OF PTL RELTONHP uczkowsk, K. nsttut of Photogrammtry and Cartography, Warsaw Unvrsty of Tchnology. TRCT Th spatal rlaton taks plac among at last two ojcts and concrns thr mutual poston n th gographcal spac. Th spatal rlatons takng plac n ralty ar prsntd n a map or n a dataas n a modl way. ccordng to th purpos, w can dstngush th spatal rlatons:! fuzzy (nar, far, hghr, lowr tc.),! drctonal (north, southast tc.),! topologcal (nghour-hood, ntrscton, ncluson, tc.),! mtrcal (dstanc, angl, lnght, tc.),! logcal (xcluson, ovrlappng, tc.). Th growng numr of ltratur wth varous mtodologcal approachs to dstngushng th spatal rlatons shows that ths knd of th clasfcaton s vry complcatd and dffcult. n ths papr w assum that th classfcaton concrns th ojcts wth th topologcal dmnsons: 0 pont, 1 ln, 2 ara, 3 sold. Thr ar 10 man groups of th topologcal rlatons: (pont pont, pont ln, pont ara, pont sold, ln ln, ln ara, ln sold, ara ara, ara sold, sold sold). W also assum that vry ojct has an ntror and an xtror. Ths classfcaton s asd on th followng rlatons n th calculus of classs: xcluson, non-xcluson, ovrlappng, nclud, possson of th common pont, possson of common ponts. 1. PTL RELTONHP n th fld of gonformaton, rlatonshps ar dstngushd twn ojct attruts and rlatonshps nvolvng th mutual poston of at last two ojcts n gographc spac. patal rlatonshps can vwd from two ponts. Th frst on concrns th rlatons twn ojcts n gographc spac. Th scond on, drvd from th frst, concrns th rlatons twn ojct rprsntatons n th data as. Usng natural languag, w dscr th spatal rlatons twn two ojcts y such trms as: low, south of, nar, nsd, tc. From a scntfc pont of vw ths st of trms should formalzd and logcally dscrd. n th ltratur on th sujct two groups of spatal rlatonshps ar gnrally mntond: mtrc rlatons - asd on calculatons mad on coordnats, and topologcal rlatons - dscrng th nghorhood, ncluson, crossovr of ojcts, tc. (Fg. 1). Topologcal rlatonshps ar rsstant to strtchng, scalng, or rotaton. D C C D M trc rlaton s T op ologcal rlaton s R gon D s ast of rgon T h ara of rgon D s largr thaan s......... R gons,, C ar adjacnt R gon D s dstant from rgons,, C Fgur 1. Mtrc and topologcal aspcts of spatal rlatonshps. Procdngs of th 21 st ntrnatonal Cartographc Confrnc (CC) Duran, outh frca, 10 16 ugust 2003 Cartographc Rnassanc Hostd y Th ntrnatonal Cartographc ssocaton (C) N: 0-958-46093-0 Producd y: Documnt Transformaton Tchnologs
2. NME OF PTL RELTONHP On of th ssntal prolms nvolvng spatal rlatonshps s thr trmnology. Th sam rlatonshp can gvn svral dffrnt nams. good xampl of th aov can th rlatonshp Covrd- y from th 4-ntrscton modl, trmd and dfnd dffrntly y dffrnt authors (Fg. 2). C onvrtd y n sd - n t rs k t Ovrlap n cd n t n trscto n Edg - nsd authors 4 -ntrscton m odl G u ttn g Pullar 1 Pullar 2 W agnr M odl R alm Fgur 2. Dffrnt nams of th sam rlaton, accordng to Gotl (2001). nothr xampl hr s th DYNMO systm of th ntrgraph company, whr th sam rlatonshp twn a pont and an ara s gvn four dffrnt trms: a pont touchs an ara, a pont ntrscts an ara, an ara contans a pont and an ara ncloss a pont (Fg. 3). Undrstandaly, ths dos not undrmn th valu of th DYNMO systm, n whch svral dozn spatal rlatonshps wr mplmntd. P o n t to u c h s a r a P o n t n t rs c ts a r a ra contans pont ra ncloss pont Fgur 3. Dffrnt nams of th sam rlatonshp n th DYNMO systm. t turns out that natural languag dos not allow to nam spatal rlatonshps logcally. t stms from th fact that thr s a larg numr of such rlatons and th dffrncs twn thm ar mnor, unnotcd y G usrs. Prhaps th soluton to th prolm would to dscr ths rlatonshps graphcally. 3. METHOD OF DEFNNG THE ET OF PTL RELTONHP Thr ar many mthods of dfnng th sts of spatal rlatonshps: runng (1996), Gotl (2001), chndr (1997), Papadas & Thodords (1997). n th prsnt papr only two wr chosn and dscrd. On of th st known s th 4-ntrscton modl proposd y Egnhofr and Fronzosa (1991). t s a modl whch maks t possl to dffrntat only spatal rlatonshps twn ojcts of th rgon typ. ts authors assumd that a rgon conssts of a oundary δ, an ntror o and an xtror. o = Fgur 4. Th ntror and oundary of an ara accordng to Egnhofr, Clmntn, Flc (1994) Th rlatonshps ar dfnd on th ass of a 2-dmntonal matrx. o o o o st of slctd rlatonshps was prsntd n Fg. 5. (1)
N am of rlaton dsjont Exam pl o o o o qual contans nsd covrs covrd y m t ovrlaps Fgur 5. patal rlatonshps n th 4-ntrscton modl accordng to Egnhofr and Franzosa. Th mthods usd n th 4-ntrscton modl wr dvlopd y Egnhofr and Hrrng (1994) n thr 9-ntrscton modl. n ths modl, pont, ln and surfac ojcts ar consdrd. Th spatal dmnsons of ths ojcts ar 0, 1, and 2, rspctvly. rgon wthout hols s a rgon wth a connctd xtror and a connctd oundary (Fg. 6a). ddtonally, th dfnton of a rgon wth hols was ntroducd. Namly, a rgon wth hols s a rgon wth dsconnctd xtror and a dsconnctd oundary (Fg. 6). Thr ar two knds of lns dffrntatd n th modl: a smpl ln - wth two dsconnctd oundars (Fg. 6c) and a complx ln - wth mor than two dsconnctd oundars (Fg. 6d). a c d Fgur 6. rgon wth a) a connctd and ) dsconnctd oundary; and c) a smpl and d) complx ln, accordng to Egnhofr and Hrrng (1994). Th topologcal dscrpton of th rlatons twn two ojcts can prsntd as a 3x3 matrx. o o o o 0 - R(, ) = whr: o ntror, oudary, - xtror. (2) o - - Th 9-ntrscton modl s now proaly th st to dscr spatal rlatonshps, not wthout ts faults, howvr. Th ntroducton of smpl and complx lns has rsultd n rasng th numr of rlatons twn thm, whch ar 57 now. o th prolm s to sort out ths rlatons and nam thm n trms of natural languag. 4. TENTTVE CLFCTON OF ELEMENTRY PTL RELTONHP Lt us consdr now 4 typs of ojcts n 3-dmnsonal spac such that:! pont P s topologcal dmnson s 0,! ln L s topologcal dmnson s 1,! rgon s, ara s topologcal dmnson s 2,! sold s topologcal dmnson s 3.
Vwd from a topologcal prspctv, a pont and a ln hav only an dg. n a rgon or sold, an dg and an ntror ar dstngushd (Fg. 7). P L P = P L = L = = Fgur 7. Th ntror and dg of an ojct n th topologcal approach. oth an ara and a sold can hav so-calld hols. From a topologcal pont of vw, w ar dalng hr wth a dsjont rgon and a dsjont sold, also havng dsjont dgs. Havng only an dg, a ln posssss so-calld not dsjont ponts (homomorphc ponts) at ts oth nds. twn ths ponts thr ar dsjont ponts. Th rmoval of an pont homomorphc dos not run th topology of an ojct whras th rmoval of a dsjont pont dos (Fg. 8). Usng th topologcal concpt of an undsjontd pont, lt us assum that ths pont s calld th dg of a ln. n ths cas, th dsjontd ponts wll th ln s ntror. f th noton of an xtror s furthr addd to thos of an xtror and dg (Fg. 9), thn th cratd modl wll comply wth th prncpls lad down for th 9-ntrscton modl. U ndsjont rgon and sold dsjontd dgs D sjontd and undsjontd ponts for a ln dsjontd ponts undsjontd ponts Fgur 8. Topologcal dosyncrass of lns, rgons and solds. L 1 E X T E R O R L2 P L P = P L = L L = = Fgur 9. Th dg, ntror and xtror of an ojct n th proposd modl. Th assumpton was that th frst stag of classfcaton of spatal rlatonshps would thr dvson n complanc wth th rlatons of xtnsons n th st thory, nrchd only th dvson of th ovrlappng class nto two: on wth a common pont and th othr wth common ponts. Ths classfcaton s common for all th comnatons that xst twn th four typs of ojcts. Fgur 10. Th st-thortc classfcaton of spatal rlatons.
Th thr fgurs low (Fgs. 11, 12 and 13) show, y way of xampl, lmntary rlatons n 3D spac xstng twn: a surfac and a ln, a ln and a ln, a surfac and a surfac. For th sak of clarty, spcfc rlatons twn th componnts of ojcts wr graphcally dscrd. Th noton of an xtror was assumd to connctd only wth th ojct whch s th doman of a rlatonshp. dsjont not dsjont n sd nsd and tougchng oudary nsd & tougchng oudary tw c on th oundary n sa d ovrlap com m on ponts passng through surfac nsd, touchng oundary at on sd & passng through th othr passng through along oundary t rm n a tn g n sd at on nd startng nsd & passng through along oundary com m on pont n t rs c tn g oundary n t rs c tn g n t ro r lnkng oudary ln k n g n t ro r l n 1 2 1 2 1 2 1 2 s u r f a c 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 Fgur 11. Elmntary spatal rlatons twn a surfac and a ln.
dsjont not dsjont n sd qual touchng oundary n sd n sd ovrlap com m on ponts trm natn g com m on pont lnkng oudary lnkng ntror n t rs c tn g n tror l n 1 2 1 2 1 1 1 2 2 2 1 2 l n 1 2 1 2 1 2 1 2 1 1 1 2 2 2 1 2 Fgur 12. Elmntary spatal rlatons twn a ln and a ln.
dsjont not dsjont n sad q ual n sd toug chng oudary nsd ovrlap com m on p on ts xtndng ovr oundary touchng xtror along on oudary sta rtn g fro m o n oudary n t rs c tn g n tror n t rs c tn g oundary com m on p on t com m on pont ln k n g n t ro r lnkng oudary s u r f a c s u r f a c Fgur 13. Elmntary spatal rlatons twn a surfac and a surfac.
Th spatal rlatons dscussd n th papr wr calld lmntary. t mans that ths rlatons can th ass for complx rlatons (Fg. 14). 6 6 6 4 3 2 2 = 5 4 = 5 3 2 4 3 5 = = = = p p 1 1 1 Fgur 14. Jonng lmntary rlatons. 5. UMMRY Th quston of spatal rlatons whch s so mportant for th dvlopmnt of gonformaton has not rcvd adquat attnton among rsarchrs yt. Dffrnt ways of dfnng spatal rlatons, trmnologcal prolms concrnng th rlatons and thr mplmntatons n Gographc nformaton ystms pont out that th prolm wll contnu to th cntr of our attnton for som tm n th futur. Th prsnt papr was to a larg xtnt nsprd y th 9-ntrscton modl (of Max J.Egnhofr and John R.Hrrng). s gonformaton taks nto account not only two ut, stll to a gratr xtnt, thr dmnsons, th proposd modl was dvlopd for 3D spac. Th ral challng n th futur wll to nclud tm as a fourth dmnson. Th proposd modl concrns only topologcal rlatons. To account for many prolms t would ncssary to nclud such non-topologcal rlatons as, for nstanc, paralllsm and prpndcularty. ras and solds whch ar not homomorphc whr not consdrd n th prsnt papr. t s a quston whch rqurs sparat attnton. 6. REFERENCE [1] runng M. (1996), ntgraton of patal nformaton for Go-nformaton ystms. prngr-vrlag, rln- Hdlrg. [2] Egnhofr M.J., Fronzosa R.D. (1991), Pont-st topologcal spatal rlatons. ntrnatonal Journal of Gographcal nformaton ystms, Taylor&Francs Ltd., vol. 5, no. 2. [3] Egnhofr M.J., Clmntn E., Flc P. (1994), Topologcal rlatons twn rgons wth hols. ntrnatonal Journal of Gographcal nformaton ystms, Taylor&Francs Ltd., vol. 8, no. 2. [4] Egnhofr M.J.,M.ark D.M, Hrrng J. (1994), Th 9-ntrscton: formalsm and ts us for natural-languag spatal prdcats. U.. Natonal Cntr for Gographc nformaton and nalyss. [5] Gotl D. (2001), Mozlwosc wykorzystana analtycznych mtod projktowana systmow nformatycznych w tworznu az danych przstrznnych na przykładz topografczngo systmu nformacyjngo. Warsaw Unvrsty of Tchnology. [6] Papadas D., Thodords Y. (1997), patal rlatons, mnmum oundng, rctangls, and spatal structurs, ntrnatonal Journal of Gographcal nformaton cnc, Taylor&Francs Ltd., vol.11, no. 2. [7] chndr M. (1997), patal Data Typs for Dataas ystms. prngr Computr cnc, Hagn, Grmany.
CLFCTON OF PTL RELTONHP uczkowsk, K. nsttut of Photogrammtry and Cartography, Warsaw Unvrsty of Tchnology. ography Krzysztof uczkowsk! orn n 1950 n Warsaw, Poland! work n nsttut of Photogrammtry and Cartography, Warsaw Unvrsty of Tchnology! doctor s dgr n 1983! ntrsts: Gographcal nformaton ystms, Cartography, Gostatstca