180
Paola Podest` a, Barbara Catania, and Alberto Belussi
We denote by r t ((d 1 , d 2 ), θ 1 , (d 3 , d 4 )) the set of all relations in T REL(d 3 , d 4 ) that
are dist-based consistent with θ 1 ∈ T REL(d 1 , d 2 ) and by r c ((d 1 , d 2 ), θ 1 ,
(d 3 , d 4 )) the set of all relations in CREL(d 3 , d 4 ) which are dist-based consistent with
θ 1 ∈ CREL(d 1 , d 2 ). Dist-based consistency satisfies several useful properties.
Proposition 8.2. Dist-based consistency satisfies the following properties:
1. It is a many-to-many relationship, that is, the cardinality of r h ((d 1 , d 2 ), θ 1 , (d 3 , d 4 ))
may be greater than one, h ∈ {t, c}.
2. It is symmetric, that is, θ 2 ∈ r h ((d 1 , d 2 ), θ 1 , (d 3 , d 4 )) if and only if θ 1 ∈
r h ((d 3 , d 4 ), θ 2 , (d 1 , d 2 )), h ∈ {t, c}.
3. It is reflexive, that is, r h ((d 1 , d 2 ), θ, (d 1 , d 2 )) = {θ} and d h (θ, (d 1 , d 2 ), θ, (d 1 , d 2 ))
= 0, h ∈ {t, c}.
Concerning item (1), it can be shown that, for example, r t ((R, R), Overlap,
(L, L)) = {Overlap, Cross} (see [2]).
Depending on the type of the spatial relations considered, there exist different
relationships between eq-based and dist-based consistency notions, as pointed out
by the following proposition.
Proposition 8.3. The following properties hold:
1. For topological relationships, eq-based consistency implies dist-based consistency, that is, θ ∈ r t ((d 1 , d 2 ), θ, (d 3 , d 4 )) except when θ = T ouch and one of
the following condition holds: (i) (d 1 , d 2 ) = (R, P) and (d 3 , d 4 ) = (P, L) or (ii)
(d 1 , d 2 ) = (L, P) and (d 3 , d 4 ) = (P, R).
2. For cardinal directional relations: (a) r c ((d 1 , d 2 ), θ, (d 3 , d 4 )) = {θ} when d 3
P or θ is a single-tile relation; (b) r c ((d 1 , d 2 ), θ, (d 3 , d 4 )) ⊆ S (θ) when d 3 = P and
θ is a multi-tile relation.
Proof Sketch. Item (1) directly follows from the results presented in [2]. This strange
behavior is probably due to boundary information, quite relevant for the T ouch relationship, that are lost when transforming a region into a point. Item (2a) follows
from the fact that, under the stated condition, eq-based consistency can always be
defined and therefore d c (θ, (d 1 , d 2 ), θ, (d 3 , d 4 )) = 0. Moreover, it is easy to show that
the distance value 0 can only be obtained when the cardinal relations coincide. Item
(2b) follows from the fact that, when d 3 = P, θ CREL(d 3 , d 4 ) and relations in S (θ)
obviously minimize the distance.
8.7 Motivating Scenarios Revisited
In the following, we complete the examples introduced in Sect. 8.3 by considering
definitions and results presented in Sects. 8.4 and 8.5. To this purpose, we consider
again the three distinct maps M 1 , M 2 , and M 3 , sketched in Fig. 8.4.
Paola Podest` a, Barbara Catania, and Alberto Belussi
We denote by r t ((d 1 , d 2 ), θ 1 , (d 3 , d 4 )) the set of all relations in T REL(d 3 , d 4 ) that
are dist-based consistent with θ 1 ∈ T REL(d 1 , d 2 ) and by r c ((d 1 , d 2 ), θ 1 ,
(d 3 , d 4 )) the set of all relations in CREL(d 3 , d 4 ) which are dist-based consistent with
θ 1 ∈ CREL(d 1 , d 2 ). Dist-based consistency satisfies several useful properties.
Proposition 8.2. Dist-based consistency satisfies the following properties:
1. It is a many-to-many relationship, that is, the cardinality of r h ((d 1 , d 2 ), θ 1 , (d 3 , d 4 ))
may be greater than one, h ∈ {t, c}.
2. It is symmetric, that is, θ 2 ∈ r h ((d 1 , d 2 ), θ 1 , (d 3 , d 4 )) if and only if θ 1 ∈
r h ((d 3 , d 4 ), θ 2 , (d 1 , d 2 )), h ∈ {t, c}.
3. It is reflexive, that is, r h ((d 1 , d 2 ), θ, (d 1 , d 2 )) = {θ} and d h (θ, (d 1 , d 2 ), θ, (d 1 , d 2 ))
= 0, h ∈ {t, c}.
Concerning item (1), it can be shown that, for example, r t ((R, R), Overlap,
(L, L)) = {Overlap, Cross} (see [2]).
Depending on the type of the spatial relations considered, there exist different
relationships between eq-based and dist-based consistency notions, as pointed out
by the following proposition.
Proposition 8.3. The following properties hold:
1. For topological relationships, eq-based consistency implies dist-based consistency, that is, θ ∈ r t ((d 1 , d 2 ), θ, (d 3 , d 4 )) except when θ = T ouch and one of
the following condition holds: (i) (d 1 , d 2 ) = (R, P) and (d 3 , d 4 ) = (P, L) or (ii)
(d 1 , d 2 ) = (L, P) and (d 3 , d 4 ) = (P, R).
2. For cardinal directional relations: (a) r c ((d 1 , d 2 ), θ, (d 3 , d 4 )) = {θ} when d 3
P or θ is a single-tile relation; (b) r c ((d 1 , d 2 ), θ, (d 3 , d 4 )) ⊆ S (θ) when d 3 = P and
θ is a multi-tile relation.
Proof Sketch. Item (1) directly follows from the results presented in [2]. This strange
behavior is probably due to boundary information, quite relevant for the T ouch relationship, that are lost when transforming a region into a point. Item (2a) follows
from the fact that, under the stated condition, eq-based consistency can always be
defined and therefore d c (θ, (d 1 , d 2 ), θ, (d 3 , d 4 )) = 0. Moreover, it is easy to show that
the distance value 0 can only be obtained when the cardinal relations coincide. Item
(2b) follows from the fact that, when d 3 = P, θ CREL(d 3 , d 4 ) and relations in S (θ)
obviously minimize the distance.
8.7 Motivating Scenarios Revisited
In the following, we complete the examples introduced in Sect. 8.3 by considering
definitions and results presented in Sects. 8.4 and 8.5. To this purpose, we consider
again the three distinct maps M 1 , M 2 , and M 3 , sketched in Fig. 8.4.