EXTENDED FORMAL SPECIFICATIONS OF 3D SPATIAL DATA TYPES

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1 - 1 - EXTENDED FORMAL SPECIFICATIONS OF D SPATIAL DATA TYPES - TECHNICAL REPORT - André Borrann Coputaton Cvl Engneerng Technsche Unverstät München INTRODUCTION Startng pont for the developent of a spatal query language s the foral defnton of the seantcs of the avalable spatal data types and the operators workng on the. Such a syste of spatal data types s also referred to as spatal algebra. It captures the fundaental abstractons of spatal enttes and ther relatonshps. Our spatal type syste conssts of the four types Pont, Lne, Surface and Body. The ncorporaton of types wth lower densonalty also allows for the utlzaton of the language n the context of densonally reduced odels whch are wdely used n cvl engneerng. The defnton of the spatal types s not lted to non-curved (plane) enttes. The data types are forally defned usng pont set theory and pont set topology (Gaal, 196). Ths ethodology s well establshed thanks to the efforts of the GIS research county. Straght-forward applcaton of pure pont set theory would ply that all ponts of a densonally reduced entty, such as a pont, a lne or a surface, would belong to the boundary of ths entty. Because ths s not sutable for the specfcaton of topologcal relatonshps, we take n general another approach here: We defne a densonally reduced geoetrc obect (Lne, Surface) as beng coposed of appngs fro obects of lower densonalty (1D-Intervall -> D-Lne ; 2D-Regon -> D- Surface) whle preservng the concept of boundary / nteror / exteror durng ths appng. Ths fnally leads to ore powerful defntons that can be used for the specfcaton of topologcal relatonshps. The GIS research county dstngushes between sple and coplex spatal types. Coplex spatal data types can be understood as ult-coponent data types,.e. a Coplex Pont can consst of an arbtrary nuber of ponts, a Coplex Lne can consst of several curves and a Coplex Body ay have a nuber of unconnected parts. Though sple data types reflect ntutve understandng, they do not ncorporate the propertes of a closed type syste, where the geoetrc set operatons unon, ntersecton, and dfference never result n an obect outsde of the type syste. Therefore and because of the exstence of bodes wth holes and cavtes wthn buldngs, we decded to use coplex spatal obects. In the an, our defntons for spatal data types n D space are algned to the odel proposed by Schneder and Wenrch (2004). One excepton s the denonaton of the data types: nstead of pontd, lned and volue we use the ters Pont, Lne, Surface and Body, as proposed n (Zlatanova, 2000). Furtherore, the type relef s otted, because t s not needed for the applcaton doan consdered here. The topologcal notons of boundary ( A), nteror(a ), exteror (A ) and closure ( A ) are gven for each spatal data type. They are requred for the foral specfcaton of topologcal relatonshps. Schneder and Wenrch provde both a structured and an unstructured defnton for each type. Whereas the unstructured defnton defnes a type as a pont set satsfyng partcular condtons, t provdes no specfcaton for the boundary, nteror or exteror of ths type. Ths s provded by the structured defnton whch odels a type as

2 - 2 - beng coposed of appngs fro obects of lower densonalty. At the sae te, the structured defntons ust not be ore restrctve than the unstructured defnton,.e. all of the pont sets classfed as belongng to a certan type by ts unstructured defnton ust also be constructble by eans of the structured defnton. Because specfcatons for boundary, nteror and exteror of each type are requred, we wll provde only the structured defnton and ot the unstructured one. Schneder et al. enforce a unqueness constrant n ther structured defntons. It allows an entty to be coposed n only one specfc way, e.g. f a Lne obect can be rendered by a sngle curve t should not be possble to create the sae Lne by two curves. We consder the unqueness constrant not necessary for our purposes and do not apply t n our defntons. POINT A value of type Pont s defned as a fnte set of solated geoetrc ponts n the D space: Pont = P P s fnte} By defnton, a Pont P = p 1,...p n } has no boundary,.e. P =, and all ponts belong to the nteror, whch s equal to the closure: P = P = P. The exteror of P s P = R P. LINE A Lne s an arbtrary collecton of D curves. It s defned as the unon of the ages of a fnte nuber of contnuous appngs fro 1D to D space. In order to be able to defne the boundary of a Lne we have to nvestgate ts coponents. A Lne s coposed of several curves. A curve results fro a sngle appng f and s by defnton non-self-ntersectng. It s defned as: curve = f ([0,1]) ( ) f :[0,1] s a contnous appng ] [ } ] [ ( ) a, b 0,1 : a b f ( a) f ( b) ( ) a 0,1 b 0,1 : f ( a) f ( b)} The appngs f(0) and f(1) are called the end ponts of the curve. Condton () avods two nteror ponts of the nterval [0,1] beng apped to a sngle pont of the curve, thereby prohbtng self-ntersecton and degenerated curves consstng of only one pont. Condton () allows loops f(0) = f(1), but forbds equalty of dfferent nteror ponts and equalty of an nteror pont wth an end pont. If f(0) = f(1), the curve fors a loop and hence has by defnton no endponts. Two curves c 1, c 2 are called quas-dsont, f ther nterors do not ntersect. They are allowed to eet n the endponts 1. Forally wrtten: quas-dsont( c, c ) a, b 0,1 : f ( a) f ( b) ] [ f (0) f ( b) f (1) f ( b) f ( a) f (0) f ( a) f (1) Ths s not a curve, because curves are not self-ntersectng. two dsont curves two quas-dsont curves two quas-dsont curves 1 Schneder et al. cla that two quas-dsont curves do not for loops n order to fulfll the unqueness constrant.

3 - - The defnton of quas-dsont curves s used to ntroduce the concept of a block 2 whch specfes a connected coponent of a Lne: block = c ( ), 1 : c curve ( ) 1 < : c and c are quas-dsont ( ) > 1 1 : 1, : c and c eet Condton () ensures that all curves are utually quas-dsont. Thereby we avod that an end pont of one curve concdes wth an nteror pont of another curve. Condton () ensures that each curve s connected to at least one other curve. } a block consstng of 8 curves a block consstng of 4 curves one Lne consstng of 2 blocks A Lne can now be defned as the unon of a nuber of dsont blocks: Lne = b () n, 1 : b block ( ), 1 : b and b are dsont Condton () ensures that a curve n one block can not ntersect wth any curve n another block. We are now able to defne the nteror, exteror, boundary and closure of a Lne. The boundary of a Lne s the set of the end ponts of all quas-dsont curves t s coposed of, nus those end ponts that are shared by several curves. These shared ponts belong to the nteror of a Lne. The closure of a Lne L s the set of all ponts of L ncludng the end ponts. For the nteror we obtan L = L L = L L, and for the exteror we get L = R L. SURFACE Snce the defnton of the Surface type s based on appngs of 2D regons to D space, the defnton of the Regon2D type as provded n (Schneder, 2006) s needed frst. A Regon2D s ebedded nto the two-densonal Eucldean space R 2 and odeled as a specal nfnte pont set. The concept of the neghborhood of a pont s used to defne the nteror, exteror and closure of the Regon2D type. Assung the exstence of a Eucldean dstance functon n 2D d: wth d( p, q) = d(( x1, y1),( x2, y2)) = ( x1 x2) + ( y1 y 2) we defne the neghborhood as follows: Let q R 2 and ε R + 2. The set Nε ( q) = p d( p, q) ε} s called the (closed) neghborhood of radus ε and center q. Let X R 2 and q R 2. q s an nteror pont of X f there s a neghborhood N ε (q) such that N є (q) X. q s an exteror pont of X f there s a neghborhood N є (q) such that N ε (q) X =. q s a boundary pont of X f q s nether an nteror nor an exteror pont of X. q s a closure pont of X f q s ether an nteror or a boundary pont of X. The set of all nteror / exteror / boundary / closure ponts of X s called the nteror / exteror / boundary / closure of X. 2 Schneder et al. cla that n a block an end pont s ether the end pont of only one sngle curve or by ore than two curves n order to fulfll the unqueness constrant. }

4 - 4 - A sngle Regon2D obect. It ay have holes and consst of several coponents. Geoetrc anoales that are excluded: Isolated or danglng pont and lnes feautures and ssng pont and lne features. Schneder et al. use the noton of regular closed pont sets for the defnton of the Regon2D type n order to avod geoetrc anoales: a set of ponts X R 2 s regular closed f, and only f, X = X. The nteror operaton excludes pont sets wth danglng ponts, danglng lnes and boundary parts. The closure operaton excludes ponts sets contanng cuts and punctures whle re-establshng the boundary that was excluded by the nteror operaton. Specfcatons for bounded and connected sets are another requreent for the defnton of the Regon2D type. Two pont sets X, Y R 2 are separated f, and only f, X Y = = X Y. A pont set X R 2 s connected f, and only f, t s not the unon of non-epty separated sets. Let q = (x, y) R 2 2 and q = x + y 2. A set X R 2 s bounded f there s a nuber r R + such that q < r for all q X. The type Regon2D can now be defned as: 2 Regon2D = R R s regular closed and bounded, the nuber of connected sets of R s fnte } Ths defnton odels coplex regons as possbly consstng of several coponents and possbly possessng holes. Based on ths defnton, the Surface data type can accordngly be defned as the unon of the ages of a fnte nuber of contnuous appngs fro a 2D regon to D space. For a structured defnton, we defne the surface obect as beng coposed of a nuber of so called superfces. A superfces s consdered a non-self-ntersectng surface coponent resultng fro one sngle appng s. It ay possess holes. Let connected_regon2d Regon2D be all sngle-coponent 2D regons possbly possessng holes. Then the set of superfces s defned as: superfces = ( s R) () R connected _ regon2 D, () s: R s a contnous appng () ( x, y ), ( x, y ) R : ( x, y ) ( x, y ) s(( x, y )) s(( x, y )) (v) ( x, y ) R ( x, y ) R : s(( x, y )) s(( x, y ))} Condton () avods two nteror ponts of R beng apped to a sngle pont n the superfces, thereby prohbtng self-ntersecton and degenerated superfces consstng of only one pont or beng a lne. Condton (v) forbds boundary ponts of R beng apped to the sae pont as an nteror pont.

5 - 5 - Ths s a superfces. Ths s a partally closed superfces. The "sea" belongs to the nteror. Ths s not a superfces, because boundary ponts of the underlyng Regon2D are apped to ponts that concde wth ponts that result fro appngs of nteror ponts. Ths s not a superfces, because condton () s volated: dfferent nteror ponts of the underlyng Regon2D are apped to the sae ponts n IR³. The defnton allows superfces to be closed or partally closed. A superfces S s closed (partally closed) f sets B 1 and B 2 exst wth R = B1 B2( R B1 B2) and B1 B2 =, so that s(b 1 ) = s(b 2 ). In ths case, the set I(s)=s(B 1 ) s part of the nteror of S; otherwse I( S ) =. Gven a superfces S, ts boundary s S = s( R) I( S) and ts nteror s S = s( R ) I( S). Hence, ts closure s S = ( S) S, and ts exteror s S = S. Let T be the set of all superfces over. Two superfces S 1, S 2 T are called quasdsont f ther nterors do not ntersect. They ay share a coon boundary. Forally wrtten: quas-dsont( S, S ) s ( R ) s ( R ) = s ( R ) s ( R ) = s ( R ) s ( R ) = The defnton allows that n case one or both of the superfces s (partally) closed they ay eet at the neat. Two superfces eet f they are quas-dsont and s1( R1) s2( R2). We use the defnton of quas-dsont superfces to ntroduce the concept of a superfces block, whch s a connected coponent of a surface. A superfces block s defned as superfces _ block = S ( ), 1 : S T 1 () 1 < : S and S are quas-dsont () > 1 1 : 1, : S and S eet Condton () ensures that all superfces are utually quas-dsont. Thereby we avod that the boundary of one superfces concdes wth the nteror of another superfces, except for the sea of a (partally) closes superfces. Condton () ensures that each superfces s connected to at least one other superfces. } Schneder et al. cla that et al. cla that n a surface block, a boundary curve belongs ether to exactly one superfces or s shared or by ore than two superfces n order to fulfll the unqueness constrant.

6 - 6 - two quas-dsont surfaces a superfces block consstng of two superfces. superfces ay eet at the 'sea' of a closed superfces, a Surface consstng of 8 superfces Two surface blocks c 1, c 2 superfces _ block are dsont f, and only f, c 1 c 2 =. We are now able to provde a structured defnton for the spatal data type Surface: n Surface = c ( ) n N, 1 n : c superfces _ block ( ) 1 n : c and c are dsont Condton () ensures that superfces n one block ay not ntersect wth superfces n any other block. In order to defne the boundary of a Surface, we use the auxlary set L(S). Let S be a surface obect wth superfces S 1,..., S, and let L(S) be the set of all curves that for the boundary of ore than one superfces 4 : LS ( ) = l, 1 : l curve 1 Then the boundary of S s S = S S, and the nteror of S s L( ) S = S ( S). Hence, the closure of S s L S = S S, and the exteror of S s = S = S. ( }) } l ( S ) card S (1 l S 2 } BODY Bodes are ebedded nto the three-densonal Eucldean space R and odeled as specal nfnte pont sets. It s possble for a Body to consst of several coponents and t ay possess cavtes. The noton of the neghborhood of a pont s eployed to defne the nteror, exteror and closure of the Body type, the defnton of whch corresponds to that gven above for 2D space. Let X R and q R. q s an nteror pont of X f there s a neghborhood N ε (q) such that N ε (q) X. q s an exteror pont of X f there s a neghborhood N ε (q) such that N ε ( q) X =. q s a boundary pont of X f q s nether an nteror nor an exteror pont of X. q s a closure pont of X f q s ether an nteror or a boundary pont of X. The set of all nteror / exteror / boundary / closure ponts of X s called the nteror / exteror / boundary / closure of X. For slar reasons as for the defnton of Regon2D, the defnton of the type Body s based on the noton of regular closed pont sets: A set of ponts X R s regular closed f, and only f, X = X. Here, the nteror operaton excludes pont sets contanng solated or danglng pont, lne, and surface features. The closure operaton excludes pont sets wth punctures, cuts or strpes and reestablshes the boundary that was reoved by the nteror operaton. By defnton, closed neghborhoods are regular closed sets. Another essental s a specfcaton for bounded and connected sets n D. Two pont sets X, Y R are separated f, and only f, X Y = = X Y. A pont set X R s connected 4 Schneder et al. use a copletely dfferent defnton for L(S).

7 - 7 - f, and only f, t s not the unon of non-epty separated sets. Let q=(x, y, z) R, and 2 2 q = x + y + z 2. A set X R s bounded f there s a nuber r R + such that q < r for all q X. The spatal data type Body can now be defned as Body = B B s regular closed and bounded, the nu ber of connected sets of B s fnte } CORRESPONDING CONVEX OBJECT The specfcaton of topologcal predcates that can be used to query relatonshps of concave obects, such as surround and encopass, s based on the concept of correspondng convex obects. 2 Frst, a defnton of convexty s needed. In general, a pont set n Eucldean space, or s convex f t contans all the lne segents connectng any par of ts ponts. In order to gan a ore powerful concept of convexty for densonally reduced enttes, we use a slghtly dfferent approach here. We defne an obect as beng convex, f t s convex n t s orgnal doan space,.e. that of a Lne n R, that of a Surface n R 2 and that of a Body n R. The operaton convex returns the convex obect correspondng to a gven obect. The correspondng convex obect s defned as follows: Body The convex Body B convex correspondng to a Body B s defned as the unon of the all convex sets correspondng to the connected sets n B. For each connected set B n B there s a correspondng convex set that s the sallest set that () contans all ponts of B and () s a convex set n R. Surface The defnton of the convex Surface S convex correspondng to the Surface S s based on the defnton of the convex set correspondng to the Regon2D obect underlyng the Surface S. Let R be a Regon2D obect. For each connected set R n R there s a correspondng convex set that s the sallest set that () contans all ponts of R and () s a convex set n R 2. The convex Regon R convex correspondng to a Regon2D R s defned as the unon of all convex sets correspondng to the connected sets n R. The convex Surface S convex correspondng to the Surface S results fro the convex Regon2D R convex by applyng the sae appngs f to R convex as appled to R to yeld S. Lne A Lne s the unon of appngs of the nterval [0;1] (1D space) nto D space. The nterval [0;1] s convex n R. Therefore, each Lne s convex.

8 - 8 - REFERENCES Cleentn, E., and Felce, P. D. (1996). "A odel for representng topologcal relatonshps between coplex geoetrc features n spatal databases." Inforaton Scences, 90(1-4), Egenhofer, M., and Franzosa, R. (1991). "Pontset topologcal spatal relatons." Internatonal Journal of Geographcal Inforaton Systes, 5(2), Gaal, S. (196). Pont Set Topology. Acadec Press, New York, USA. Melton, J. (200). Advanced SQL:1999. Understandng Obect-Relatonal and Other Advanced Features, Morgan Kaufann, San Francsco, USA. Open Geospatal Consortu (OGC) (2005). OpenGIS Ipleentaton Specfcaton for Geographc nforaton. Docuent Schneder, M., and Wenrch, B. E. (2004). "An abstract odel of three-densonal spatal data types." 12th ACM Intern. Worksh. on Geographc Inforaton Systes. Washngton, DC, USA. Schneder, M., and Behr, T. (2006). "Topologcal relatonshps between coplex spatal obects." ACM Transactons on Database Systes. In Press. Zlatanova, S. (2000). "On D Topologcal Relatonshps." 11th Int. Workshop on Database and Expert Systes Applcatons, Greenwch, London, UK.

Introduction. Leslie Lamports Time, Clocks & the Ordering of Events in a Distributed System. Overview. Introduction Concepts: Time

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