Spatial topology can be defined in a number of different ways. The approach used here was chosen because of its support for a number of queries and its grounding in set theory. The attribute spatialTopologicRelationship takes it value from an enumeration, whose values are defined below. The values are defined by direct comparisons of the primary object, termed A, to a second object, referred to as B. disjoint: A and B do not share any space.
intersects: This term is the compliment of disjoint, i.e. not disjoint.
adjacent: A and B intersect ,but their interiors are disjoint. overlaps: The interior of A is partially included in the interior of
B and vice versa.
contains: The interior of B is included entirely in the interior of
A, but the reverse is not true.
includes: All of B is part of A, but the reverse is not true. (Note
that includes is not mutually exclusive with respect to both adjacent and
contains.)
equals: The locus of points corresponding to A equals the locus of
points corresponding to B.
By the above definitions, a polygon includes an arc on its boundary, but
does not contain it; additionally, the arc and the polygon are adjacent to one
another. Of the seven relationships above, all of them are symmetric with the
exception of contains and includes.
The definitions above can be defined formally using set theory and clear
definitions of the boundary and interior of each object. These concepts in
turn are based on the notion that every object may be considered as a set of
points in space. The set of points corresponding to an object A is referred to
simply as A, and for purposes here is completely equivalent to object A.
Consider a neighbourhood around a point, within a space of the same
dimensionality as the object in question, defined such that: (i), a subset of
the points included in the neighbourhood is a subset of the points
constituting the object in question, and (ii), a subset of the points included
in the neighbourhood is not a subset of the points constituting the object.
The boundary of the object is defined as the union of the limiting case of all
such neighbourhoods as their radii approach a length of zero. (modified from
Egenhofer and Franzosa, 1991). The boundary of the object (i.e., the boundary
of A) is Ab.
The interior of an object is the set of points defining the object which
are not on the object's boundary. (The interior of A, Ai = A - Ab). An
alternative but equivalent definition is based on the concept of
neighbourhood. The interior of an object is the union of the limiting case, as
the length of the radii approach zero, of all neighbourhoods whose
corresponding sets of points are subsets of the set of points representing the
object.
The boundary and interior of objects whose geometry belongs to different
classes is provided in the table below. These results stem from the point set
approach being taken here.
Note 1: A point, ring, and surface of a volume are topologically
similar in that they all have an interior, but no boundary.
Note 2: For components of a geometric aggregate which are not
disjoint, any shared portions of space are considered as part of the interior.
Let X and Y represent the boundary or interior of A or B. That is, X may
be the boundary of A, the interior of A, the boundary of B, or the interior
of B; Y is the part of the other object to which X is being compared. When we
compare X and Y there exist three possible outcomes.
1 X and Y are disjoint, that is they do not intersect, or put another
way, they do not share any part of space.
2 X and Y intersect such that some part of X shares the same space as
some part of Y, but some part of X is outside of Y.
3 X and Y intersect such that all of X shares the same space as some
part of Y. This simply means that X is a subset of Y and X may in fact equal
Y.
Note that each of the above three set theory statements may be evaluated as
either true or false. Also, these comparisons can be applied to objects of any
dimensions in any dimensional space. The table below addresses all of the
comparisons involving two objects. The values in the table are always true or
false, and for each row, one of the values must be true and the other two
false. Note that two of the rows don't have to be completed. In addition to
boundary and interior comparisons, the objects in their entirety are compared.
The shaded areas in the preceeding table are the only comparisons which
must be completed in order to determine the higher level spatial topological
relationships. This is clarified on the following table.