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Eliseo ClementiniUniversity of L’[email protected]
2nd International Workshop on Semantic and Conceptual Issues in GIS (SeCoGIS 2008) – 20 October 2008, Barcelona
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Presentation summaryPresentation summary
1. Introduction2. The geometry of the sphere3. The 5-intersection on the plane4. Projective relations among points
on the sphere5. Projective relations among regions
on the sphere6. Expressing cardinal directions7. Conclusions & Future Work
1. Introduction2. The geometry of the sphere3. The 5-intersection on the plane4. Projective relations among points
on the sphere5. Projective relations among regions
on the sphere6. Expressing cardinal directions7. Conclusions & Future Work
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IntroductionIntroduction
• A flat Earth: – most spatial data models are 2D– models for spatial relations are 2D
•Do these models work for the sphere?• Intuitive facts on the Earth surface cannot be represented:
– A is East of B, but it could also be A is West of B (Columbus teaches!)– any place is South of the North Pole (where do we go from the North Pole?)
• A flat Earth: – most spatial data models are 2D– models for spatial relations are 2D
•Do these models work for the sphere?• Intuitive facts on the Earth surface cannot be represented:
– A is East of B, but it could also be A is West of B (Columbus teaches!)– any place is South of the North Pole (where do we go from the North Pole?)
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IntroductionIntroduction
• state of the art–qualitative spatial relations
• 2D or 3D topological relations • 2D or 3D projective relations• topological relations on the sphere (Egenhofer 2005)
•proposal • projective relations on the sphere
–JEPD set of 42 relations
• state of the art–qualitative spatial relations
• 2D or 3D topological relations • 2D or 3D projective relations• topological relations on the sphere (Egenhofer 2005)
•proposal • projective relations on the sphere
–JEPD set of 42 relations
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The geometry of the sphere The geometry of the sphere
• The Earth surface is topologically equivalent to the sphere
• Straight lines equivalent to the great circles
• For 2 points a unique great circle, but if the 2 points are antipodal there are infinite many great circles through them.
• The Earth surface is topologically equivalent to the sphere
• Straight lines equivalent to the great circles
• For 2 points a unique great circle, but if the 2 points are antipodal there are infinite many great circles through them.
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The geometry of the sphere The geometry of the sphere
• Two distinct great circles divide the sphere into 4 regions: each region has two sides and is called a lune.
• What’s the inside of a region?
• Two distinct great circles divide the sphere into 4 regions: each region has two sides and is called a lune.
• What’s the inside of a region?
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• The convex hull of a region A is the intersection of all the hemispheres that contain A
• The convex hull of a region can be defined if the region is entirely contained inside a hemisphere.
• A convex region is always contained inside a hemisphere.
• The convex hull of a region A is the intersection of all the hemispheres that contain A
• The convex hull of a region can be defined if the region is entirely contained inside a hemisphere.
• A convex region is always contained inside a hemisphere.
The geometry of the sphereThe geometry of the sphere
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• It is a model for projective relations
• It is based on the collinearity invariant
• It describes ternary relations among a primary object A and two reference objects B and C
• It is a model for projective relations
• It is based on the collinearity invariant
• It describes ternary relations among a primary object A and two reference objects B and C
The 5-intersection on the planeThe 5-intersection on the plane
BC
Between(B,C)
Rightside(B,C)
Leftside(B,C)
Before(B,C)
After(B,C)
A Leftside(B,C)
A Before(B,C)
A Between(B,C
)
A After(B,C)
A Rightside(B,
C)
A Leftside(B,C)
A Before(B,C)
A Between(B,C
)
A After(B,C)
A Rightside(B,
C)18/04/2318/04/23 88
• Special case of intersecting convex hulls of B and C
• 2-intersection
• Special case of intersecting convex hulls of B and C
• 2-intersection
The 5-intersection on the planeThe 5-intersection on the plane
BC
Inside(B,C)
Outside(B,C)
A Inside(B,C)
A Outside(B,C)
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• case of points• P1 can be between,
leftside, before, rightside, after points P2 and P3
• P1 can be inside or outside points P2 and P3 if they are coincident
• case of points• P1 can be between,
leftside, before, rightside, after points P2 and P3
• P1 can be inside or outside points P2 and P3 if they are coincident
The 5-intersection on the planeThe 5-intersection on the plane
P2P3
P1P1
P1P1
P1
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• case of points• P1 can be
between, leftside, rightside, nonbetween points P2 and P3
• Special cases:– P2, P3 coincident
» Relations inside, outside
– P2, P3 antipodal» Relations
in_antipodal, out_antipodal
• case of points• P1 can be
between, leftside, rightside, nonbetween points P2 and P3
• Special cases:– P2, P3 coincident
» Relations inside, outside
– P2, P3 antipodal» Relations
in_antipodal, out_antipodal
Projective relations for points on the sphere
Projective relations for points on the sphere
y
zls
rs
nonbtbt
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– Plain case:• External tangents exist if B
and C are in the same hemisphere
• Internal tangents exist if convex hulls of B and C are disjoint
• Relations between, rightside, before, leftside, after
– Plain case:• External tangents exist if B
and C are in the same hemisphere
• Internal tangents exist if convex hulls of B and C are disjoint
• Relations between, rightside, before, leftside, after
Projective relations for regions on the sphereProjective relations for regions on the sphere
B
C
ls
bt
rs
af
bf
A Leftside(B,C)
A Before(B,C)
A Between(B,C
)
A After(B,C)
A Rightside(B,
C)
A Leftside(B,C)
A Before(B,C)
A Between(B,C
)
A After(B,C)
A Rightside(B,
C)18/04/2318/04/23 1212
– Special cases:• reference regions B, C
contained in the same hemisphere, but with intersecting convex hulls (there are no internal tangents)
•Relations inside and outside
– Special cases:• reference regions B, C
contained in the same hemisphere, but with intersecting convex hulls (there are no internal tangents)
•Relations inside and outside
Projective relations for regions on the sphereProjective relations for regions on the sphere
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A Inside(B,C)
A Outside(B,C)
– Special cases:• reference regions B, C are not
contained in the same hemisphere, but they lie in two opposite lunes (there are no external tangents but still the internal tangents subdivides the sphere in 4 lunes)
• It is not possible to define a between region and a shortest direction between B and C
• relations B_side, C_side, BC_opposite
– Special cases:• reference regions B, C are not
contained in the same hemisphere, but they lie in two opposite lunes (there are no external tangents but still the internal tangents subdivides the sphere in 4 lunes)
• It is not possible to define a between region and a shortest direction between B and C
• relations B_side, C_side, BC_opposite
Projective relations for regions on the sphereProjective relations for regions on the sphere
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– Special cases:• If B and C’s convex hulls
are not disjoint and B and C do not lie on the same hemisphere, there are no internal tangents and the convex hull of their union coincides with the sphere.
• Relation entwined
– Special cases:• If B and C’s convex hulls
are not disjoint and B and C do not lie on the same hemisphere, there are no internal tangents and the convex hull of their union coincides with the sphere.
• Relation entwined
Projective relations for regions on the sphereProjective relations for regions on the sphere
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Projective relations for regions on the sphere
Projective relations for regions on the sphere
• The JEPD set of projective relations for three regions on the sphere is given by all possible combinations of the following basic sets:
– between, rightside, before, leftside, after (31 combined relations);– inside, outside (3 combined relations);– B_side, C_side, BC_opposite (7 combined relations);– entwined (1 relation).
• In summary, in the passage from the plane to the sphere, we identify 8 new basic relations. The set of JEPD relations is made up of 42 relations.
• The JEPD set of projective relations for three regions on the sphere is given by all possible combinations of the following basic sets:
– between, rightside, before, leftside, after (31 combined relations);– inside, outside (3 combined relations);– B_side, C_side, BC_opposite (7 combined relations);– entwined (1 relation).
• In summary, in the passage from the plane to the sphere, we identify 8 new basic relations. The set of JEPD relations is made up of 42 relations.
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Expressing cardinal directionsExpressing cardinal directions
• Set of relations (North, East, South, West) applied between a reference region R2 and a primary region R1.
• Possible mapping:– North = Between(R2, North Pole). – South = Before(R2, North Pole)– East = Rightside (R2, North Pole)– West = Leftside (R2, North Pole)– undetermined dir= After(R2, North Pole)
• Alternative mapping:– North = Between(R2, North Pole) – CH(R2)– …
• Set of relations (North, East, South, West) applied between a reference region R2 and a primary region R1.
• Possible mapping:– North = Between(R2, North Pole). – South = Before(R2, North Pole)– East = Rightside (R2, North Pole)– West = Leftside (R2, North Pole)– undetermined dir= After(R2, North Pole)
• Alternative mapping:– North = Between(R2, North Pole) – CH(R2)– …
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South
West East
North
ConclusionsConclusions• Extension of a 2D model for projective relations to the sphere
– For points, no before/after distinction– For regions, again 5 intersections plus 8 new specific relations
• Mapping projective relations to cardinal directions
• Extension of a 2D model for projective relations to the sphere– For points, no before/after distinction– For regions, again 5 intersections plus 8 new specific relations
• Mapping projective relations to cardinal directions
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• Spatial reasoning on the sphere• Refinement of the basic geometric categorization in four
directions, taking also into account user and context-dependent aspects that influence the way people reason with cardinal directions
• Integration of qualitative projective relations in web tools, such as Google Earth
• Spatial reasoning on the sphere• Refinement of the basic geometric categorization in four
directions, taking also into account user and context-dependent aspects that influence the way people reason with cardinal directions
• Integration of qualitative projective relations in web tools, such as Google Earth
Further work
Thank YouThank You
Any Questions?Thanks for your
Attention!!!Eliseo Clementini
Any Questions?Thanks for your
Attention!!!Eliseo Clementini
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