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.

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

disconnected

concave

square

Introduction (2)

• Deriving other projective properties from collinearity

before after

between nonbetween rightside leftside

collinear aside

U

Introduction (3)

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

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

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

Collinearity among three regions (2)

coll_1(A,B,C) =def xA [yB[zC [coll(x,y,z)]]];

coll_2(A,B,C) =def xA [yB[zC [coll(x,y,z)]]];

coll_3(A,B,C) =def xA [yB[zC [coll(x,y,z)]]];

coll_4(A,B,C) =def xA [yB[zC [coll(x,y,z)]]];

coll_5(A,B,C) =def xA [yB[zC [coll(x,y,z)]]];

coll_6(A,B,C) =def xA [yB[zC [coll(x,y,z)]]];

coll_7(A,B,C) =def xA [yB[zC [coll(x,y,z)]]];

coll_8(A,B,C) =def xA [yB[zC [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)

degenaratecases

Collinear_1• coll_1(A,B,C) =def xA [yB[zC [coll(x,y,z)]]]

• symmetric

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

Collinear_2• coll_2(A,B,C) =def xA [yB[zC [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 xA [yB[zC [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 xA [yB[zC [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

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

Collinear_6 xA yB zC

Collinear_7 xA yB zC Collinear_8 xA yB zC

Collinear_4 xA yB zC

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

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

Step-wise collinearity (1)

– Step-wise collinearity can be defined for four and more regions for all the eight kinds of collinearity

),,(1_),,(1_,...),,,,(1__ DCBcollCBAcollEDCBAcollsw ...),,(1_ EDCcoll

rather weak

A

B

C

D

Step-wise collinearity (2)

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

),,(2_),,(2_,...),,,,(2__ DCBcollCBAcollEDCBAcollsw ...),,(2_ EDCcoll

“local” collinearity for each triplet, but the global arrangement may be curvilinear

N-ary collinearity (1)

• Collinear_1:

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

coll_1(A,B,C,D,E,…) =def xA yB zC tD uE,… coll(x,y,z,t,u,…)

symmetric

N-ary collinearity (2)

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

coll_2(A,B,C,D,E,…) =def xA yB zC tD uE,… coll(x,y,z,t,u,…)

• Collinear_2: » primary object A, ref. objects B,C,D,E,…

Collinearity zone with3 reference objects(set intersection of collinearity zones of pairs of reference objects)

N-ary collinearity (3)

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

Collinearity zone with3 reference objects

Collinearity zone with4 reference objects(narrower)

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= )

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

3

n

Categorizing configurations using relation collinear_1

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

• Case of four regions … k+1=5

aside(A,B,C)

coll_1(B,C,D) aside(C,D,A) aside(A,B,D)

aside(A,B,C)

coll_1(B,C,D) coll_1(C,D,A) aside(A,B,D)

aside(A,B,C)

coll_1(B,C,D) coll_1(C,D,A) coll_1(A,B,D)

coll_1(A,B,C)

coll_1(B,C,D) coll_1(C,D,A) coll_1(A,B,D)

aside(A,B,C)

aside(B,C,D) aside(C,D,A) aside(A,B,D)

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, R.Billen & E. Clementini, SebGIS 2005