semantics of collinearity among regions
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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 - PowerPoint PPT PresentationTRANSCRIPT
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