Determining Spatial Relationships

Determining Spatial Relationships

Spatial relationships indicate how two geometries interact with one another. They are a fundamental capability for querying geometry.

Dimensionally Extended 9-Intersection Model

According to the OpenGIS Simple Features Implementation Specification for SQL, "the basic approach to comparing two geometries is to make pair-wise tests of the intersections between the Interiors, Boundaries and Exteriors of the two geometries and to classify the relationship between the two geometries based on the entries in the resulting 'intersection' matrix."

In the theory of point-set topology, the points in a geometry embedded in 2-dimensional space are categorized into three sets:

Boundary : The boundary of a geometry is the set of geometries of the next lower dimension. For POINTs, which have a dimension of 0, the boundary is the empty set. The boundary of a LINESTRING is the two endpoints. For POLYGONs, the boundary is the linework of the exterior and interior rings.

Interior : The interior of a geometry are those points of a geometry that are not in the boundary. For POINTs, the interior is the point itself. The interior of a LINESTRING is the set of points between the endpoints. For POLYGONs, the interior is the areal surface inside the polygon.

Exterior : The exterior of a geometry is the rest of the space in which the geometry is embedded; in other words, all points not in the interior or on the boundary of the geometry. It is a 2-dimensional non-closed surface.

The Dimensionally Extended 9-Intersection Model (DE-9IM) describes the spatial relationship between two geometries by specifying the dimensions of the 9 intersections between the above sets for each geometry. The intersection dimensions can be formally represented in a 3x3 intersection matrix.

For a geometry g the Interior, Boundary, and Exterior are denoted using the notation I(g), B(g), and E(g). Also, dim(s) denotes the dimension of a set s with the domain of {0,1,2,F}:

• 0 => point
• 1 => line
• 2 => area
• F => empty set

Using this notation, the intersection matrix for two geometries a and b is:

Visually, for two overlapping polygonal geometries, this looks like:

Reading from left to right and top to bottom, the intersection matrix is represented as the text string '212101212'.

For more information, refer to:

• OpenGIS Simple Features Implementation Specification for SQL (version 1.1, section 2.1.13.2)
• Wikipedia: Dimensionally Extended Nine-Intersection Model (DE-9IM)
• GeoTools: Point Set Theory and the DE-9IM Matrix

Named Spatial Relationships

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.

General Spatial Relationships

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

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,*}:

• T => intersection dimension is non-empty; i.e. is in {0,1,2}
• * => don't care

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 '111****':

-- 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');