Projective relations in a 3D environment

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Projective relations in a 3D environment

• Roland Billen1 Eliseo Clementini2
• 1 University of Liège (Belgium)
• 2 University of LAquila (Italy)

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TOC

• Background and motivations
• Ternary proj. relationships among points in R²
• Ternary proj. relationships among regions in R²
• Ternary proj. relationships among points in R³
• Ternary proj. relationships among bodies in R³
• Quaternary proj. relationships among points in R³
• Quaternary proj. relationships among bodies in R³
• Further research

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Background and Motivations

• Qualitative Spatial Reasoning
• 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

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Background and Motivations

• Why projective geometry?
• Definition of many qualitative relations
• Topological
• Lakes inside Scotland
• Projective
• Cities between Glasgow and Edinburgh
• Lakes surrounded by mountains
• Shops on the right of the road
• Flags above the tree
• Metric
• Edinburgh is east of Glasgow
• Edinburgh is not far from Glasgow

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Background and Motivations

• Projective invariants
• Collinearity properties
• e.g., three points belong to the same line

RO2
RO1
PO
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Background and Motivations

• We wished to extend our model in 3D
• Could be used in
• 3D GIS
• Virtual Reality
• Augmented Reality
• Robot Navigation
• Navigation in Geographic environment

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among points in
R²

• Deriving other projective properties from
  collinearity

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among points in
R²

• Partition of R² based on the two reference points
• Set of JEPD relationships (7)

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among regions in
R²

• Still based on collinearity and reference objects
  shapes
• Set of JEPD relationships (34)

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among regions in
R²
ls(A,B,C) (1 0 0 0 0 0 0), bf(A,B,C) (0 1 0
0 0 0 0)
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among regions in
R²
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among points in
R³

• Almost the same that in R²
• Except that

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among points in
R³

• The specialisation of the aside relation is not
  possible in R³
• Set of JEPD relations (6)

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among bodies in
R³

• The relation collinear among bodies is the
  generalisation of the same relation among points
• The partition of the space is based on tangent
  planes (similarity with regions in R²)

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among bodies in
R³

• A collinearity subspace can be defined
• The space is divided into a between subspace, a
  non-between subspace and an aside subspace

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among bodies in
R³

• Same basic relationships than for points
• set of JEPD relationships (18)

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Ternary projective relationships among bodies in
R³
bf(A,B,C)
bt(A,B,C)
bfas(A,B,C)
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among points
in R³

• Three non collinear points define one an only one
  plane in the space
• ? concept of coplanarity
• Such a plane (called hyperplane) divides the
  whole space in two regions , called halfspaces
• Depending on the order of the three reference
  points, the plane can be oriented in R³
• ? Positive and negative halfspaces
• Based on this partition, one can define
  projective relations between a point and three
  reference points
• ? These relations are therefore quaternary
• above, below, internal, external, inside and
  outside is a JEPD set of relations in R³ (6)

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among points
in R³
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among points
in R³
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among bodies
in R³

• The concept of coplanarity between four bodies
  can be introduced as a generalisation of the same
  relation among points
• We end up the same basic relationships than for
  points, and a set of JEPD relationships (18)

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among bodies
in R³

• To Build the coplanarity subspace
• We consider 8 internal and external tangent
  planes to the three reference bodies

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among bodies
in R³
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among bodies
in R³
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among bodies
in R³

• The all set of quaternary relations can be
  obtained based on the empty / non-empty
  intersections of the primary body A with the
  subspaces which satisfy the basic quaternary
  relations

int(A,B,C,D) (1 0 0 0 0 0), ext(A,B,C,D) (0
1 0 0 0 0), ab(A,B,C,D) (0 0 1 0 0
0), be(A,B,C,D) (0 0 0 1 0 0), in(A,B,C,D)
(0 0 0 0 1 0), ou(A,B,C,D) (0 0 0 0 0 1)
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among bodies
in R³
ext(A,B,C,D)
int(A,B,C,D)
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Quaternary projective relationships among bodies
in R³
ab(A,B,C,D)
extab(A,B,C,D)
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Further research

• (at SDH 04)
• Algorithms for the computation of projective
  relations. (done)
• Reasoning system for all ternary relations,
  composition tables and proofs. (on going)
• Extensions to n-ary relations surrounded by, in
  the middle of, etc.
• Extensions to other geometric types region/line,
  line/line, etc.
• Extensions to 3D relations.

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Further research

• (currently)
• Algorithms for the computation of projective
  relations. (done)
• Reasoning system for all ternary relations,
  composition tables and proofs. (almost done)
• Extensions to n-ary relations surrounded by, in
  the middle of, etc. (partially done)
• Extensions to other geometric types region/line,
  line/line, etc.
• Extensions to 3D relations. (done)
• Reasoning system for all quaternary relations,
  composition tables and proofs.
• Mapping these concepts to specific environment

Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Mapping in 2D
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Mapping in 3D
Projective relations in a 3D environment, Billen
R. Clementini E., GIScience 06, Muenster
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Thanks for attention Questions ????