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```					                             Eliseo Clementini
University of L’Aquila
eliseo@ing.univaq.it

2nd International Workshop on Semantic and Conceptual
26/02/2013   Issues in GIS (SeCoGIS 2008) – 20 October 2008, Barcelona   1
Presentation summary
1. Introduction
2. The geometry of the sphere
3. The 5-intersection on the plane
4. Projective relations among points
on the sphere
5. Projective relations among regions
on the sphere
6. Expressing cardinal directions
7. Conclusions & Future Work
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Introduction
• 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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Introduction
• 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 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
• 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 geometry of the sphere
• 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.
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The 5-intersection on the plane
Leftside(B,C)
• It is a model for
projective
relations            Before(B,C)
C
• It is based on the                   B    Between(B,C)
After(B,C)
collinearity
invariant                          Rightside(B,C)
• It describes
ternary relations
A
among a primary                          Leftside(B,C)
object A and two
reference objects          A                A              A
B and C                Before(B,C)       Between(B,C)     After(B,C)
A
Rightside(B,C)
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The 5-intersection on the plane
Outside(B,C)
• Special case of
intersecting
convex hulls of B                                C
and C                                BInside(B,C)
• 2-intersection

A            A
Inside(B,C)   Outside(B,C)

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The 5-intersection on the plane
• case of points             P1
• P1 can be                                  P1
P3
between, leftside,              P1
before, rightside,        P2
after points P2      P1              P1
and P3
• P1 can be inside
or outside points
P2 and P3 if they
are coincident

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Projective relations for points on
the sphere
• case of points
• P1 can be                     ls
nonbt
between, leftside,            bt
z
rightside,                y
rs
nonbetween
points P2 and P3
• Special cases:
– P2, P3 coincident
» Relations
inside,
outside
– P2, P3 antipodal
» Relations
in_antipodal,
26/02/2013   out_antipodal                         11
Projective relations for regions on the sphere

– Plain case:                                      ls

• External tangents                                  bt        C    af
exist if B and C
B
bf
are in the same                                         rs
hemisphere
• Internal tangents
exist if convex
hulls of B and C                        A
are disjoint                       Leftside(B,C)
• Relations
A            A                           A
between,             Before(B,C)   Between(B,C)                  After(B,C)
rightside, before,
A
leftside, after
Rightside(B,C)
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Projective relations for regions on the sphere

– Special cases:
• reference regions
B, C contained in
the same
hemisphere, but
with intersecting
convex hulls
(there are no
internal
tangents)
•Relations                A            A
Inside(B,C)   Outside(B,C)
inside and
outside
26/02/2013
13
Projective relations for regions on the sphere

– 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,
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BC_opposite
Projective relations for regions on the sphere

– 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

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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.

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Expressing cardinal directions
• Set of relations (North, East, South, West)
applied between a reference region R2 and
North
a primary region R1.
• Possible mapping:                               West            East

–   North = Between(R2, North Pole).
–   South = Before(R2, North Pole)
–   East = Rightside (R2, North Pole)                  South
–   West = Leftside (R2, North Pole)
–    undetermined dir= After(R2, North Pole)
• Alternative mapping:
– North = Between(R2, North Pole) – CH(R2)
– …

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Conclusions
• 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
Further work
• 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

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Thank You

Any Questions?
Thanks for your Attention!!!
Eliseo Clementini
eliseo@ing.univaq.it

26/02/2013                                  19

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