Spatial Information Systems (SIS) COMP Spatial relations

Spatial Relations Topological Relations: containment, overlapping, etc. [Egenhofer et al. 1991]Topological Relations: containment, overlapping, etc. [Egenhofer et al. 1991] Metric Relations: distance between objects, etc. [Gold and Roos 1994]Metric Relations: distance between objects, etc. [Gold and Roos 1994] Direction Relations: north of, south of, etc.Direction Relations: north of, south of, etc. [Hernandez et al. 1990; Frank et al. 1991] A B B A B A 1 Km AB

Spatial Relations Spatial objects (in a vector-based representation) can be characterised in terms of their spatial relationsSpatial objects (in a vector-based representation) can be characterised in terms of their spatial relations 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).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). When stored in a spatial database, usually, lines are approximated by polylines and regions by polygonsWhen stored in a spatial database, usually, lines are approximated by polylines and regions by polygons

Topological Relations Topological relations are defined using point-set topology concepts, such as boundary and interiorTopological relations are defined using point-set topology concepts, such as boundary and interior For example:For example: –the boundary of a region consists of a set of curves that separate the region from the rest of the coordinate space –The interior of a region consists of all points in the region that are not on its boundary Given this, two regions are said to beGiven this, two regions are said to be adjacent if they share part of a boundary but do not share any points in their interior

“Topology matters, metric refines”“Topology matters, metric refines” 4-intersection matrix for topological relations between regions (polygons)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 involvedDefined on the basis of intersections between boundary and interior of the two regions A and B involved b(A)  b(B) b(A)  i(B) i(A)  b(B) i(A)  i(B) Each entry in the matrix is either empty or non-empty Example: ¬       Topological Relations () )( AB

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 holesOf 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 4-intersection matrix (Egenhofer et al.) disjointcontainsequalinside meet coverscoveredByoverlap )(     )( ¬ ¬¬ ¬¬ ¬¬ ¬¬ ¬¬ ¬¬ ¬¬ ¬ )( ¬  ¬  )(  ¬  ¬  ¬  ¬  ¬  ¬  ¬  ¬ )( ¬    ¬   ¬  )( ¬       )( ¬  ¬   ¬   ¬  )( ¬   ¬  ¬ 

Pros:Pros: –simple model –well accepted Cons:Cons: –Does not distinguish between conceptually different situations Example:Example: All three situations correspond to the same matrixAll three situations correspond to the same matrix Pros & Cons )( ¬ ¬¬ ¬¬ ¬¬ ¬¬ ¬¬ ¬¬ ¬¬ ¬ AB A B A B

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) - adding the dimension of each component would distinguish from case (3) AB A B (1)(2) A B (3)

9-intersection matrix (Egenhofer et al.) 9-intersection matrix for topological relations between generic sets of spatial entities (not just region/region relations): considers interior, boundary, exterior9-intersection matrix for topological relations between generic sets of spatial entities (not just region/region relations): considers interior, boundary, exterior NOTE: the boundary of a line consists of its endpoints, the interior of a line consists of all points composing the line excluding its endpointsNOTE: the boundary of a line consists of its endpoints, the interior of a line consists of all points composing the line excluding its endpoints b(A)  b(B) b(A)  i(B) b(A)  e(B) i(A)  b(B) i(A)  i(B) i(A)  e(B) e(A)  b(B) e(A)  i(B) e(A)  e(B) ()

9-intersection matrix cont.d Entries in the matrix can assume values empty/non- empty or correspond to other properties as seen beforeEntries in the matrix can assume values empty/non- empty or correspond to other properties as seen before Several other variations have been definedSeveral other variations have been defined b(A)  b(B) b(A)  i(B) b(A)  e(B) i(A)  b(B) i(A)  i(B) i(A)  e(B) e(A)  b(B) e(A)  i(B) e(A)  e(B) ()

Overlayed and non-overlayed sets of entities We can consider generic sets of entities (any topological relation allowed) or sets with specific properties/structuresWe 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 entitiesOverlayed 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 entitiesAll relations are possible between pairs of entities No specific structure characterises these sets of entitiesNo specific structure characterises these sets of entities Inefficient to maintain topologyInefficient to maintain topology Layered model: use of different levels corresponding to different “meaning”Layered model: use of different levels corresponding to different “meaning” Each layer (stored separately) is an overlayed set but different layers can intersect each otherEach layer (stored separately) is an overlayed set but different layers can intersect each other Example: one layer for road network, one layer for hydrographyExample: one layer for road network, one layer for hydrography intersection (bridge)

Overlayed sets of entities If we consider overlayed sets of entities only disjoint and meet relations are possible between two polygonsIf 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 edgesOverlayed 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 f1f2 n7 n1 n2 n8 n4 n3 n5n6 n9 n10 n11 e1 = (n1,n2) e2 = …