Semantics of Collinearity Among Regions

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Semantics of Collinearity Among Regions

• R.Billen E. Clementini, SebGIS 2005

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

• Introduction
• Semantics of collinearity among three regions
• Semantics of collinearity of four and more
  regions
• Categorizing configurations using relation
  collinear_1

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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
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Introduction (2)

• Deriving other projective properties from
  collinearity

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Introduction (3)

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

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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

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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

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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)

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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
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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

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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

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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

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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.

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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
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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

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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
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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
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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
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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)
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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)
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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 k1 different cases
• (with k )

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Semantics of Collinearity Among Regions, R.Billen
E. Clementini, SebGIS 2005
Categorizing configurations using relation
collinear_1

• Case of four regions k15

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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