1369 algebraic transformations of cadastral topological data

1369 algebraic transformations of cadastral topological data

ISSN 2029-7106 print / ISSN 2029-7092 online
ISBN 978-9955-28-829-9 (3 Volume)
ISBN 978-9955-28-827-5 (3 Volumes)
ENVIRONMENTAL ENGINEERING
th
The 8 International Conference
May 19–20, 2011, Vilnius, Lithuania
Selected papers
http://enviro.vgtu.lt
© Vilnius Gediminas Technical University, 2011
ALGEBRAIC TRANSFORMATIONS OF CADASTRAL TOPOLOGICAL DATA
ElŜbieta Lewandowicz
University of Warmia and Mazury in Olszty, Faculty of Geodesy and Land Management, Department of Surveying
12 Jana Heweliusza St., 10-957 Olsztyn, Poland. E-mail: [email protected]
Abstract. In this paper we introduce algebraic transformations of cadastral topological data. The traditional description of this data in attribute tables is transferred to the matrix forms. This allowed to introduce algebraic transformations. As an outcome one obtains new forms of description of cadastral structures. In the second part of the paper new
topological data were transformed to quantum metric and this lead to new description of the neighbourhood of sets of
geometrical cadastral data elements. They can be used in analytical processes, e.g. in neighbourhood analysis and
aggregation of areas.
Keywords: cadastral data, topology models, algebraic transformation topology data.
2. Description of topology of cadastral data in tabular
and matrix form
1. Introduction
In the research on structure of geographical space
one of the important issues is the analysis of the connections between geographical objects. The description of
this connections is difficult because of their complexity.
Part of this connections comes from the position of the
objects in space. A description of those connections using
geometrical data is given by topology. In the Geographic
Information System (GIS), topologies are described in
connected attribute tables. The detailed way of their description was given in norms (ISO 19107), described in
the papers (Molenaar 1998, Longley i in. 2006, Gaździcki
1990, Eckes 2006) and is used in applications (Esri 2003,
Autodesk 2000, Bentley 2000). This tabular description
can be transformed into algebraic form – matrix
(Lewandowicz 2007). It can be also introduced in the
form of a geometric graph (Gould 1988, Kulikowski
1986), whose elements: nodes (N), edges (E) and faces
(F) describe points, lines and areas. Algebraic transformations of the obtained matrices can create new topological models which describe spatial structures in a new
way (Lewandowicz 2007, 2010).
The main aim of this paper is to look for models
which allow to supplement the descriptions of spatial
connections between cadastral objects, based on geometrical relations. In realisation of this aim, a simple geometrical example of cadastral data is used. The main point is
to write topological data in a tabular way and then to
transform them into algebraic form. Graphic visualization
of new forms of description of topologies, using new
graphs, supplements results of transformations. The
proposition to use new ways of description of topology in
spatial analysis represents a practical aspect of the conducted research.
Let's take the part of cadastral data visualized geometrically on Fig 1, as the simple example, on which the
theoretical part of the paper will be based.
Fig 1. The sketch of parcels with identifiers of parcels,
border points and borderlines (1,2,3,4,5, ... , is id. of border points, 1,2,3,4,5, … - id. of borderlines, 221, 222/1,
222/2, 223...-id. of parcels)
The presented geometrical data can be written in the
form of topological data in traditional tabular way:
• table 1 with data of connections between
borderlines and border points,
• table 2 with parcel and borderlines data.
The matrix description of these data is presented by
matrix SE-N, which corresponds to the table 1 and by matrix SF-E representing data from the table 2. The matrix SEN contains data on connections of borderlines and border
points.
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Elements of matrix (1) (S E − N )ij take the values 1, if
S NE − N = ( S E − N ) T ( S E − N ) ,
(4)
S EN− E = ( S E − N )(S E − N ) T ,
(5)
S (EF − F ) = ( S F − E )( S F − E ) T ,
(6)
the borderline i has the beginning or the end at the border
point j:
Table 1. Connections between lines and points
id line
id points
5
2, 6
6
2, 5
7
1, 2
8
3, 5
9
5, 7
10
4, 10
…
…
Basing on
S FE− N = ( S F − E )( S E − N ) ,
(7)
S FN− N = ( S F − E )( S E − N ) / 2 .
(8)
S FE− N (7) one can determine other rela-
tions between (F-F) than the ones given in (6)
S (NF − F ) = (( S FN− N )( S FN− N ) T ) .
Table 2. Connections between areas and lines
id polygon
222/2
223
221/1
…
id lines
5, 17, 9, 6,
8, 9, 16, 13
12, 7, 6, 8,
..., ..., ...,
(9)
Relations between (E-E), determined from F, one
can generate basing on SF-E (2):
S (FE − E ) = ( S F − E ) T ( S F − E ) .
Similarly using
tions (N-N) (1)
(10)
S FE− N one can determine new rela-
S (FN − N ) :
S (FN − N ) = ( S FN− N ) T ( S FN− N )
(11)
The values of those matrices are interesting. In
Elements of matrix (1) (S E − N )ij take the values 1, if
S
E
(N −N )
E
the borderline i has the beginning or the end at the border
point j.
The matrix SF-E has the information on connections
of parcels with borderlines. Elements of the matrix (2)
(S F − E )ij take the values 1, if the area of parcel i is limited
(3), diagonal values ( s ( N − N ) ) ii , contain information about the number of borderlines arriving at the
border points with identifier i (12). The off-diagonal values (12) indicate connections of border points with borderlines and they take the values from two-element set
{0,1}. In our case, if the border points i, j are connected
by borderline j, as shown in (Fig 1) accordingly to the
notation in table 2.
by borderline, elements of the matrix
value 1.
(2)
(12)
3. New algebraic form of description for topological
model of cadastral data
The above introduced matrices were used to algebraic transformations what allowed to obtain new description for topologies.
Basis matrix ={ SE-N , SF-N }
(3)
( s NE − N ) ij take the
In the matrix
S (NE − E ) (4), the diagonal elements
( s (NE − E ) ) ii (13) take the values equal 2, indicating that
borderline is connected with two border points. The offdiagonal values, equal 1, indicate on connection between
two neighbouring borderlines with one common border
point. The values equal 0 point out that edges ij are not in
the neighbourhood of each other.
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N
(13)
The matrix SF −N contains information on geometrical relations between parcels and border points, and as
the result of multiplying (9) by itself, one obtains the relations (19):
Next matrix S(EF−F) (14) contains information related
(19)
with geometry of parcels.
(14)
In the ( S(NF −F ) ) matrix, the diagonal elements (19),
The diagonal values (s(EF−F) )ii (14) give information
describe the number of border points determining the
areas of parcels. The off-diagonal values describe the
number of mutual border points of neighbouring parcels.
The values equal 1 indicate that the parcels are connected
with only one border point.
about the number of borderlines describing the area of a
parcel. The off-diagonal elements take the values equal 0,
if the parcels with identifiers i and j are not directly connected with each other by one borderline. The offdiagonal values
( s(EF −F ) ) ij = n, n > 0 ,
(15)
The relations between (E-E) generated from (10)
and described in (20) point out the connections of (E-E)
coming from the borders belonging to the same areas of
parcels.
constitute the number of intervals of borderlines of two
neighbouring parcels, with identifiers i, j.
(20)
(16)
Values on the diagonal
E
The elements of the S F − N matrix take the values
from two-element set {0,2}. Values 2 are result of connections between lines and points (a line is determined by
two points). Dividing them by two – gives as the result
the matrix
S
N
F −P
( s(FE − E ) ) ii
N
take the values equal 1, if the borderline i is
the border of maximal complex.
The off-diagonal elements
(17)
*
If an element of the matrix S F − N (18) ( s F − N ) ij
takes the value 1, this means that the border point j determines the borderlines of the area of the parcel i.
indicate the
number of parcels, which have the borderlines i in the
assumed set of cadastral data. Usually it is the number 2.
, with the values from the set {0,1}.
S FN− N = ( S F − E )( S E − N ) / 2 .
( s(FE − E ) ) ii
( s(FE −E ) )ij take the val-
ues 1, if two borderlines i,j determine the border of the
same parcel. The value 0 indicate that there is no such
connection.
Similarly the relations (N-N) - S(FN − N ) generated
from (11) one can introduce in the following form (21).
(18)
(21)
1371
F
Diagonal values S ( N − N ) indicate that the border
point i is an element of description of the borders of
n = ( s (FN − N ) ) ii parcels. The off-diagonal ones show in
how many descriptions of parcels the border is based on
two border points ij.
4. Transformation of topological data to quantum
metric
The above mentioned matrices contain the numerical
data on the forms of connections of the objects from cadastral object sets (F, E, N). By taking the zero values on
the diagonal and transforming the off-diagonal elements
into quantum metric {0,1} we obtain a new description of
topological data.
transformacja
S
 → S Q
(22)
Fig 2. Graph representing chosen connections between
elements N,E,F, describing border points, borderlines and
parcels
This transformation follows from:
Z
(S XX
) Q = ( S (ZXX ) ) − Diag((S (ZXX ) )
Z
(S XY
) Q = (S (ZXY ) ) if X ≠Y ,
where:
and if:
or
(23)
X ∈{N , E, F}, Y ∈{N , E, F} Z ∈{empty, N , E, F}
Z
( s (ZXY ) ) ij > 0 ⇒ ( s XY
) ij = (( s (ZXY ) ) ij /( s(ZXY ) ) ij ) .
As
the result we obtain the matrix description of neighbourhood of cadastral objects.
5. Geometrical visualization of generated topological
models
Fig 3. Graph of connections visualizing the relations between (F-E) and (E-F), can be used in the neighbourhood
analysis including borderlines
The introduced matrix description of topological relations, obtained from the transformations (23), one can
show graphically in the form of graphs. In the Fig 2, there
are shown only chosen relations between sets: N, E, F.
Graph of connections of parcels one can obtain from the
relations (F-E), given in the table 2 and in the matrix SF-E
(Fig 3). It includes the borderlines. The parcels'
neighbourhood graph described in the matrix
( S (EF − F ) ) Q is shown in Fig 4.
F
Q
Using the matrix ( S ( E − E ) ) one can generate the
subset representing the connections (E-E)F described in
one row of the
( S (FE − E ) ) Q matrix. It contains the set of
borderlines of the aggregated area obtained after removing the edge Ei.
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Fig 4. Graph of connections (F-F)E which can be used in
the neighbourhood of parcels analysis
They enrich the description of cadastral structures. Another 8 matrices, after transformation into quantum metric, introduce relations which are useful in practical solution
They are particularly helpful in the neighbourhood
analysis and in the processes of object's aggregation. The
obtained solutions can be used in the modification of already existing algorithms as long as they will accelerate
already existing numerical processes.
References
Fig 5. Digraph representing connections
(E-E)F, described in one row of the matrix, for id E-21, it is possible
to be used in aggregation of areas of parcels 136 and 138
Fig 5 shows such subgraph for E21, generated from
the row of matrix i=21.
The set of border points of aggregated areas one can
obtain from the matrix
(S(FN −N ) )Q as the result of re-
moving the border point – node (N) and the borderlines –
edges (E) connected with it .
6. Conclusions
In this paper we have introduced the transformations
of topological data of cadastral map based on two tables
(table 1, 2), described in the algebraic form. The results
written in the form of 8 matrices introduce new ways of
descriptions of connections between cadastral objects.
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