Transcript Document
Spatial Information Systems (SIS)
COMP 30110 Spatial relations
Spatial Relations
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Topological Relations: containment, overlapping, etc. [Egenhofer et al. 1991] A B A B
• •
Metric Relations: distance between objects, etc. [Gold and Roos 1994]
1 Km
A B Direction Relations: north of, south of, etc.
[Hernandez et al. 1990; Frank et al. 1991] A B
Spatial Relations
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Spatial objects (in a vector-based representation) can be characterised in terms of their spatial relations
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Spatial data in vector format are collections of points, lines, and regions (i.e., subsets of the Euclidean plane). Examples of vector datasets are thematic maps, city maps, digital terrain models (DTMs).
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When stored in a spatial database, usually, lines are approximated by polylines and regions by polygons
Topological Relations
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Topological relations are defined using point-set topology concepts, such as boundary and interior
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For example:
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the boundary of a region consists of a set of curves that separate the region from the rest of the coordinate space
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The interior of a region consists of all points in the region that are not on its boundary
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Given this, two regions are said to be adjacent if they share part of a boundary but do not share any points in their interior
Topological Relations
• • •
“Topology matters, metric refines” 4-intersection matrix for topological relations between regions (polygons) Defined on the basis of intersections between boundary and interior of the two regions A and B involved
(
b(A)
b(B) i(A)
b(B) b(A)
i(A)
i(B)
)
i(B)
Each entry in the matrix is either empty or non-empty
Example:
(
¬
)
A B
4-intersection matrix (Egenhofer et al.)
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Of the 16 (2 4 ) configurations we can obtain by assigning values empty/non-empty to each entry in the matrix only 8 are possible for regions without holes
disjoint
( )
contains
(
¬
¬
) (
inside
¬
¬
)
equal
(
¬
¬
)
meet
(
¬
)
covers
(
¬
¬
¬
)
coveredBy
(
¬
¬
¬
)
overlap
(
¬
¬
¬
¬
)
Pros & Cons
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Pros:
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simple model
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well accepted
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Cons:
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Does not distinguish between conceptually different situations
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Example:
B B A B A A
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All three situations correspond to the same matrix
(
¬
¬
¬
¬
)
Possible extension
Use different values for matrix entries - for example, number of connected components of the intersections can be used to distinguish (1) and (2)
B A B A
(1) (2) - adding the dimension of each component would distinguish from case (3)
B A
(3)
9-intersection matrix (Egenhofer et al.)
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9-intersection matrix for topological relations between generic sets of spatial entities (not just region/region relations): considers interior, boundary, exterior
(
b(A)
i(A)
e(A)
b(B) b(B) b(B) b(A)
i(A)
e(A)
i(B) b(A)
i(B) i(A)
i(B) e(A)
e(B) e(B) e(B)
) •
NOTE: the boundary of a line consists of its endpoints, the interior of a line consists of all points composing the line excluding its endpoints
9-intersection matrix cont.d
(
b(A)
i(A)
e(A)
b(B) b(B) b(B) b(A)
i(A)
e(A)
i(B) b(A)
i(B) i(A)
i(B) e(A)
e(B) e(B) e(B)
) •
Entries in the matrix can assume values empty/non empty or correspond to other properties as seen before
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Several other variations have been defined
Overlayed and non-overlayed sets of entities
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We can consider generic sets of entities (any topological relation allowed) or sets with specific properties/structures Overlayed sets: we do not allow for proper intersections among entities Non-overlayed set: A and B overlap Overlayed set: the intersection becomes a new polygon and A and B change their shape More on overlay operations later
Generic sets of entities
• • • • • •
All relations are possible between pairs of entities No specific structure characterises these sets of entities Inefficient to maintain topology Layered model: use of different levels corresponding to different “meaning” Each layer (stored separately) is an overlayed set but different layers can intersect each other Example: one layer for road network , one layer for hydrography
intersection (bridge)
Overlayed sets of entities
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If we consider overlayed sets of entities only disjoint and meet relations are possible between two polygons Overlayed sets of entities correspond to plane graphs in which we consider not only nodes (also called vertices) and edges but also the polygons (also called faces) bounded by closed cycles of edges n1 n8 e1 = (n1,n2) e2 = … n2 n9
f1 f2
n3 n10 n7 n6 n5 n4 n11