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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
26/02/2013                                  2
                              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?)

26/02/2013                                          3
                                   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


26/02/2013                                          4
      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.


26/02/2013                            5
      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?




26/02/2013                       6
        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.
26/02/2013                      7
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)
26/02/2013                                                          8
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)



26/02/2013                                                    9
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



26/02/2013                                          10
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)
 26/02/2013                                                                 12
  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,
 26/02/2013                               14
     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



 26/02/2013                               15
         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.




26/02/2013                                                                     16
            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)
   – …



    26/02/2013                                                           17
                                              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


26/02/2013                                                              18
                     Thank You

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




26/02/2013                                  19

				
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