Metric details for natural-language spatial relations.
Spatial relations often are desired answers that a geographic information system (GIS) should generate in response to a user's query.  Current GIS's provide only rudimentary support for processing and interpreting natural-language-like spatial relations, because their models and representations are primarily quantitative, while natural-language spatial relations are usually dominated by qualitative properties. Studies of the use of spatial relations in natural language showed that topology accounts for a significant portion of the geometric properties. This article develops a formal model that captures metric details for the description of natural-language spatial relations.  The metric details are expressed as refinements of the categories identified by the   9-intersection, a model for topological spatial relations, and provide a more precise measure than does topology alone as to whether a geometric configuration matches with a spatial term or not. Similarly, these measures help in identifying the spatial term that describes a particular configuration. Two groups of metric details are derived: splitting ratios as the normalized values of lengths and areas of intersections; and closeness measures as the normalized distances between disjoint object parts. The resulting model of topological and metric properties was calibrated for 64 spatial terms in English, providing values for the best fit as well as value ranges for the significant parameters of each term.  Three examples demonstrate how the framework and   its calibrated values are used to determine the best spatial term for a relationship between two geometric objects.
Figure

1: Geometric interpretations of the 19 line-region relations that can be realized from the
9-intersection (Egenhofer and Herring 1991).
The only other topological invariant used here is the concept of the number of component. A
component is a separation of any of the nine intersections (Egenhofer and Franzosa 1995). The
number of components of an intersection is denoted by #(A. B). For example, for line-region
relation LR 14, #(L. R)  2, whereas for LR 10, #(L. R) =1.
The 19 line-region relations can be arranged according to their topological neighborhoods
(Egenhofer and Mark 1995a) based on the knowledge of the deformations that may change a
topological relation by pulling or pushing the line's boundary or interior (Figure 2). The
topological neighborhoods establish similarities that were shown to correspond to groupings
people frequently make when using a particular natural-language term (Mark and Egenhofer
1994b). For example, the term crosses was found to correspond to the five relations located in the
diagonal from the lower left to the upper right of the conceptual neighborhood diagram (LR 8 to
LR 14 in

Figure

2). Such groupings of the 9-intersection relations in the conceptual neighborhood
diagram may serve as a high-level measure to define the meaning of natural-language spatial
relations. However, topology per se may be insufficient as the only measure, particularly in
border-line cases where small metric changes have a significant influence on topology.

Figure

2: The conceptual neighborhood graph of the nineteen line-region relations(Egenhofer
and Mark 1995a).
The following sections define two metric concepts-splitting and nearness-that apply to
topological relations and may enhance each of the nineteen topological relations to distinguish more
details.
Splitting
Splitting determines how a region's interior, boundary, and exterior are divided by a line's interior
and boundary, and vice versa. To describe the degree of a splitting, the metric concepts of the
length of a line and the area of a region are used. In the context of topological relations between
lines and regions, length applies to the line's interior, any non-empty intersection with a line's
interior, or their components; and to region boundaries, any non-empty intersection between a
region's boundary and a line's exterior, or their components. Area applies to the interior or
regions, the intersections between a line's exterior and a region's interior or exterior, and their
components. Among the entries of the 9-intersection for a line and a region, there are seven
intersection that can be evaluated with a length or an area (Table 1). Only the three intersections
between the line's boundary cannot be evaluated with a length or area measure, because these
intersections are 0-dimensional (i.e., points).
R R R-
length(L. R) length(L. R) length(L. R- )
L- area(L- . R) length(L- . R) area(L- . R- )

Table

1: Area and length measures applied to the nine intersections of the line's interior
boundary (L), and exterior ( L- ) with the region's interior ( R), boundary
(R), and exterior (R- ).
To normalize these lengths and areas, each of them is put into perspective with the line and the
region: The two area intersections are compared with the area of the region, resulting in two
splitting measures. Another ten splitting measures are obtained by comparing the four length
intersections with the length of the line, and the length of the region's perimeter.
3.1 Inner Area Splitting
Inner area splitting describes how the line's interior divides the region's interior. With this
separation a one-dimensional object splits a two-dimensional object into two (or more) parts such
that parts of the region's interior are on one side of the line, and others are located on the opposite
side of the line (Figure 3). Inner area splitting only applies to a subset of the 19 region-line
relations. Those relations for which the line's interior intersects with the region's interior
L. R=  ), but the line's boundaries is outside of the region's interior (L . R=  ),
always have a value for inner area splitting. In addition, inner area splitting may apply if the line's
interior intersects with the region's boundary and interior ( L. R=  and L.
and the line's boundary intersects with the region's interior (L . R=  ). In such situations it
is necessary that there are more components in the interior-interior intersection than there are
components of the intersection between the line's boundary and the region's interior, i.e.,
#(L. R) > #(L . R).

Figure

3: Inner area splitting: the line's interior divides the region's interior into parts on two
opposite sides (more complex configurations may have multiple separations on
either side of the line).
A normalized measure of this property is the inner areasplitting ratio (IAS) as the smaller sum
of the areas on either side of the line-left and right are chosen arbitrarily and their choice does not
influence the measure-over the total area of the region (Eq. 2). The range of IAS is
It would reach 0 if the interior-interior intersection between the line and the region
was empty, and is 0.5 if the line separates the region's interior into areas that total the same size on
the left-hand side and the right-hand size.
min(area(leftComponents(L- . R)), area(rightComponents(L- . R)))
area(R)
3.2 Outer Area Splitting
Outer area splitting occurs if the line's interior interacts with the exterior of the region such that it
produces separations of the exterior between the interior of the line and the boundary of the region.
This involves a one-dimensional object that splits a two-dimensional object (the region's exterior)
into two (or more) two-dimensional parts: (1) parts of the region's exterior that are bounded
because they are completely surrounded by the line's interior and the region's boundary, and (2)
parts of the region's exterior that are unbounded (Figure 4).

Figure

4: Outer area splitting: the line's interior divides the region's exterior into bounded and
unbounded areas (more complex configurations may have multiple areas that are
bounded by the same line).
Outer area splitting requires that the line's interior intersects with the region's exterior
and that the line's boundary is located in the region (L .
splitting also may apply to configurations for which line interiors intersect with both the region's
interior and boundary ( L. whose line boundaries intersect
with the region's exterior (L .  For these situations, it is necessary that the region's
exterior contains more components of the line's interior than of the line's boundary
(#(L. R- ) > #(L . R-)). A normalized measure of outer area splitting is the outerareasplitting
ratio (OAS) as the ratio of the sum of the region's area and the bounded exterior, which is the part
of the exterior that is enclosed by the line's interior and the region's boundary, over the region's
area (Eq. 3). It is greater than zero such that the larger the bounded area, the larger the splitting
ratio. It would reach 0 if the bounded area was non-existent (i.e., either an empty intersection
between the line's interior and the region's exterior, or an insufficient number of components in the
intersection between the line's interior and the region's exterior.
area(boundedComponents(L- . R- )
area(R)
3.3 Inner Traversal Splitting
The region's interior separates the line's interior into inner and outer line segments. This involves a
two-dimensional object splitting a one-dimensional object into two one-dimensional parts (or sets
of parts): line parts that are inside the closure of the region, and line parts outside of the region

Figure

5).

Figure

5: Inner traversal splitting: the region's interior divides the line into parts of inner and
outer segments (more complex configurations may have multiple inner and outer
segments for a line).
Inner traversal splitting applies to relations in which the line's interior is located at least partially
in the region's interior ( L. R=  ). A normalized measure for the traversal is the inner
traversal splitting ratio (ITS) between the length of the inner parts of the line and the length of the
total line (Eq. 4). Its range is 0 < ITS 1. ITS would be 0 if the interior-interior intersection
between the line and the region was empty. The greatest value is reached if the line's interior is
completely contained in the region's interior.
length(L. R)
length(L)
3.4 Entrance Splitting
While the inner traversal splitting normalizes the common interiors with respect to the line's length,
the entrance splitting compares the length of the common interiors to the length of the region's
boundary. It applies under the same conditions as the inner traversal splitting. Its measure, called
the entrancesplittingratio (ENS), captures how far the line enters into the region (Eq. 5). All
values of the entrance splitting ratio are greater than zero, but no upper bound exists.
length(L. R)
length(R)
3.5 Outer Traversal Splitting
While the inner traversal splitting describes how much of the line is in the region's interior, the
outer traversal splitting refers to the part of the line that is in the region's exterior. Outer traversal
splitting applies to relations in which the line's interior is located at least partially in the regions'
exterior (L. normalized measure for the traversal is the outer traversal splitting
ratio (OTS) between the length of the outer parts of the line and the length of the total line (Eq. 6).
length(L. R- )
length(L)
3.6 Exit Splitting
Analog to the pair of inner traversal splitting and entrance splitting, the outer traversal splitting has
a dual, the exit splitting. It captures how far the line exits the region, and applies under the same
conditions as the outer traversal splitting. The exit splitting ratio (EXS) normalizes the length of the
line's interior that lays in the region's exterior with respect to the length of the region's boundary
(Eq. 7). It is greater than 0 and has no upper bound.
length(L. R- )
length(R)
3.7 Line Alongness
The region's boundary interacts with the line's interior such that it separates the line into two sets
of line parts: line segments that are outside of the region's boundary (i.e., either in the region's
interior or exterior), and line segments that are contained in the boundary. This separation makes a
one-dimensional object splitting another one-dimensional object into two or more one-dimensional
parts (

Figure

6).

Figure

Line alongness: the region's boundary separates the line's interior into parts of
outer and inner segments (more complex configurations may have multiple
components in the intersection between the region's boundary and the line's
interior).
In order to consider line alongness, the line's interior must intersect with the region's boundary
As the measure for the separation, we introduce the notion and concept line
alongness ratio (LA) as the ratio between the length of all line parts contained in the boundary, and
the total length of the line (Eq. 8). The range of the line alongness ratio is 0  LA 1. LA is 0 if
the line intersects the region's boundary exclusively in 0-dimensional components, and it reaches 1
if L R.
length(L. R)
length(L)
3.8 Perimeter Alongness
The line's interior separates the region's boundary into two sets of objects, one that coincides whit
the line's interior, and another that is disjoint from the line's interior. The separation is such that a
one-dimensional object splits another one-dimensional object into two (or more) one-dimensional
objects. The perimeter alongness can be measured for relations in which the line's interior
intersects with the region's boundary ( L. R = ). The perimeter alongness is measured by
the ratio between the length of coinciding parts between the line's interior and the region's
boundary and the perimeter, called the perimeteralongnessratio (PA) (Eq. 9). The range of the
perimeter alongness ratio is 0  PA < 1. PA is 0 if the interior-boundary intersection between the
line and the region consists exclusively of disconnected 0-dimensional components. PA would
reach the maximum of 1 if cycles were permitted as lines and such a cycle would coincide with the
region's boundary.
length(L. R)
length(R)
3.9 Perimeter Splitting
Perimeter splitting occurs if the line splits the region's boundary into two or more parts. This
involves two (or more) zero-dimensional or one-dimensional objects-the line's boundary or
interior-cutting another one-dimensional object (the region's boundary) (Figure 7).

Figure

7: Perimeter splitting: the line separates the region's boundary into segments (more
complex configurations may create multiple segments in the region's boundary).
Perimeter splitting requires that the line intersects the region's boundary
such that the region's boundary is split into at least two
components (#(R- L) 2). The perimeter splitting ratio (PS) is the ratio between the longest of
these components and the region's perimeter (Eq. 10). Its range is 0 < PS < 1.
max(length(components(L- . R)))
length(R)
3.10 Length Splitting
While the perimeter splitting compares the length of the longest perimeter component with the total
length of the perimeter, the length splitting compares it with the length of the line. The metric
measure is the line splitting ratio (LS) (Eq. 11), which is great than 0 without an upper bound.
max(length(components(L- . R)))
length(L)
3.11 Comparison of the Splitting Ratios
Each splitting ratio applies to several different topological relations. Figure 8 shows how the
criteria for the ten splitting ratios map onto the conceptual neighborhood graph of the line-region
relations (Egenhofer et al. 1993). Each constraint covers a contiguous area.
IAS ITS and ENS LA and PA
OAS OTS and EXS LS and PS

Figure

8: The relations that qualify for inner area splitting (IAS), outer area splitting (OAS),
inner traversal splitting (ITS), entrance splitting (ENS), outer traversal splitting
(OTS), exit splitting (EXS), line alongness (LA), perimeter alongness (PA), line
splitting (LS), and perimeter splitting (PS). Black, gray, and white indicate that the
metric measure applies always, sometimes, and never, respectively.
4 Closeness
Unlike splitting, which requires coincidence and describes how much is in common between two
objects, closeness describes how far apart disjoint parts are. The object parts involved are the
boundary and the interior of the line, and the boundary of the region. There is no need to consider
the region's interior, since it is delineated by its boundary, and therefore no additional information
in could be found by considering it in addition to the region's boundary.
Closeness involves considerations of distances among points and lines. For the configurations
considered, there are four types of closeness measures of interest (the metric axioms for distances
apply, i.e., there is a null element, distances are symmetric, and the triangle inequality holds):
(1) the distance between a line's boundary and the region's boundary if the line's boundary is
located in the exterior of the region;
(2) the distance between a line's boundary and the region's boundary if the line's boundary is
located in the interior of the region;
(3) the distance of the shortest path between a line's interior and the region's boundary if the
line's interior is located in the exterior of the region; and
(4) the distance of the shortest path between a line's interior and the region's boundary if the
line's interior is located in the interior of the region.
The closeness measures are not completely orthogonal, since depending on the shape of the
line or the region, they may have the same values. For instance, for the configuration in Figure 9a,
the distance from the region's boundary to the line's boundary (i.e., its two endpoints as defined in
Section 2) is the same as the distance from the region's boundary to the line's interior, since the
line's boundary is the line's closest part to the region's boundary; however, in Figure 9b, the same
parameters have different values because the line's interior is closer to the region's boundary than
the line's boundary.
(a) (b)

Figure

9: Two configurations with (a) identical and (b) different values for the distance
measures from the line's boundary and interior to the region's boundary.
Distances are commonly defined between points; however, the closeness measures require
distance measures between a point and a line, or between two lines.
1: The distance between a point p and the boundary of a region (R) is defined as
the length of the shortest path from p to R (Eq. 12).
$/ q  R | dist( p,q) < dist( p,r)
Therefore, there is no other point on the region's boundary that would be closer to p
Definition 2: The distance between the interior of a line ( L) and the boundary of a region
(R) is defined as the length of the shortest path from L to R (Eq. 13).
Therefore, there are no other parts in the line's interior that would be closer to any point on the
region's boundary
4.1 Outer Closeness
The outer closeness describes the remoteness of the region's boundary R from p, a boundary
point of a line located in the exterior of the region (Figure 10a). Outer closeness only applies to
those line-region relations with at least one point of the line's boundary being located in the
region's exterior (L . purely quantitative measure for the remoteness would be the
distance between the region's boundary and the line's boundary point(s) in the region's exterior

Figure

10b). It is the shortest connections between the line's boundary and the region, i.e., there
exists no other point in the region's boundary that would be closer to the line's boundary (Eq. 14).
Since this measure is only applicable if L . can never be 0.
(a) (b) (c)

Figure

10: Outer closeness: (a) the line's boundary in the region's exterior, (b) the remoteness
measure BE from the region's boundary to the line's boundary, and (c) the
region's outer buffer zone as an equi-distant enlargement of the region.
While the actual distance between the two boundaries is a precise measure, it varies
significantly with the scale of the representation. For instance, a scaling by a factor of 2 would
make any two objects be twice as much remote. A variety of dimension-independent measures
could be thought of, such as the proportion by which the line would have to be extended, or
shrunken, so that its boundary coincides with the region's boundary. We selected two outer
closeness measures: (1) the outer line closeness as the ratio between the distance from the line's
boundary to the region's boundary, and the line's length (Figure 10b), and the outer area closeness
as the ratio between the area made up by an equi-distant enlargement of the region-also known as
a buffer zone (Laurini and Thompson 1992)-and the actual area (Figure 10c).
We define the outer area closeness measure (OAC) in terms of the area of the region R and the
area made up by the buffer zone, denoted by D (R). It is of width BE and extends into the
region's exterior (Eq. 15). OAC is greater than 0 with no upper bound, and would be 0 if BE
were 0. The normalization area(D (R))/(area (D (R))+area (R)) would produce values between
0 and 1, however, the distribution would be non linear, particularly for area(R)  area(D (R) ).
area(D (R))
area(R)
The outer line closeness measure (OLC) is defined in terms of BE, the distance from the line's
boundary to the region's boundary, and the line's length (Eq. 16). Its values are greater than 0
without an upper bound. It would be 0 if BE were 0.
length(L)
4.2 Inner Closeness
Analogous to the outer closeness, the inner closeness captures the remoteness of the line's
boundary, located in the interior of the region (criterion: L . R=  ), from the region's
boundary (Figure 11a). The mere distance between the boundaries of the region and the line are
captured by a quantitative measure BI (Eq. 17). This distance is greater than 0, because the line's
boundary must be located in the region's interior. If both boundary points of the line are inside R,
then BI is the distance of that boundary point closest to the region's boundary.
BI
BI
DBI
(a) (b)

Figure

11: Inner closeness: (a) the line's boundary in the region's interior and (b) the region's
inner buffer zone as an equi-distant reduction of the region.
min(dist ( p,R)) | p  (L . R) (17)
The innerareacloseness (IAC) is then defined as the ratio between the area made up by an
equi-distant reduction of the region and the actual area (Figure 11b). The buffer zone D (R) has
BI
the width b and is taken from the region's boundary into the region's interior (Eq. 18). Its rage is
D (R)
area(R)
The inner line closeness (ILC) refers to the relative amount the line has to be extended or
shortened to coincide with the region's boundary. The increment is normalized with respect to the
line's actual length (Eq. 19).
BI
length(L)
4.3 Outer Nearness
The outer nearness describes how far the line's interior is from the region's boundary (Figure
12a). It only applies to one line-region relation, namely the one with the line's boundary and
interior completely contained in the region's exterior (L  R- ). The quantitative measure for outer
nearness is the length of the shortest connection between the line and the region (Figure 12b). It is
always greater than zero, because L must be completely contained in R's exterior (Eq. 20).
(a) (b) (c)

Figure

12: Outer nearness: (a) the line is completely contained in the region's exterior, (b) the
remoteness measure IE from the region's boundary to the line's interior, and (c) the
region's outer buffer zone as an equi-distant enlargement of the region.
The outerareanearness (OAN) is then defined as the ratio between the area made up of an
equi-distant reduction of the region of width IE, denoted by D (R) , and the actual area of the
II
region R (Eq. 21). OAN's values are greater than 0, with no upper bound. OAN would be 0 if IE
were 0.
area(D (R))
area(R)
The outer line nearness (OLN) normalizes the length by which the line would have to be
extended or shortened such that its boundary would coincide with the region's boundary, with
respect to the length of the initial line (Eq. 22). The values of the outer line nearness are greater
than 0 and increase linearly with the length of IE.
length(L)
4.4 Inner Nearness
Complementary to the outer nearness, the inner nearness describes how far the line's interior,
located in the interior of the region (criterion: L  R), is from the region's boundary (Figure 13a)
This distance is greater than zero, because the line must be completely contained in the region's
interior (Eq. 23).
I
DII
(a) (b)

Figure

13: Inner nearness: (a) the line completely contained in the region's interior and (b) the
region's inner buffer zone as an equi-distance reduction of the region.
The inner area nearness (IAN) is then defined as the ratio between the area made up by a buffer
zone of width II, denoted by D (R) , that extends from the boundary into the region's interior
II

Figure

13b). Its range is 0 < IAN <1 (Eq. 24).
area(D (R))
area(R)
The inner line nearness (ILN) captures by how much the line would have to be extended in
order to intersect with the region's boundary. It is measured as the ratio between the distance to the
region's boundary and the length of the line (Eq. 25). The values of the inner line nearness must be
greater than zero.
II
length(L)
4.5 Comparison of the Closeness Measures
From the criteria for the closeness measures, one can derive which topological relations may be
refined by the corresponding measures (Figure 14). Except for the six topological relations in the
bottom triangle of the neighborhood graph, all relations have at least one closeness measure. Those
six relations without a closeness measure are such that both line boundaries coincide with the
region's boundary, therefore, the distances from the line's parts to the boundaries region are all
zero and no refinements can be made to these relations.
5 Parsing and Translating a Graphical Relation into a Verbal
Expression
With the two sets of parameters we can perform a detailed analysis of a simple spatial configuration
with a line and a region, capturing the configuration's topology and analyzing it according to its
metric properties. This per se would provide the basis for a computational comparison of two or
more spatial configurations for similarity (Bruns and Egenhofer 1996). Here we pursue a different
path by mapping the parsed configuration onto a natural-language term that would best describe the
spatial relation between the two geometric objects. For the time being, any semantic or
presentational aspects (Mark et al. 1995) are ignored in this mapping.
The mappings from the topological and metric measures onto corresponding natural-language
terms are based on results from human-subject experiments (Shariff 1996). A total of sixty-four
English-language terms were tested, for which subjects sketched a road with respect to a given
outline of a park such that the sketch would match the corresponding natural-language term that
describes the spatial relation. By analyzing the sketches' topological relations and their splitting and
closeness measures, we obtained the mappings from the geometry of a configuration onto the
corresponding, significant parameters and their values. Significant parameters were distinguished
from non-significant ones through a cluster analysis (Shariff 1996). The criterion for a parameter
to be considered significant for a specific spatial term was that its standard score was greater than
one (i.e., the mean of such a parameter is at least one standard deviation higher than the mean of
the entire data set). To demonstrate how the model developed here can be used for such
translations, we give three examples in which the spatial relation of a geometric configuration is
translated into a natural-language spatial term.
5.1 Example 1

Figure

15 shows a configuration in which a line (e.g., a road) crosses the boundary of a region
(e.g., a park). Based on the topology (LR 18), the applicable metric parameters for splitting and
closeness are found in Figures 8 and 14, respectively.
IAC and ILC IAN and ILN
OAC and OLC OAN and OLN

Figure

14: The relations that qualify for inner area closeness (IAC), inner line closeness (ILC),
outer area closeness (OAC), outer line closeness (OLC), inner area nearness (IAN),
inner line nearness (ILN), outer area nearness (OAN), and outer line nearness
(OLN).
The human-subject tests found that only a subset of these parameters-inner traversal splitting,
outer traversal splitting, inner area closeness, and outer area closeness-are significant for the
terms that are represented by LR 18. Table 2 shows a sample of eight terms-ends at, ends in,
ends just inside, ends outside, enters, goes into, goes out, and goes to-that apply to LR 18,
together with the significant parameters. For each parameter, the mean value (i.e., the best fit) and
the range of values is given. The value range of a metric parameter refers to the minimum and
maximum value obtained from the subjects' sketches. The goal is now to determine which of these
terms are a better match for the particular configuration, and which do not convey the meaning the
meaning of the configuration.
Topological
Relation
Spatial Term ITS ETS IAC OAC
mean range mean range mean range mean range
LR ends at 0.24 0.02-0.65 0.76 0.36-0.98 0.79 0.19-0.98 7.69 2.09-20.73
LR ends in 0.51 0.17-0.91 0.49 0.09-0.83 0.81 0.18-0.99 3.66 0.59-9.94
LR just inside 0.16 0.05-0.71 0.84 0.29-0.95 0.55 0.05-0.89 3.95 0.86-10.27
LR
LR
LR
LR
LR goes to 0.20 0.04-0.57 0.81 0.44-0.96 0.75 0.18-0.99 7.37 1.63-12.04

Table

2: Spatial terms of topological relation LR 18, with means and value ranges of their
significant parameters for splitting and closeness measures.

Table

3 summarizes for the four parameters how they are calculated and provides the values
obtained for the configuration in Figure 15.
inner traversal
splitting
length(L)
I
R
outer traversal
splitting
outer area
closeness
length(L)
LI
R
area(D
area(R)
R
inner area area(D )
closeness
area(R)
BI
R
BI

Table

3: Calculating the inner traversal splitting, the outer traversal splitting, the outer area
closeness, and the inner area closeness for the configuration displayed in Figure 15.
By comparing these values with the calibrated model, the terms are ranked according to best fit.
The terms ends in, ends outside, enters, goes into, and goes out fall outside of the value ranges of
at least two parameters (Table 2) and, therefore, these terms are not considered for this
configuration. Among the remaining three terms, ends just inside is the best fit for three
parameters; goes to is the second best for three parameters, and ends at ranks third in three out of
four times. Therefore, the sentence, The road ends just inside the park would be selected as the
best fit, while valid alternatives would be, The road goes to the park or The road ends at the
park.
R

Figure

15: Does the line enter or end just inside the region?
5.2 Example 2

Figure

shows a configuration in which a line intersects a region such that it is close to the
region's boundary from the inside and farther from the region's boundary in the exterior. A sample
of terms that may fit this description are crosses, cuts through, goes through, runs into, and splits.
R

Figure

Does the line cross or cut through the region?
For the configuration's topological relation, LR 14, the human-subject tests found two metric
parameters to be significant: inner area splitting and outer area closeness. Table 4 displays the mean
and the value range for each parameter.
Topological Spatial Term IAS OAC
Relation mean range mean range
LR 14 cuts through 0.32 0.01-0.50 1.75 0.41-6.04
LR 14 runs into 0.13 0.09-0.44 3.42 0.54-11.96

Table

4: Spatial terms of topological relation LR 14, with means and value ranges of their
significant parameters for splitting and closeness measures.
For the configuration in Figure 16, the term runs into does not qualify, because the
configuration is not located within the range of the inner area splitting. From among the remaining
four spatial terms, splits comes closest to the mean values of inner area splitting and outer area
closeness; therefore, it is selected as the term to describe the configuration. The ranking of the
terms in between is more difficult, because they are subject to more subtle differences. Certainly,
crosses would be better to describe the scene than cuts through, since both parameters have values
that are closer to the mean of crosses than to the mean of cuts through. The term goes through,
however, has a better match with the inner area closeness than both crosses and goes through
have, however, it ranks considerably lower in the outer area closeness.
inner area
splitting
R 2outer area
closeness
area(D
area(R)
R

Table

5: Calculating the inner area splitting and the outer area closeness for the configuration
displayed in Figure 16.
5.3 Example 3
The following characteristics describe the configuration in Figure 17, in which a line is outside of
the region, but follows the shape of the region. Candidate terms to describe this configuration are
bypasses, goes up to, and runs along (Table 6).
R

Figure

17: Does the line run along or bypass the region?
Topological Spatial Term OAN OAC
Relation mean range mean range
goes up to 0.33 0.03-0.75 0.39 0.03-4.79
runs along 0.33 0.16-1.29 1.06 0.30-7.13

Table

Spatial terms of topological relation LR 1, with means and value ranges of their
significant parameters for splitting and closeness measures.
Based on the topological relation, LR 1, the significant parameters are outer area closeness and
outer area nearness. The term bypasses does not fall within the ranges of outer area nearness or
outer area closeness (Table 7), and is therefore not considered. Both terms goes up to and runs
along have the same values for outer area nearness, but since runs along has a significantly lower
value for the outer area closeness, it is chosen as the better term to describe the configuration than
goes up to.
outer area
closeness
area(D
area(R)
R
outer area
nearness
area(D
area(R) IE
R

Table

7: Calculating the outer area closeness and the outer area nearness for the
configuration displayed in Figure 17.
6 Conclusions
This paper developed a computational model to describe the semantics of natural-language spatial
terms based on their geometry. The model is based on the 9-intersection topological model and
refines is with metric details in the form of splitting and closeness ratios. Splitting ratios describe
the proportion of an intersection with respect to the interior or boundary of the two objects. Their
normalized values all fall within the interval between 0 and 1 and grow linearly with the size of the
intersection. Closeness ratios specify distances between boundaries and interiors. For inclusion or
containment relations, the (inner) closeness ratios are normalized to range between 0 and 1, while
closeness ratios for disjoint relations are greater than zero with no upper limit. While this may
appear to be an inconsistency in the model, it is necessary to obtain measures that grow linearly
with the distance between the parts. The model was only developed for relations between a region
and a line, however, the concepts generalize to relations between other geometric types, such as
two regions or two lines.
Splitting and closeness measures can be implemented with standard GIS software. A prototype
implementation with the Arc/Info GIS, however, requires the separation of the two objects into
different layers (Shariff 1996). A method for computing the intersections necessary to determine
the topological relation, using the described by Mark and Xia (1994). In
order to determine the metric parameters, AMLs were written to compute intersections, lengths,
and areas. Although this method demonstrated the feasibility of implementing the required
operators with a commercial GIS, it was cumbersome, because Arc/Info does not support an object
concept, and performance was slow. The use of GIS data structures that support an object model,
and the integration of algorithms that are tailored to the operations necessary for efficient
implementations of the 9-intersection and the metric refinements, are subjects for future
investigations.
The model developed applies to a number of applications in the area of spatial reasoning, such
as similarity retrieval and intelligent spatial query languages. We demonstrated how to use the
model to generate natural-languages terms for simple spatial configurations. Based on a calibration
of the 9-intersection with splitting ratios and closeness ratios, using human-subjects experiments
for sixty-four English-language (Shariff 1996), we showed how a geometric configuration with a
linear and an areal object can be analyzed to determine the pertinent features of their spatial
relations. Values obtained from this method lead to the selection of appropriate natural-language
spatial terms for such spatial scenes.
While the splitting and closeness ratios as refinements of topology cover much of the critical
properties of the spatial relations, there are other parameters left that may make additional
contributions to better choices of natural-language terms. Further investigations-both
formalizations and human-subject tests-are necessary to develop a comprehensive and robust set
of definitions of the semantics of natural-languages spatial relations. Some of these considerations
were outlined in a larger-scale research plan (Mark et al., 1995). The most obvious aspect to study
is the influence of the meanings of the objects on the choice of the spatial terms. Whether the
objects or concern are roads and parks vs. hurricanes and islands, may lead to different mappings
from topology and metric refinements onto the same spatial terms. With respect to geometry, the
current model abstracts away all influences of orientation. This is a valid approach for modeling all
those concepts and terms that are independent of orientation (such as those based primarily on
containment, neighborhood, and closeness; however, orientation, is another parameter that may be
critical for those relations expressing information about direction. For example, orientation may be
important to distinguish north from south (or above from underneath) Orientations are invariant
under translations and scaling, but they may change under rotation. The orientation of the objects
can be assessed in several different ways: (1) the global cardinal relation between two objects, i.e.,
a relation with respect to a fixed orientation framework; (2) the orientation of an individual object,
i.e., the cardinal relation between the object's major axis and a global reference frame; and (3) a
local relation, i.e., the cardinal direction with respect to the framework established by one of the
two objects' orientations. Similar to the metric properties, one could consider purely quantitative
measures, e.g., in the form of degrees. Since people usually do not make such a fine distinction,
coarser, qualitative models are necessary to formalize the properties of the three orientation
concepts.


--R

Similarity of Spatial Scenes.
Cognitive Distance in Intraurban Space.
Problems in Cognitive Distance: Implications for Cognitive Mapping.
Symbolic Projection for Image Information Retrieval and Spatial Reasoning.

A Calculus of Individuals Based on
Representing and Acquiring Geographic Knowledge.
On the Equivalence of Topological Relations.
Categorizing Binary Topological Relationships Between Regions
Modeling Conceptual Neighbourhoods of Topological Line-Region Relations
Naive Geography.
A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis
Qualitative Spatial Reasoning about Distances and Directions in Geographic Space.
Using Orientation Information for Qualitative Spatial Reasoning.



Language and Spatial Cognition-An Interdisciplinary Study of the Prepositions in English
VIsual TRAnslator: Linking Perceptions and Natural Language Descriptions.
On the Robustness of Qualitative Distance- and Direction Reasoning
A System for Translating Locative Prepositions from English into French.
Modeling Spatial Knowledge.
Fundamentals of Spatial Information Systems.
The Image of a City.
Evaluating and Refining Computational Models of Spatial Relations Through Cross-Linguistic Human-Subject Testing
Calibrating the Meanings of Spatial Predicates from Natural Language: Line-Region Relations
Modeling Spatial Relations Between Lines and Regions: Combining Formal Mathematical Models and Human Subjects Testing.
Research Initiative 13 Report on the Specialist Meeting: User Interfaces for Geographic Information Systems.
Interaction with Geographic Information: A Commentary.
Determining Spatial Relations Between Lines and Regions in Arc/Info Using the 9-Intersection Model
Mental Representations of Spatial and Nonspatial Relations.

Human Factors in GeographicalInformation Systems.
The Measurement of Cognitive Distance: Methods and Construct Validity.
The Geometry of Environmental Knowledge.
Cognitive Aspects of Human-Computer Interaction for Geographic Information Systems: An Introduction
An Algorithm to Determine the Directional Relationship Between Arbitrarily-Shaped Polygons in the Plane
The Child's Conception of Space.
A Spatial Logic Based on Regions and Connection.
A Model of the Human Capacity for Categorizing Spatial Relations.

On the Internal Structure of Perceptual and Semantic Categories.
Principles of Categorization.
Natural Language Spatial Relations: Metric Refinements of Topological Properties.
Parts: A Study in Ontology.
Algebraic Topology.
How Language Structures Space.
--TR
Representing and acquiring geographic knowledge
An algorithm to determine the directional relationship between arbitrarily-shaped polygons in the plane
Qualitative Representation of Spatial Knowledge
Language and Spatial Cognition
Cognitive Aspects of Human-Computer Interaction for Geographic Information Systems
Human Factors in Geographical Information Systems
The Geometry of Environmental Knowledge
Using Orientation Information for Qualitative Spatial Reasoning
Natural-language spatial relations

--CTR
Josef Benedikt , Sebastian Reinberg , Leopold Riedl, A GIS application to enhance cell-based information modeling, Information SciencesInformatics and Computer Science: An International Journal, v.142 n.1, p.151-160, May 2002
Tiago M. Delboni , Karla A. V. Borges , Alberto H. F. Laender, Geographic web search based on positioning expressions, Proceedings of the 2005 workshop on Geographic information retrieval, November 04-04, 2005, Bremen, Germany
Haowen Yan , Yandong Chu , Zhilin Li , Renzhong Guo, A Quantitative Description Model for Direction Relations Based on Direction Groups, Geoinformatica, v.10 n.2, p.177-196, June      2006
Salvatore Rinzivillo , Franco Turini, Knowledge discovery from spatial transactions, Journal of Intelligent Information Systems, v.28 n.1, p.1-22, February  2007
Guoray Cai, Contextualization of Geospatial Database Semantics for Human---GIS Interaction, Geoinformatica, v.11 n.2, p.217-237, June      2007
geographic information systems;topological relations;spatial relations;metric refinements;GIS