Chapter 5. Interrogazioni spaziali

Secondo la specifica OpenGIS Simple Features Implementation Specification for SQL, "l'approccio di base per comparare due geometrie è quello di determinare l'interesezione tra gli interni, i confini e gli esterni di ognuna, classificando così la relazione tra le due geometrie attraverso la risultante matrice di 'intersezione'."

Nella teoria della topologia degli insiemi di punti, i punti di una geometria nello spazio bidimensionale sono categorizzate in 3 insiemi:

L'articolo Dimensionally Extended 9-Intersection Model (DE-9IM) descrive la relazione spatiale tra due geometrie specificando le dimensioni delle 9 intersezioni tra gli insiemi di punti sopracitati di ogni geometria. La dimensione dell'intersezione può essere formalmente rappresentata in una matrice d'intersezione 3x3.

Per una geometria g Interno, Confine e Esterno si denotano usando la notazione I(g), B(g) ed E(g). Inoltre, dim(s) denota la dimensione di un insieme s nel dominio {0,1,2,F}:

Usando questa notazione, la matrice d'intersezione per due geometrie a e b è:

Visivamente, per due geometrie poligonali parzialmente sovrapposte, avrà questo aspetto:

	Interno Confine Esterno Interno dim( I(a) ∩ I(b) ) = 2 dim( I(a) ∩ B(b) = 1 dim( I(a) ∩ E(b) ) = 2 Confine dim( B(a) ∩ I(b) ) = 1 dim( B(a) ∩ B(b) ) = 0 dim( B(a) ∩ E(b) ) = 1 Esterno dim( E(a) ∩ I(b) ) = 2 dim( E(a) ∩ B(b) ) = 1 dim( E(a) ∩ E(b) = 2

Leggendola da sinistra a destra e dall'alto verso il basso, la matrice d'intersezione viene rappresentata come la stringa di testo '212101212'.

Per maggiori informazioni si veda:

To make it easy to determine common spatial relationships, the OGC SFS defines a set of named spatial relationship predicates. PostGIS provides these as the functions ST_Contains, ST_Crosses, ST_Disjoint, ST_Equals, ST_Intersects, ST_Overlaps, ST_Touches, ST_Within. It also defines the non-standard relationship predicates ST_Covers, ST_CoveredBy, and ST_ContainsProperly.

Spatial predicates are usually used as conditions in SQL WHERE or JOIN clauses. The named spatial predicates automatically use a spatial index if one is available, so there is no need to use the bounding box operator && as well. For example:

SELECT city.name, state.name, city.geom
FROM city JOIN state ON ST_Intersects(city.geom, state.geom);

For more details and illustrations, see the PostGIS Workshop.

In some cases the named spatial relationships are insufficient to provide a desired spatial filter condition.

For example, consider a linear dataset representing a road network. It may be required to identify all road segments that cross each other, not at a point, but in a line (perhaps to validate some business rule). In this case ST_Crosses does not provide the necessary spatial filter, since for linear features it returns true only where they cross at a point. A two-step solution would be to first compute the actual intersection (ST_Intersection) of pairs of road lines that spatially intersect (ST_Intersects), and then check if the intersection's ST_GeometryType is 'LINESTRING' (properly dealing with cases that return GEOMETRYCOLLECTIONs of [MULTI]POINTs, [MULTI]LINESTRINGs, etc.). Clearly, a simpler and faster solution is desirable.

A second example is locating wharves that intersect a lake's boundary on a line and where one end of the wharf is up on shore. In other words, where a wharf is within but not completely contained by a lake, intersects the boundary of a lake on a line, and where exactly one of the wharf's endpoints is within or on the boundary of the lake. It is possible to use a combination of spatial predicates to find the required features: ST_Contains(lake, wharf) = TRUE ST_ContainsProperly(lake, wharf) = FALSE ST_GeometryType(ST_Intersection(wharf, lake)) = 'LINESTRING' ST_NumGeometries(ST_Multi(ST_Intersection(ST_Boundary(wharf), ST_Boundary(lake)))) = 1 ... but needless to say, this is quite complicated.

These requirements can be met by computing the full DE-9IM intersection matrix. PostGIS provides the ST_Relate function to do this:

SELECT ST_Relate( 'LINESTRING (1 1, 5 5)',
                  'POLYGON ((3 3, 3 7, 7 7, 7 3, 3 3))' );
st_relate
-----------
1010F0212

To test a particular spatial relationship, an intersection matrix pattern is used. This is the matrix representation augmented with the additional symbols {T,*}:

Using intersection matrix patterns, specific spatial relationships can be evaluated in a more succinct way. The ST_Relate and the ST_RelateMatch functions can be used to test intersection matrix patterns. For the first example above, the intersection matrix pattern specifying two lines intersecting in a line is '1*1***1**':

-- Find road segments that intersect in a line
SELECT a.id
FROM roads a, roads b
WHERE a.id != b.id
      AND a.geom && b.geom
      AND ST_Relate(a.geom, b.geom, '1*1***1**');

For the second example, the intersection matrix pattern specifying a line partly inside and partly outside a polygon is '102101FF2':

-- Find wharves partly on a lake's shoreline
SELECT a.lake_id, b.wharf_id
FROM lakes a, wharfs b
WHERE a.geom && b.geom
      AND ST_Relate(a.geom, b.geom, '102101FF2');