Modeling and Querying History of Movement

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Modeling and Querying History of Movement

• Discrete Model

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4.3 A Discrete Model

• The task Define finite representations for all
  the types of the abstract model (and map these
  into data structures later).

Base Types int Dint int ?? ? real ... string
bool Time Type instant Dinstant real ?? ?
Spatial Types point trivial points trivial line a
finite set of line segments (which may intersect
in end-points)
abstract
discrete
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A Discrete Model

• Spatial Types (cont.)
• region

a finite set of polygons, each with polygonal
holes
Seg ... (def. of line segments) Cycle S ?
Seg ... (def. of simple polygon) Face (c,
H) c ?? Cycle, H ? Cycle, such that
... Dregion F ? Face f1, f2 ? F ? f1 ? f2
? edge-disjoint(f1, f2)
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A Discrete Model

• Type Constructors
• a an ordered domain
• range(a) A value is a finite set of disjoint,
  non-adjacent intervals over a. Each interval can
  be closed, open, or half-open
• a a standard or a spatial type
• moving(a) A value is a partial function f
  Ainstant ? Aa
• intime(a) A value is a pair from Ainstant Aa

straightforward interesting trivial
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A Discrete Model

• Representation of types moving(a) Represent the
  temporal development of the value of type a by
  decomposing the time dimension into a set of
  disjoint time intervals (slices) such that
  within each slice the development can be
  described by some simple function. Called the
  sliced representation.

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A Discrete Model

• In terms of data types, the sliced representation
  is realized by a generic type constructor mapping
  applicable to unit types, where a unit value is a
  pair, consisting of a time interval and a
  description of a simple function.

BASE ? SPATIAL ? TEMPORAL moving of abstract
model replaced by BASE ? SPATIAL ?
UNIT const ? UNIT ureal, upoint, upoints,
uline, uregion UNIT ? MAPPING mapping in the
discrete model.
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A Discrete Model - Sketch of Definitions

• Sliced representation (generic definition)
• Let (V, lt) a set with a total order.
• Interval(V)
• (s, e, lc, rc) s, e ? V, lc, rc ? bool, s ?
  e, (s e) ? (lc rc true)
• Temporal unit
• Unit(S) Interval(Instant) ? S (S a set)
• Mapping

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A Discrete Model - Sketch of Definitions

• Real Units

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A Discrete Model - Sketch of Definitions

• Point Units

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A Discrete Model - Sketch of Definitions

• Points Units

Semantics (value at instant t)
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A Discrete Model - Sketch of Definitions

• Region Units

First define a moving segment
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A Discrete Model - Sketch of Definitions

• Region Units

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A Discrete Model - Sketch of Definitions

• Line Units

Semantics (value at instant t)