IZ ZNANOSTI IN STROKE 
UDC 347.235.11 (084.3):528.063.1 
NU ERICAL P OCESS OF 
STRAIG TENI G THE EDGES 
OF GRAP ICALLY-BASED 
CAD ST L APS 
Zmago Fras, M.Se., Tomaž Gvozdanovic, M.Se. 
Monolit d.o.o., Ljubljana 
Received March 7, 1995 
Revised lune 7, 1995 
Abstract 
This article describes a· solution fara problem we faee 
combining maps of graphically-based cadastral maps. 
Convex or concave borders are aligned through a special 
transformation which changes the parcel geomehy. Minimal 
geometrie distortion is aehieved through the use of earefully 
seleeted eurves. 
Keywords: digital land cadastre, edges slraightening, graphie 
survey, maps 
l STARTING POINTS 
In the process of setting up the digital land cadastre (DLC) one uses two main sources of data that define the links between the topology and the geometry of 
individual parts of parcels in the land cadastre. These are numerically- and 
graphically-based cadastral maps. The basic units of management in the DLC are 
cadastral communes, so the sheets (into which the cadastral commune is currently 
divided in the current documentation will have to be joined together). We start with 
the fact that both numerically and graphically-based sheets were rectangular in their 
basic form. Somewhcre in the process of transformation from the analogue to the 
digital form, this condition therefore needs to be fulfilled, the gcometrical structure 
must not be altered, i.e. the relative relationships in the DLC must be preserved in 
the present documentation (the basic starting point for the DLC). After the 
transformation of sheets, their present contents must correspond to their original 
contents (there should be no transfer of parcels from one sheet to another). 
2 PROBLEM 
This transformation does not require any additional work nor does it present a problem in the case of numerically-based maps, since their maintenance has 
become coordinate, whereby the edges of the sheets had a status of immovable lines 
(they were a component of the cadastre, so to speak). In graphically-based maps this 
is different. Through maintenance proccdures of the graphica!ly-based sheets, their 
edges acquired a convex/concave form (the rule of absolute untouchability of the 
Geodetski vestnik 39 (1995) 2 
sheet edges was not taken into account for historical reasons) and the sheets are no 
longer rectangular (Figure 1 ). 
D 
actual stote theore'rical form of a sheet 
Figure 1: Actual stale and theoretical fonns of sheets from cadastral maps 
ue to this new form of sheets overlapping, holes can occur when joining sheets 
(Figure 2) and individual parts of parcels and the parcels themselves can be 
lost Sometimes it is impossible to join the same parcels and their topological 
connection is thus lost. 
o '\3140 ' 
+ 140 
Figure 2: Theoretically combining sheets and the case of a problematic connection of aclual sheets 
The problem of concave or convex forms unfortunately is not solved by any of the known general plane linear transformations (Helmert, afine, bilinear). All 
authors who have dealt with the transformation of graphically-based maps (Čuček, 
1979, Mivšek, 1991, Oven, 1993, Wiens, 1984) mainly discussed the appearent forms 
of errors within sheets and (the distribution of) their influence transformed 
data/parcels. The basic building elements of the DLC, however, are sheets/cadastral 
maps. In order to successfully join sheets within cadastral communes it is necessary 
to search for/define a new transformation or procedure for graphically-based sheets 
to salve the problems of sheet form. 
3 IDEA 
he presented starting points and problem were first written as a list of rules or 
boundary conditions: 
• Deformations on individual sheet edges are independent of transformation 
on other edges. 
• The corners of sheets sections are without deformations (the only known 
points). 
• The transformation function must be continuous and continuously derivab!e. 
Geodetski vestnik 39 (1995) 2 
• Forced correction (straightening) of sheet edges or the influence of errors 
which accumulate at sheet edges should have a minimum influence on parcel 
shape; the influence dcpends on the degree of deformations. 
The following solution attempts to fulfil the above boundary conditions: 
• influential areas of corrections due to deformations of individual sheet edges 
were limited by sheet diagonals "watersheds" (Figure 3) - sheets were divided 
to 4 parts; it follows from this that: 
- there are no corrections of diagonals due to sheet edgc deformations (the 
influences of two different edges converge here ), 
- at the diagonal intersection = sheet centre we therefore obtained points 
without deformations (same as corners of sheet sections), 
- sheets were roughly divided into 4 circular cutouts. 
influential areas of corrections 
due to right sheet edge 
deformations 
Figure 3: Detennination of influential areas of corrections due 
to individual sheet edge defonnations 
• Within influential areas corrections due to sheet edge deformations cxtend 
radially from the sheet centre (Figure 4) and are gradually reduced from the 
sheet edgc towards the sheet centre. It follows from this that: 
Figure 4: Direction of the influence of corrections due to sheet edge deformations 
• The correction function must fulfil two conditions that partially exdude each 
other (rcducing the influence of sheet-edge deformations as you move 
further from the sheet edge and maintaining the relative relationships among 
the sheet contents - Figure 5: discrete transformation). 
Geodetski vestnik 39 (1995) 2 
.----· 
1 
~ 
dR 
sheet-edge 
deformation 
R 
theoretical 
sheet edge 
area of influence of corrections 
due to sheet-edge deformations 
poin'r of major changes 
of relative relationships 
middle 
of the sheet 
Figure 5: Example of the discrete transformation funclion /o transfer the influence 
of sheet-edge defonnalions 
• For the correction function for deformations in radia! directions, we have 
chosen part of a sinusoidal curve sinn (Figure 6), which is continuous ancl 
continuously derivable, to ensure a continuous ancl smooth transformation 
ancl a more even distribution of the influence of sheet-edge deformations. 
dR 
sinusoidal curve 
r 
correction curves 
cR 
Figure 6: The selected type of correction curve 
• The degree of n = 2 was selectcd. (On the basis of empirical finclings, it 
clistributes the influence of sheet eclge deformations most "naturally"; in 
order to determine optimum degrees of the sinusoidal curve it would be 
necessary to perform a more extensive analysis of concrete data) - Endosure 
1 illustrates curves for the transfer of corrections from the sheet edge 
towarcls the centre of the sheets for different n values. 
• The spatial deformation plane above the sheet is formed on the basis of five 
points without deformations, sheet-edge deformations and the correction 
function (Figure 7). 
Geodetski vestnik 39 (1995) 2 
7: Spatial defonnation plane above the sheet 
o Radial correction is calculated on the basis of the spatial plane for each 
x, y, i.e. according to the following equation: 
[ sm (; . n - ~) + 1] . dt 
This equation can be solved at the vector or raster level; transformation at the vector 
leve! is recommended, since 
o it is less difficult, 
o it is simpler to program, 
o in the phase of data acquisition (sheet vectorization) it is not necessary to 
know the "true" coordinates, 
o raster data are the basic archives and must not be corrected since the 
historical link would be !ost. 
Graphical presentation of the procedure: 
.~ 
_.,.,..,.._,_~-.-,-, 
•- conditions J 1 
* bi!inear transformation,_· - • 
of corners 
* the corners and centre of 
the map are without deformation 
* corrections in the radia! 
direction 
Geodetski vestnik 39 (1995) 2 
3D correction plane 
above sheels 
map edgc map middlc 
form of radia! correction profiles 
D 
!;. 
Required input data: 
• the collection of most equidistant points (thumb distribution, detail, optional 
points) on sheet edges, 
• the coordinates of sheet-content corners, 
• the form of the function for radia! correction. 
Enclosure 2 presents different forms of 3D corrcction planes above shects. 
4 CONClLUSlION 
he present transformation solution (preservation of the sheet form) for 
graphically-based sheets is the rcsult of the theoretical and practical experience 
of the authors in their work for the DLC. This solution is simple and quick. A more 
extensive test is needcd for systemic usc, which would yield the answer to the 
question: Does the presented solution fulfil the DLC project requirements in every 
way? 
Literature: 
Čuček, I., Transfonnacija načrtov zemljiškega katastra 1:2 880 v načrte nove izmere 1:2 500. 
Raziskovalna naloga, JGF, Ljubljana, 1979 
Mivšek, E, Uporaba podatkov katastrskih načrtov grafične izmere v infonnacijskem sloju 
zemljiškega katastra. 24. Geodetski dan, Bovec, Geodetski vestnik, 1991, Vol. 35, No. 3, p. 
169-173 
Oven, K, Določitev homogenih con katastrskega načrta grafične izmere. Diplomska naloga. 
Univerza v Ljubljani, FAGG OGG, Ljubljana, 1993 
Wiens, H., Flurkarlenemeuerung mittels Digitalisierung und numerischer Bearbeitung unter 
besonderer Berucksichtigung des Zusammenschlusses von Jnselkarten zu einem homogenen 
Rahmenkartenwerk. Kirschbaum Verlag, Bonn, 1984 
Review: Dr. Radoš Šumrada 
Joe Triglav 
Geodetski vestnik 39 (1995) 2 
Enclosure 1: Cwves for the transfer of conections from the sheet cdges towards the centre of the shects 
Endos111re 2: Different forms correction planes above the sheets