Topological Relationships for Temporal Points

Topological Relationships for Temporal Points
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The topological relationships such as ST_Intersects and ST_Relate can be generalized for temporal points. The arguments of these generalized functions are either a temporal point or a base type (that is, a geometry or geography), but these functions do not allow a base type in both arguments. Furthermore, both arguments must be of the same base type, that is, these functions do not allow to have a temporal geometry point (or a geometry) and a temporal geography point (or a geography) as arguments.

There are two versions of the temporal topological relationships:

The first version applies the traditional topological function to the union of all values taken by the temporal point (which is a geometry or geography) and returns a boolean or a text. Examples are the intersects and relate functions.
The second version is defined with the temporal semantics, that is, the traditional topological function is computed at each instant and results in a tbool or a ttext. Examples are the tintersects and trelate functions.

All spatial relationships in the two versions are defined for temporal geometry points, while only four of them are defined for temporal geography points, namely, covers, coveredby, intersects, and dwithin, and the corresponding temporal versions.

The semantics conveyed by the first version of the relationships varies depending on the relationship and the type of the arguments. For example, the following query

SELECT intersects(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
  tgeompoint '[Point(0 1)@2012-01-01, Point(1 1)@2012-01-03)');

tests whether the temporal point ever intersected the geometry, since the query is conceptually equivalent to the following one

SELECT ST_Intersects(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
  geometry 'Linestring(0 1,1 1)');

where the second geometry is obtained by applying the trajectory function to the temporal point. On the other hand, the query

SELECT contains(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
  tgeompoint '[Point(0 1)@2012-01-01, Point(1 1)@2012-01-03)');

tests whether the geometry always contains the temporal point. Finally, the following query

SELECT intersects(tgeompoint '[Point(0 1)@2012-01-01, Point(1 0)@2012-01-03)',
  tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)');

tests whether the temporal points may intersect, since the query above is conceptually equivalent to the following one

SELECT ST_Intersects('Linestring(0 1,1 0)', 'Linestring(0 0,1 1)');

The first versions of the relationships are typically used in combination with a spatio-temporal index when computing the temporal relationships. For example, the following query

SELECT T.TripId, R.RegionId, tintersects(T.Trip, R.Geom)
FROM Trips T, Regions R
WHERE intersects(T.Trip, R.Geom)

which verifies whether a trip T (which is a temporal point) intersects a region R (which is a geometry), will benefit rom a spatio-temporal index on the column T.Trip since the intersects function will automatically perform the bounding box comparison T.Trip && R.Geom. This is further explained later in this document.

Three topological relationships available in PostGIS are not provided in the temporal version.

tcontainsproperly since it would always be equal to tcontains: ST_Contains returns true if and only if no points of B lie in the exterior of A, and at least one point of the interior of B lies in the interior of A. ST_ContainsProperly returns true if B intersects the interior of A but not the boundary (or exterior).
tcrosses since it would always returns false: ST_Crosses returns true if the supplied geometries have some, but not all, interior points in common.
toverlaps since it would always returns false: ST_Overlaps returns true if the geometries share space, are of the same dimension, but are not completely contained by each other.

Similarly, only a few temporal topological relationships are meaningful when the two arguments are temporal points. Therefore, the relationships supported for two temporal geometry points are tdisjoint, tequals, tintersects, tdwithin, and trelate (with 2 and 3 arguments), while only tintersects and tdwithin are supported for two temporal geography points.

The relate and the trelate functions have two forms with either two or three arguments. The two-argument forms consider the spatial relationship between the interior, the boundary, and the exterior of the arguments and return a text or a ttext value representing the maximum intersection matrix pattern. This pattern is defined using the Dimensionally Extended 9 Intersection Model or DE-9IM (see the PostGIS documentation for more details). The three-argument forms determine whether the first two arguments satisfy the intersection matrix pattern given as third argument (a text value) and return a Boolean or a temporal Boolean.

Possible Spatial Relationships

May contain

contains({geo, tgeompoint}, {geo, tgeompoint}): boolean

SELECT contains(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)');
-- true

May contain properly

containsproperly({geo, tgeompoint}, {geo, tgeompoint}): boolean

SELECT containsproperly(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)');
-- false

May cover

covers({geo, tpoint}, {geo, tpoint}): boolean

SELECT covers(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)');
-- true

May be covered by

coveredby({geo, tpoint}, {geo, tpoint}): boolean

SELECT coveredby(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)');
-- false

May cross

crosses({geo, tgeompoint}, {geo, tgeompoint}): boolean

SELECT crosses(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(2 2)@2012-01-03)');
-- true

May be disjoint

disjoint({geo, tgeompoint}, {geo, tgeompoint}): boolean

SELECT disjoint(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)');
-- false

May be equal

equals({geo, tgeompoint}, {geo, tgeompoint}): boolean

SELECT equals(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)');
-- false

May intersect

intersects({geo, tpoint}, {geo, tpoint}): boolean

SELECT intersects(geometry 'Polygon((0 0 0,0 1 0,1 1 0,1 0 0,0 0 0))',
tgeompoint '[Point(0 0 1)@2012-01-01, Point(1 1 1)@2012-01-03)');
-- false
SELECT intersects(geometry 'Polygon((0 0 0,0 1 1,1 1 1,1 0 0,0 0 0))',
tgeompoint '[Point(0 0 1)@2012-01-01, Point(1 1 1)@2012-01-03)');
-- true

May overlap

overlaps({geo, tgeompoint}, {geo, tgeompoint}): boolean

SELECT overlaps(geometry 'Linestring(1 1,3 3)',
tgeompoint '[Point(0 0)@2012-01-01, Point(2 2)@2012-01-03)');
-- true

May touch

touches({geo, tgeompoint}, {geo, tgeompoint}): boolean

SELECT touches(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(0 1)@2012-01-03)');
-- true

May be within

within({geo, tgeompoint}, {geo, tgeompoint}): boolean

SELECT within(geometry 'LineString(1 1,2 2)',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-03)');
-- true

May be at distance within

dwithin({geo, tpoint}, {geo, tpoint}, double): boolean

SELECT dwithin(geometry 'Polygon((0 0 0,0 1 1,1 1 1,1 0 0,0 0 0))',
tgeompoint 'Point(0 2 1)@2000-01-01,Point(2 2 1)@2000-01-02', 1)
-- true
SELECT dwithin(geometry 'Polygon((0 0 0,0 1 1,1 1 1,1 0 0,0 0 0))',
tgeompoint 'Point(0 2 2)@2000-01-01,Point(2 2 2)@2000-01-02', 1)
--	false

May relate

relate({geo, tgeompoint}, {geo, tgeompoint}): text

relate({geo, tgeompoint}, {geo, tgeompoint}, text): boolean

SELECT relate(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)');
-- "1F2F01FF2"
SELECT relate(geometry 'Polygon((0 0,0 1,1 1,1 0,0 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(1 1)@2012-01-03)','1F2F01FF2');
-- true

Temporal Spatial Relationships

Temporal contains

tcontains({geo, tgeompoint}, {geo, tgeompoint}): tbool

SELECT tcontains(geometry 'Polygon((1 1,1 2,2 2,2 1,1 1))',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-04)');
-- "{[f@2012-01-01, f@2012-01-02], (t@2012-01-02, f@2012-01-03, f@2012-01-04)}"

Temporal covers

tcovers({geo, tpoint}, {geo, tpoint}): tbool

SELECT tcovers(geometry 'Polygon((1 1,1 2,2 2,2 1,1 1))',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-04)');
-- "{[f@2012-01-01, t@2012-01-02, t@2012-01-03], (f@2012-01-03, f@2012-01-04]}"

Temporal covered by

tcoveredby({geo, tpoint}, {geo, tpoint}): tbool

SELECT tcoveredby(geometry 'Point(1 1)',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-04)');
-- "{[f@2012-01-01, t@2012-01-02], (f@2012-01-02, f@2012-01-04)}"

Temporal disjoint

tdisjoint({geo, tgeompoint}, {geo, tgeompoint}): tbool

SELECT tdisjoint(geometry 'Polygon((1 1,1 2,2 2,2 1,1 1))',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-04)');
-- "{[t@2012-01-01, f@2012-01-02, f@2012-01-03], (t@2012-01-03, t@2012-01-04]}"
SELECT tdisjoint(tgeompoint '[Point(0 3)@2012-01-01, Point(3 0)@2012-01-05)',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-05)');
-- "{[t@2012-01-01, f@2012-01-03], (t@2012-01-03, t@2012-01-05)}"

Temporal equals

tequals({point, tgeompoint}, {point, tgeompoint}): tbool

SELECT tequals(geometry 'Point(1 1)',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-04)');
-- "{[f@2012-01-01, t@2012-01-02], (f@2012-01-02, f@2012-01-04]}"
SELECT tequals(tgeompoint '[Point(0 3)@2012-01-01, Point(3 0)@2012-01-05)',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-05)');
-- "{[f@2012-01-01, t@2012-01-03], (f@2012-01-03, f@2012-01-05)}"

Temporal intersects

tintersects({geo, tpoint}, {geo, tpoint}): tbool

SELECT tintersects(geometry 'MultiPoint(1 1,2 2)',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-04)');
-- "{[f@2012-01-01, t@2012-01-02], (f@2012-01-02, t@2012-01-03],
(f@2012-01-03, f@2012-01-04]}"
SELECT tintersects(tgeompoint '[Point(0 3)@2012-01-01, Point(3 0)@2012-01-05)',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 3)@2012-01-05)');
-- "{[f@2012-01-01, t@2012-01-03], (f@2012-01-03, f@2012-01-05)}"

Temporal touches

ttouches({geo, tgeompoint}, {geo, tgeompoint}): tbool

SELECT ttouches(geometry 'Polygon((1 0,1 2,2 2,2 0,1 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 0)@2012-01-04)');
-- "{[f@2012-01-01, t@2012-01-02, t@2012-01-03], (f@2012-01-03, f@2012-01-04]}"

Temporal within

twithin({geo, tgeompoint}, {geo, tgeompoint}): tbool

SELECT twithin(geometry 'Point(1 1)',
tgeompoint '[Point(0 0)@2012-01-01, Point(2 2)@2012-01-03)');
-- "{[f@2012-01-01, t@2012-01-02], (f@2012-01-02, f@2012-01-03]}"

Temporal distance within

tdwithin({geo, tpoint}, {geo, tpoint}, double): tbool

SELECT tdwithin(geometry 'Polygon((1 1,1 2,2 2,2 1,1 1))',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 0)@2012-01-04)', 1);
-- "{[f@2012-01-01, t@2012-01-02, t@2012-01-03], (f@2012-01-03, f@2012-01-04)}"
SELECT tdwithin(tgeompoint '[Point(1 0)@2000-01-01, Point(1 4)@2000-01-05]',
tgeompoint 'Interp=Stepwise;[Point(1 2)@2000-01-01, Point(1 3)@2000-01-05]', 1);
-- "{[f@2000-01-01, t@2000-01-02, t@2000-01-04], (f@2000-01-04, t@2000-01-05]}"

Temporal relate

trelate({geo, tgeompoint}, {geo, tgeompoint}, text): tbool

trelate({geo, tgeompoint}, {geo, tgeompoint}): ttext

SELECT trelate(geometry 'Polygon((1 0,1 2,2 2,2 0,1 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 0)@2012-01-04)');
-- "{[FF2FF10F2@2012-01-01, FF20F1FF2@2012-01-02, FF20F1FF2@2012-01-03],
(FF2FF10F2@2012-01-03, FF2FF10F2@2012-01-04]}"
SELECT trelate(geometry 'Polygon((1 0,1 2,2 2,2 0,1 0))',
tgeompoint '[Point(0 0)@2012-01-01, Point(3 0)@2012-01-04)', 'FF20F1FF2');
-- "{[f@2012-01-01, t@2012-01-02, t@2012-01-03], (f@2012-01-03, f@2012-01-04]}"
2012-01-04)}"

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