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the solution is
respectively as the problem of finding of the visible and the
invisible lines in computer graphics. In general case the
solution is very complicated but applying the restriction of
relatively small angles between the projection rays and vertical
planes (as walls) the presentation could be simplified. Taking
into account such restriction the following groups of arcs could
be formulated:
I. Arcs without changes in height between two areas or with
relatively low difference in slope - a type arcs separating
different areas.
II. Arcs with jump in the height between two adjacent areas:
r - roof/terrain border (or roof/roof);
v - wall edge / terrain border (or wall edge/roof);
p - roof / wall border;
w - wall / wall border (between different buildings);
III. Arcs defining break in surface slope, not in height:
h - wall / roof edge;
t - terrain / wall edge (or roof / wall edge in situation when a
part of the building is lying over the roof or terrace);
e - visible edge between surrounding walls.
In situation if nodes are marked only at that points where more
than two arcs are connected some combination of height arcs
could appear and should be added to terminal arcs set. The arcs
of this type are:
j - right arc of type wall / terrain (sequence v,t);
| - left arc of type wall / roof or terrain (sequence t,v);
u - arc of type single wall (sequence of v,t,v arcs);
n - arc of type internal roof (sequence p,t,p);
f - left part of internal roof (sequence p.t);
g - right part of internal roof (sequence t,p);
The different topological type of arcs is established by usage of
different subscript index with following meaning:
k - main external arc of leading contour of last cluster;
m - main external arc of leading contour of non-last isolated
cluster;
c - main arc of external contour of non-leading contour;
b - non-main outside contour arc of isolated cluster;
i - internal arc between two different contour from equal level;
S - non-main arc of contours from different levels;
d - main arc of internal contour, oriented in direction to point
the contour of upper level.
o - single or last isolated contour;
q non-last isolated contour from set of isolated contours or area
clusters.
It is necessary to be mentioned that not all topological types for
different height types of arcs exist. According to this the
following extended set of topological and height types of arcs
that are possible are forming the set of finite elements of
grammar.
1. Non connected elements:
Se - empty initial area;
2. First order connected elements:
sgrinitial outside area, containing sub areas (isolated areas or
area clusters.
3. Second order connected elements (one pointer to inside
cluster list and one implied connection to own contour):
a, - separate (or last) isolated area.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
4. Third order connected elements containing two pointers to
the next arcs of the own and adjacent contour and one pointer to
the main arc of the own contour, or to the main arc of the upper
level contour for an element of type d.
5. Fourth order connected elements (two pointers to internal
and external contour and one pointer to internal isolated cluster
list and one implied connection to own contour.
6. Fifth order connected elements (two pointers to internal and
external contour, one implied connection to own contour, one
pointer to internal isolated cluster list and one pointer to the
next cluster.
The non-terminal elements set could be defined based on
formulation of the main processes of generation of the
topological configuration. The generation processes could be
separated in several main steps. The image generation starts
with transaction from initial element to sets of isolated
contours. Every contour could converts to cluster of adjacent
contours by attaching of areas (only by outside not self-closed
arcs following clockwise direction). This process allows to
form not only outside ring of contours but inside contours of
arbitrary levels. To model the process of generating of hanging
contours (attached in point) a zero length arc is involved (with
coincide head and tail node points). The process of growing of
cluster stop when the main element of leading contour changes
to round state. As a result the main arc element of every contour
is defined and non-main arc elements transfer to terminal. The
last step is transition to mode of generation of isolated sub-
contours or conversion to terminal elements.
The presentation of different generation processes is done by
usage of additional subscript index corresponding to the special
activated state of the arc:
p - generation state of isolated contour;
© - generation state of the external contour arc;
£ - last isolated arc by the moving external arc;
V - scanning state of the new external arc over the isolated arcs;
® - isolated arcs scanning state:
% - generation of hidden contours state.
The main group of non-terminal elements could be defined as
set of elements, corresponding to the terminal ones. It had to be
enlarged to take into account their different states of generation.
It is convenient for that purposes to use subscript index. To
every one generation process corresponds a set of the rewriting
rules. To restrict the number of the non-terminal elements not
only context free but context dependent rules are used too.
The first set of rules formulate the generation of isolated
contours and sets of isolated contours, transition between
hanging contours and main arc element in generation state.
Only for initial few production rules the full form of
presentation will be used because it takes a lot of space. The
generation of initial contours and the chains of isolated
contours is describe by the rules:
So 9 sgO So () > SA 0 (11)(02)
ACTOR (3.33
AYANT 2) ROC) Rosi)
Ao(12) 9 AgA o 110,30,02).
Aq(1,23) 9 Aq, (12,3)
Ag (1,2,3) > Rap (1,2,3)
(1)
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