Spatial relation
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A spatial relation[1] specifies how some object is located in space in relation to some reference object. Since the reference object is usually much bigger than the object to locate, the latter is often represented by a point. The reference object is often represented by a bounding box (oblique or axis-parallel).
In Anatomy it might be the case that a spatial relation is not fully applicable. Thus, the degree of applicability is defined which specifies from 0 till 100% how strongly a spatial relation holds. Often researchers concentrate on defining the applicability function for various spatial relations.
In spatial databases and Geospatial topology the spatial relations are use for spatial analysis and constraint specifications.
Commonly used spatial relations are: topological, directional and distance relations.
Topological relations
- Main article DE-9IM.
The DE-9IM model express important space relations because they are formally specified, and are invariant to rotation, translation and scaling transformations.
For any two spatial objects a and b, that can be points, lines and/or polygonal areas, there are 9 relations derived from DE-9IM:
Equals | a = b Topologically equal. Also (a ∩ b = a) ∧ (a ∩ b = b) |
Disjoint | a ∩ b = ∅ a and b are disjoint, have no point in common. They form a set of disconnected geometries. |
Intersects | a ∩ b ≠ ∅ |
Touches | (a ∩ b ≠ ∅) ∧ (aο ∩ bο = ∅) a touches b, they have at least one boundary point in common, but no interior points. |
Covers | b lies in the interior of a (extends Contains). Other definitions: "no points of b lie in the exterior of a", or "Every point of b is a point of (the interior of) a". |
Contains | a ∩ b = b |
CoveredBy | Covers(b,a) |
Within | a ∩ b = a |
Directional relations
Directional relations can again be differentiated into external directional relations and internal directional relations. An internal directional relation specifies where an object is located inside the reference object while an external relations specifies where the object is located outside of the reference objects.
- Examples for internal directional relations: left; on the back; athwart, abaft
- Examples for external directional relations: on the right of; behind; in front of, abeam, astern
Distance relations
Distance relations specify how far is the object away from the reference object.
- Examples are: at; nearby; in the vicinity; far away
Relations by class
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Reference objects represented by a bounding box or another kind of "spatial envelope" that encloses its borders, can be denoted with the maximum number of dimensions of this envelope: 0 for punctual objects, 1 for linear objects, 2 for planar objectss, 2 for volumetric objects. So, any object, in a 2D modeling, can by classifyed as point, line or area according to its delimitation. Then, a type of spatial relation can be expressed by the class of the objects that participate in the relation:
- point-point relations: ...
- point-line relations:
- point-area relations:
- line-line relations:
- line-area relations:
- area-area relations:
More complex modeling schemas can represent a object as a composition of simple sub-objects. Examples: represent in a astronomical map a star by a point and a binary star by two points; represent in geographical map a river with a line, for its source stream, and with an strip-area, for the rest of the river. These schemas can use the above classes, uniform composition classes (multi-point, multi-line and multi-area) and heterogeneous composition (points+lines as "object of dimension 1", points+lines+areas as "object of dimension 2").
See also
- Anatomical terms of location
- Water-level task
- Dimensionally Extended nine-Intersection Model (DE-9IM)
References
- ^ J Freeman (1975), "The modelling of spatial relations", Computer Graphics and Image Processing, Elsevier. Freeman75.pdf[dead link]