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GI Systems 1991 Egenhofer, FranzosazH AsH AH AoH As V f   )      RCCF RCC = Region Connection Calculus Randell, Cui, Cohn topological relations of polygons qualitative modelling disjoint, meet, overlap conceptual neighbourhood-diagram zH A.H A#H APH A."O   &!    RCC - diagramF  Qualitative Calculi The success of qualitative reasoning relies on choice of good qualitative calculi A q-calculus is generated by a set of jointly exhaustive and pairwise disjoint (JEPD) relations on the domain of discourse&ZT o6RCC8 Topological AlgebraRegions are nonempty regular closed subsets of a connected regular topological space (e.g. Euclidean space) Basic RCC8 relations are defined on regionsRCC8 Basic Relations DC EC PO TPP NTPP EQ disjoint meet overlap covered_by contained_by equal"Z=C4\   Directional ModelePoint-based Projection-based Cone-based Region-based Rectangle Algebra Direction-Relation Matix Model  1 $ $0,G Cone-Based Cardinal Directions !Projection-Based Card. Directions""(bFrank AU. Qualitative spatial reasoning: Cardinal directions as an example. IJGIS, 1996,10:269~290ccRectangle Algebra|The rectangle relation between two boxes A,B is described by the interval relations between the x- and y-projections of A, B&}!gDirection-Relation MatrixPR.K. Goyal, M.J. Egenhofer: Similarity of Cardinal Directions. SSTD 2001: 36-58 QQ, 7Direction-Relation Matrix Reasoning with q-CalculiConstraints Given a q-calclus R, and a relation r in R, (x r y) express a constraint between two variables x and y Composition Given two constraints (x r y) & (y s z), we want to know the strongest relation t s.t. (x t z) is possible t is called the weak composition of r and sN g  A A 7  , , BReasoning with q-CalculiGiven a constraint network N={xi rij xj}, is N satisfiable? This is the satisfaction problem. For S a subset of the whole algebra, we write RSAT(S) for the problem of deciding satisfaction of networks with constraints over S.4             .   O >! 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W~v0___PPT106___PPT9 X  bbjb" @` Y <h*?  f    @` X <F?/  ~v0___PPT106___PPT9 X  bbjb" @`V W <P? /~v0___PPT106___PPT9 \I1( b" @` V <T?W ~v0___PPT106___PPT9  QS(f,ti+1)X  bbjb"  @` U <Th?   f    @` T <u?/ ~v0___PPT106___PPT9  QS(f,ti,ti+1)z bbjbjb" @`v S <?/ ~v0___PPT106___PPT9 |I lbc( b" @`HB { C WHB | C 1W1HB } C 1HB ~ C WW1H  0޽h ? ̙33d i0  4(  x  c $P   ^B  6D>P@P 4 <̣r   QSIM{lS[|~vL:NۏL['`Nw0HQ\RYr`eQ ACTIVEh-N, 6qT͑ Y (1) (6)v ACTIVE hzz:Nbk0 {l 4.1 QSIM{l0 (1) N ACTIVEh-N Nr`0 (2) [k*NSpe clbch~bQ@b gSvlbc0 (3) [~_g-NSCQvlbcubNCQ~, NCQ~ƖT, O~_gsQ|ZPN'`nl0 (4) [ glQqQSCQv~_g, [CQ~ۏL~[, Q[~[vCQ~ZPN'`nl0    H  0޽h ? ̙33 j0 x (  x  c $P   ^B  6D>P@P  <P  (5) NiR NvCQ~ub@b gSvhQ@\ʑ0k*Nʑub N*Ner`\O:NS_MRr`vT~r`0 (6) [er`ZPhQ@\nl, iR Nvr`eQ ACTIVE h0hQ@\ nlcd NRr` " eSS`b_: Y I1, I4, I7; " _s`b_: er`Ng*NMRr`v T; " Sce`b_: gSpeP@P^  <n  nzz['`ct/f[QUOb_rbЏR'`(ۏL['`ct HQ[zzMOnSЏRe_ۏL['`h:y ۏ [QUO b_rSЏR'`(ۏLctxvzSKmRg v^\OQ; ʑ0zz['`ct/fǏ[INN~zzv^[~bُNsQ| vT|egۏLv0vMR;Nvxvz/f[zz['`^!je _0 zzb_rSsQ|v['`h:yT['`b/gvb__SI{, Nuʑt, FO;`SOeg wN㉳Q] zݍy\܏02 .H  0޽h ? ̙33 n0  @(   x   c $hP   ^B   6D>P@P"   <pT \Allenve`;[Allen 1984]NRandallvzz; [Randall 1992]~Tweg b_bzz0e0ޏ~ЏRv h;0 SYNzz['`ct>muQzzĉRt, S(uN:NN~ QUO[a[~bnN~~_gvR^, gsQel;N(uN ꁨRS0['`^!jI{W0 (WُNW]S_NNN g[EaINvbg, Y~_gn(CSP)Blt, [ E N_Yzz['`ĉR/fN*NQUO~_gn(GCSP)0  /-H   0޽h ? ̙33f o0 ((  (x ( c $pP   ^B ( 6D>P@P ( <  *1992t^1uRandellI{N^zwegvRCC zze;/f(uN [zzۏLS'`vU\g, v^] z^[s0NKuipersv QSIMel{|+B#style.visibility<*p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(+8+0+p0 + 0 x/(  xx x c $@   x x c $@   x 0@  L"0 2CX x 6W) @  NGISRUK 2004  University of East Anglia*(' F V  x 6  } Slide 6 of 20*  F "H x 0޽h ? 33___PPT10i.&0+D=' = @B +! 0 %  ))(  x  c $W)@     02@  L"0 2CX  6X4 @  NGISRUK 2004  University of East Anglia*(' F V   6x:  } Slide 7 of 20*  F "  0? p@0 h Conceptual neighbourhood diagram!IA! ! .F j  j2   f.e.e8c?"6@`NNN?N    Z.e.e1?#" ` k j2    f.e.e8c?"6@`NNN?N    Z.e.e1?#" ` M 2    f.e.e8c?"6@`NNN?N1 { 2    f.e.e8c?"6@`NNN?No =   Z.e.e1?#" ` j 2   f.e.e8c?"6@`NNN?No o= 2   f.e.e8c?"6@`NNN?N3 }   Z.e.e1?#" `k M 2   f.e.e8c?"6@`NNN?N 2   f.e.e8c?"6@`NNN?N   Z.e.e1?#" `M  2   f.e.e8c?"6@`NNN?Nh :t E2   f.e.e8c?"6@`NNN?N : E  Z.e.e1?#" `M N 2   f.e.e8c?"6@`NNN?N  2   f.e.e8c?"6@`NNN?N  B   fD8c?"0@NNN?N B  TD8c?"0@NNN?N m  B B TD8c?"0@NNN?N"m H   Z.e.e1?#" `M  E 2   f.e.e8c?"6@`NNN?N  2   f.e.e8c?"6@`NNN?Nt     Z.e.e1?#" `M 2 !  f.e.e8c?"6@`NNN?N P  \ B "  `D8c?"0@NNN?N o B #  `D8c?"0@NNN?N$ $s B $B TD8c?"0@NNN?NL r BB % TD8c?"0@NNN?N CB &  fD8c?"0@NNN?NH M B '  `D8c?"0@NNN?N  TB (  `D8c?"0@NNN?N  B )  `D8c?"0@NNN?N B  H  0޽h ? 33___PPT10i.&0+D=' = @B + 0N0  % `(    s *tRP<$0     c $Y'h <$0  *p)`=P7p,H  0޽h ? 33@ 8 ___PPT10 .. @A+wDp ' = @B D+ ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D%' =%(D' =%(Du' =A@BB BB0B%(E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<*RD' =1:Bvisible*o3>+B#style.visibility<*R%(D' =-g6B fade*<3<*RD%' =%(D' =%(Du' =A@BB BB0B%(E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<*RD' =1:Bvisible*o3>+B#style.visibility<*R%(D' =-g6B fade*<3<*R+p+0+0 ++0+0 + 0N0 h(  ~  s *kP     s *l<$0  H  0޽h ? 33N F ___PPT10& .. P{K+[W_D ' m= @B Dq ' = @BA?%,( < +O%,( < +DP' =%(D' =%(D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =-g6B fade*<3<*lD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*lDP' =%(D' =%(D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =-g6B fade*<3<*lD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*l+8+0+0 +$ 0N0 #{##(  ~  s *qP   ~  s *r      6t wG  + F -'j   -'j   # B)CDEFbԔ@@ 6 ,S{NkC*0b /QM8^l8`AVM`x]>;^nme{J>))s%@# |cX.~NL:3 .0@                     `" '  C @9#BCDEHFzԔRR<bMMd x.6g,+.HZT`puF\4O( 5b/qlcXPJIC7{qlOE5-s bOA & &}J',/*3S wofYG(=,9JJJ A<:<@                           `#" ` ~ -0  # B)CDEFbԔ@@ 6 ,S{NkC*0b /QM8^l8`AVM`x]>;^nme{J>))s%@# |cX.~NL:3 .0@                     `" }b   3 zBECDEFJԔ..i 1iEV]8h+&zS^ju|%E`j}\8'40ElM! vi "$@               `#" ` W    BxԔ" l =B 0C   BP~Ԕ" k? =A 0C   # B)CDEFbԔ@@ 6 ,S{NkC*0b /QM8^l8`AVM`x]>;^nme{J>))s%@# |cX.~NL:3 .0@                     `" o T   C >BECDEFJԔ..i 1iEV]8h+&zS^ju|%E`j}\8'40ElM! vi "$@               `#" `  `   BԔ"  =B 0C  B }Ԕ" i =A 0C  3 <B)CDEFbԔ@@ 6 ,S{NkC*0b /QM8^l8`AVM`x]>;^nme{J>))s%@# |cX.~NL:3 .0@                     `"  CT (  C  BECDEFJԔ..i 1iEV]8h+&zS^ju|%E`j}\8'40ElM! vi "$@               `#" `    HԔ"   v =B 0C  H(Ԕ"   =A 0C  # B)CDEFbԔ@@ 6 ,S{NkC*0b /QM8^l8`AVM`x]>;^nme{J>))s%@# |cX.~NL:3 .0@                     `" 6 A~&  B0Ԕ" ~ 3R  =B 0C  BԔ"    =A 0C  6Ԕ"  3 ?A=B 0C  C @wBCDEHFzԔRR<bMMd x.6g,+.HZT`puF\4O( 5b/qlcXPJIC7{qlOE5-s bOA & &}J',/*3S wofYG(=,9JJJ A<:<@                           `#" ` mj   # B)CDEFbԔ@@ 6 ,S{NkC*0b /QM8^l8`AVM`x]>;^nme{J>))s%@# |cX.~NL:3 .0@                     `" 6~  6PԔ" e =B 0C  BԔ" P7 =A 0C  6 wG    6` wG  H  0޽h ? 9ff3___PPT10i.P+D=' ,= @B +y 0N0 $(  r  S pP   r  S 4  <  c $޽h? 33y___PPT10Y+D=' ,= @B + 0N0 }(    lA ?cone-dir"`/:  ~  s *P     0   Y cFrank AU. Qualitative spatial reasoning: Cardinal directions as an example. IJGIS, 1996,10:269~290 dbH  0޽h ? 9ff3___PPT10i.pB+D=' ,= @B +i 0N0 h`(    lA proj-dir"`  ~  s *P   x  c $ q  H  0޽h ? 9ff3___PPT10i. +D=' ,= @B + 0N0   (  ~  s *$P    0 HA ?I  ~  s *xz     <Ԕ" @ ?A"0C  <4Ԕ#" ` 3]  ?B"0C  <|Ԕ" a ?A"0C  <Ԕ#" ` 0 ?B"0C   6Ԕ"   `A(o,oi)B" 0 2 C   6Ԕ" &)  _A(m,d)B"0 2CH  0޽h ? 9ff3___PPT10i.g+D=' ,= @B +e 0N0 d\@ (  x  c $<Iqu   x  c $   *  # BCDEFjԔFFWoRXx  ($.,/76?mEbeLA :10Lbhqv}_9O0( z^,GT5EfW24@                       `"     BԔ" 6  =A 0C`B  s *DԔ" ( `B  s *DԔ"   `B  s *DԔ" ?E `B   s *DԔ" ?E    6Ԕ" ` ,  For each reference object A, the plane is decomposed into 9 tiles, Ai (i=0,& ,8), with A0 the MBR of A."h0 2hC   6Ԕ"   >A5 0C   6 Ԕ" 5    >A8 0C   6Ԕ" 5   >A7 0C  6 Ԕ" 5 m  >A6 0C  6tԔ"   >A3 0C  6 Ԕ" % >A2 0C  6Ԕ" m >A1 0C  6xԔ" ? >A4 0C  6$#Ԕ"   Y "0 2CH  0޽h ? 9ff3___PPT10i.++D=' ,= @B +~ 0N0 }u` (  ~  s *4P    F y   '9  2 N ? ?y   2  # BCDEFjԔFFWoRXx  ($.,/76?mEbeLA :10Lbhqv}_9O0( z^,GT5EfW24@                       `" ;   3 BCDEFԔaa!7Qe i8 "(17?&D,K?aBv[mooo\cjp w| .;I[xx,xv x\l&P+I4;4;<?VHgH>7-Y(@  }Nl:`WCAS.L,F*?(1$$ _!DH@                                `#" ` J# #   B6Ԕ" "  =A 0C  B<Ԕ"   =B 0ChB   s *DԔ"  hB   s *DԔ" 6 6 hB   s *DԔ"  hB   s *DԔ"     3 ZBCDEFBԔ((')j'N,?/t ",?BY#<_qy~|kd`a_$"vdQ?& $,&'' @             `#" `  m   6@@Ԕ" =   =C 0C  6DԔ" ]   bxThe direction relation between A, B is expressed as a 3*3 Boolean matrix M=(mij), where mij=0 iff Ai is disjoint from B.y0 2MCK CKCKC>L  0 TA ? ?K    H  0޽h ? 9ff3___PPT10i.++D=' ,= @B + 0N0 h(  ~  s *RP     s *SPe<$0  H  0޽h ? 33F>___PPT10.. 2+FD' T= @B Di' = @BA?%,( < +O%,( < +D ' =%(D ' =%(D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<* s%(D' =-g6B fade*<3<* sD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* sD' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* sD' =%(DH' =%(D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*s%(D' =-g6B fade*<3<*sD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*sD' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*sD' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*+8+0+0 + 0N0 h(  ~  s * ^P     s * _<$0  H  0޽h ? 33___PPT10v.. +FD' l= @B D' = @BA?%,( < +O%,( < +D' =%(DH' =%(D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-g6B fade*<3<**D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<**D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<**D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<**^%(D' =-g6B fade*<3<**^D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<**^D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<**^DP' =%(D' =%(D' =AB*BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =-g6B fade*<3<*^D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*^D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*^+8+0+0 + 0N0 (    s *sP<$0     s *t<$0  H  0޽h ? 33@ 8 ___PPT10 .. /+wDp ' ,= @B D+ ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D%' =%(D' =%(Du' =A@BB BB0B%(E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<*RD' =1:Bvisible*o3>+B#style.visibility<*R%(D' =-g6B fade*<3<*RD%' =%(D' =%(Du' =A@BB BB0B%(E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<*RD' =1:Bvisible*o3>+B#style.visibility<*R%(D' =-g6B fade*<3<*R+p+0+0 ++0+0 + 0N0 <(  ~  s *\P   ~  s *4  H  0޽h ? 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zw33fYyy___PPT10i.E0sD+D=' ,= @B + 0  (  x  c $6Z  H  0޽h ? zw33fYyy80___PPT10.EwD 0  @(  x  c $I,p  H  0޽h ? 3ur3f}}80___PPT10.]~ 0  `(  x  c $\  H  0޽h ? zw33fYyy80___PPT10.ED 0  (  x  c $`k  H  0޽h ? zw33fYyy80___PPT10.ED 0  (  x  c $=  H  0޽h ? zw33fYyy80___PPT10.ED 0  (  x  c $Z  H  0޽h ? zw33fYyy80___PPT10.ED 0 *"(    BA & H  0޽h ? 3ur3f}}80___PPT10._`* 0  (  x  c $Z  H  0޽h ? zw33fYyy80___PPT10.Ep D 0   (  x  c $Z  H  0޽h ? zw33fYyy80___PPT10.E`VD 0  @(  x  c $Z  H  0޽h ? zw33fYyy80___PPT10.EPD 0 `<(  ~  s *(GP   ~  s *P<  H  0޽h ? zw33fYyy___PPT10i.E@=D+D=' ,= @B +  0  (  x  c $H,  H  0޽h ? zw33fYyy___PPT10i.E@D+D=' ,= @B + 0   (   x   c $Lp  H   0޽h ? 3ur3f}}80___PPT10.c!0k 0  ((  (x ( c $  H ( 0޽h ? zw33fYyy80___PPT10.E0iD 0  0(  0x 0 c $@  H 0 0޽h ? zw33fYyy80___PPT10.EPD 0  8(  8x 8 c $ ,  H 8 0޽h ? zw33fYyy80___PPT10.E|D 0  (  x  c $,  H  0޽h ? zw33fYyy80___PPT10.E|D 0   @(  @x @ c $$  H @ 0޽h ? zw33fYyy80___PPT10.E6D` 0 @H`(  Hx H c $&  ^ H 6j H S A ??"?F   ^ H 6j H S A ??"?F   H H 0޽h ? 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GI Systems 1991 Egenhofer, FranzosazH AsH AH AoH As V f   )      RCCF RCC = Region Connection Calculus Randell, Cui, Cohn topological relations of polygons qualitative modelling disjoint, meet, overlap conceptual neighbourhood-diagram zH A.H A#H APH A."O   &!    RCC - diagramF  Qualitative Calculi The success of qualitative reasoning relies on choice of good qualitative calculi A q-calculus is generated by a set of jointly exhaustive and pairwise disjoint (JEPD) relations on the domain of discourse&ZT o6RCC8 Topological AlgebraRegions are nonempty regular closed subsets of a connected regular topological space (e.g. Euclidean space) Basic RCC8 relations are defined on regionsRCC8 Basic Relations DC EC PO TPP NTPP EQ disjoint meet overlap covered_by contained_by equal"Z=C4\   Directional ModelePoint-based Projection-based Cone-based Region-based Rectangle Algebra Direction-Relation Matix Model  1 $ $0,G Cone-Based Cardinal Directions !Projection-Based Card. Directions""(bFrank AU. Qualitative spatial reasoning: Cardinal directions as an example. IJGIS, 1996,10:269~290ccRectangle Algebra|The rectangle relation between two boxes A,B is described by the interval relations between the x- and y-projections of A, B&}!gDirection-Relation MatrixPR.K. Goyal, M.J. Egenhofer: Similarity of Cardinal Directions. SSTD 2001: 36-58 QQ, 7Direction-Relation Matrix Reasoning with q-CalculiConstraints Given a q-calclus R, and a relation r in R, (x r y) express a constraint between two variables x and y Composition Given two constraints (x r y) & (y s z), we want to know the strongest relation t s.t. (x t z) is possible t is called the weak composition of r and sN g  A A 7  , , BReasoning with q-CalculiGiven a constraint network N={xi rij xj}, is N satisfiable? This is the satisfaction problem. For S a subset of the whole algebra, we write RSAT(S) for the problem of deciding satisfaction of networks with constraints over S.4             .   O >! Reasoning with q-CalculiThe satisfaction problems are often NP hard, e.g. RSAT(IA) and RSAT(RCC8) are NPC Reasoning problems in IA and RCC8 have been investigated thoroughly, and many useful techniques have been developed. 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GI Systems 1991 Egenhofer, FranzosazH AsH AH AoH As V f   )      RCCF RCC = Region Connection Calculus Randell, Cui, Cohn topological relations of polygons qualitative modelling disjoint, meet, overlap conceptual neighbourhood-diagram zH A.H A#H APH A."O   &!    RCC - diagramF  Qualitative Calculi  The success of qualitative reasoning relies on choice of good qualitative calculi A q-calculus is generated by a set of jointly exhaustive and pairwise disjoint (JEPD) relations on the domain of discourse*ZT o6RCC8 Topological AlgebraRegions are nonempty regular closed subsets of a connected regular topological space (e.g. Euclidean space) Basic RCC8 relations are defined on regionsRCC8 Basic Relations DC EC PO TPP NTPP EQ disjoint meet overlap covered_by contained_by equal"Z=C4\   Directional ModelePoint-based Projection-based Cone-based Region-based Rectangle Algebra Direction-Relation Matix Model  1 $ $0,G Cone-Based Cardinal Directions !Projection-Based Card. Directions""(bFrank AU. Qualitative spatial reasoning: Cardinal directions as an example. IJGIS, 1996,10:269~290ccRectangle Algebra|The rectangle relation between two boxes A,B is described by the interval relations between the x- and y-projections of A, B&}!gDirection-Relation MatrixPR.K. Goyal, M.J. Egenhofer: Similarity of Cardinal Directions. SSTD 2001: 36-58 QQ, 7Direction-Relation Matrix Reasoning with q-CalculiConstraints Given a q-calclus R, and a relation r in R, (x r y) express a constraint between two variables x and y Composition Given two constraints (x r y) & (y s z), we want to know the strongest relation t s.t. (x t z) is possible t is called the weak composition of r and sN g  A A 7  , , BReasoning with q-CalculiGiven a constraint network N={xi rij xj}, is N satisfiable? This is the satisfaction problem. For S a subset of the whole algebra, we write RSAT(S) for the problem of deciding satisfaction of networks with constraints over S.4             .   O >! Reasoning with q-CalculiThe satisfaction problems are often NP hard, e.g. RSAT(IA) and RSAT(RCC8) are NPC Reasoning problems in IA and RCC8 have been investigated thoroughly, and many useful techniques have been developed. 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