Additional topologic information may be gained by describing the dimensionality of the intersection of two objects, as well as two other comparisons: encircles and surrounds. These additional comparisons are each boolean in nature, providing four combinations of which three are possible. They are defined by the enumeration, EnclosingStatus, the domain for enclosingStatus. As is described below, determination of encircles and surrounds depends on positing a third object with specific spatial topologic relationships to the two objects of interest.
intersection: this attribute describes the nature of the intersection of objects A and B in geometric terms.
empty: if they do not intersect, intersection has the value empty.
notEmpty: if they do intersect, intersection is notEmpty.
If the dimensionality of the intersection is known, intersection may take on one of the following values.
zero: the intersection is one or more discrete points.
one: the intersection is one or more arcs.
two: the intersection is one or more areas.
three: the intersection is one or more volumes. If the intersection results in objects of different dimensionality (e.g. a point and an area), as may be the case where the geometry of one or both objects is an instance of GeometricAggregate, the largest value is provided.
encircles: a portion of the boundary of A, or A itself, can serve as the boundary of a third object, C, which includes B, as defined under SpatialTopology. For example, the ring defining the boundary of a polygon encircles the polygon as well as any arbitrary set of points in the interior of the polygon.
surrounds: the (outer) boundary of A, or A itself, can serve as the boundary of a third object, C, which contains B, as defined under SpatialTopology. For example, the ring defining the boundary of a polygon surrounds any arbitrary sets of points in the interior of the polygon, but does not surround the polygon itself. If A encircles B, then it must also surround it.
If the space in which the objects reside is finite (a closed set), then the definitions above lead to the result that within that space every object with a boundary: (i), encircles every other object, and (ii), surrounds every object to which it is not adjacent. For example, the Arctic Circle may be considered to encircle all point sets to its north because a polygon can be defined covering the northern part of the earth with the Arctic Circle as its boundary. However, in a similar fashion the Arctic Circle encircles all point sets to its south. Consequently, encircles and surrounds are considered as meaningful only where the space can be reasonably treated as infinite, as would be the case with true three dimensional space or projections onto a two dimensional plane. The possible values for the attribute, enclosingStatus, as found in the enumeration EnclosingStatus, are:
notApplicable: the relationship is considered as not reasonable to determine.
noEnclosure: A does not enclose B and B does not enclose A.
encircles: A encircles B.
surrounds: A surrounds B (and also A encircles B).