Semantics of Collinearity Among Regions

Semantics of Collinearity Among Regions R.Billen & E. Clementini, SebGIS 2005

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 TOC • Introduction • Semantics of collinearity among three regions • Semantics of collinearity of four and more regions • Categorizing configurations using relation collinear_1

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Introduction (1) • Collinearity: a basic projective property from which all other projective properties can be derived • What is projective geometry? A geometry more specific than topology and less specific than metric. • E.g., topological property: • E.g., projective property: • E.g., metric property: • Why projective geometry? Definition of many qualitative relations; e.g., right_of, after, between, surrounded_by. disconnected concave square

U collinear aside between nonbetween rightside leftside before after Introduction (2) • Deriving other projective properties from collinearity

Introduction (3) • Visual emergent features (from Pomerantz et al. 2003)

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Collinearity among three points • Collinearity among points is an elementary concept of projective geometry • Three points x,y,z are said to be collinear if they lie on the same line; we write coll(x,y,z) • It is a ternary relation: • Symmetry property: • order of arguments in the relation can be exchanged • Transitivity property: • given four points and two collinear relations holding among them, we can infer collinearity for any triplet of points out of that set of four points

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Collinearity among three regions (1) • The extension of this concept to regions is important to support relations among “extended objects”. • The concept of collinearity among regions is intrinsically approximate. • Our definition is based on the collinearity of points belonging to the three regions. • We explore various possibilities by different combinations of universal and existential quantifiers: • 8 different definitions

Collinearity among three regions (2) coll_1(A,B,C) =defxA[yB[zC[coll(x,y,z)]]]; coll_2(A,B,C) =defxA[yB[zC[coll(x,y,z)]]]; coll_3(A,B,C) =defxA[yB[zC[coll(x,y,z)]]]; coll_4(A,B,C) =defxA[yB[zC[coll(x,y,z)]]]; coll_5(A,B,C) =defxA[yB[zC[coll(x,y,z)]]]; coll_6(A,B,C) =defxA[yB[zC[coll(x,y,z)]]]; coll_7(A,B,C) =defxA[yB[zC[coll(x,y,z)]]]; coll_8(A,B,C) =defxA[yB[zC[coll(x,y,z)]]];

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Structure of relations • Collinear_1 is the weaker relation. • all the other cases are specialisations, e.g.: • coll_2(A,B,C) coll_1(A,B,C) • coll_5(A,B,C) coll_2(A,B,C) coll_3(A,B,C) degenarate cases

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Collinear_1 • coll_1(A,B,C) =defxA[yB[zC[coll(x,y,z)]]] • symmetric

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Collinear_2 • coll_2(A,B,C) =defxA[yB[zC[coll(x,y,z)]]] • Primary object A, reference objects B and C • Partially symmetric • Collinearity_2 zone • 5-intersection model

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Collinear_3 • coll_3(A,B,C) =defxA[yB[zC[coll(x,y,z)]]] • Bounded or unbounded

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Collinear_5 • coll_5(A,B,C) =defxA[yB[zC[coll(x,y,z)]]] • Same zone as coll_3 but region A must be completely contained.

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Other collinear relations Collinear_6 xAyB zC Collinear_4 xAyB zC Collinear_7 xAyB zC Collinear_8 xAyB zC

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Collinearity of more regions • step-wise collinearity • apply the definition to groups of three regions in sequence. • n-ary collinearity • different combinations of existential and universal quantifiers for points of every region

Step-wise collinearity (1) • Step-wise collinearity can be defined for four and more regions for all the eight kinds of collinearity B C A D rather weak

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Step-wise collinearity (2) “local” collinearity for each triplet, but the global arrangement may be curvilinear

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 N-ary collinearity (1) • Collinear_1: coll_1(A,B,C,D,E,…) =defxAyB zCtDuE,…coll(x,y,z,t,u,…) symmetric

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 N-ary collinearity (2) • Collinear_2: • primary object A, ref. objects B,C,D,E,… coll_2(A,B,C,D,E,…) =def xAyB zCtDuE,…coll(x,y,z,t,u,…) Collinearity zone with 3 reference objects (set intersection of collinearity zones of pairs of reference objects)

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 N-ary collinearity (3) Collinearity zone with 3 reference objects Collinearity zone with 4 reference objects (narrower)

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Categorizing configurations using relation collinear_1 • Collinearity is a high-level primitive that can be used to formulate a qualitative description of the configuration of many regions in the plane • We consider the primitive relation collinear_1 and its negation aside • For n regions, the relation collinear_1 can be checked on various combinations of three regions obtaining a range of k+1 different cases (with k= )

Ù Ù Ù Ù Ù as ide(A,B,C) aside(A,B,C) aside(A,B,C) aside(A,B,C) coll_1(A,B,C) Ù Ù Ù Ù Ù aside(B,C,D) coll_1(B,C,D) coll_1(B,C,D) coll_1(B,C,D) coll_1(B,C,D) Ù Ù Ù Ù Ù aside(C,D,A) aside(C, D,A) coll_1(C,D,A) coll_1(C,D,A) coll_1(C,D,A) aside(A,B,D) aside(A,B,D) aside(A,B,D) coll_1(A,B,D) coll_1(A,B,D) Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Categorizing configurations using relation collinear_1 • Case of four regions … k+1=5

Semantics of Collinearity Among Regions, R.Billen & E. Clementini, SebGIS 2005 Conclusions • Focus on extending the concept of collinearity from points to regions • Collinearity of more than three regions • Qualitative description of the arrangement of many regions

Semantics of Collinearity Among Regions