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Braid et al.
1980
; Mäntylä
1988
) and extended Euler operators (Masuda
1993
).
First, two separate cubes are created (see Fig.
6
a). Then, they are linked by a
shared face (see Fig.
6
b). It is possible to define different sequences which results
in the same model. It should be noted that the external cell and dual graph are
present at each step of the process, but for the sake of clarity the external cell and
dual graph are not shown. The final model consists of three cells: two internal
cubes and one external cell (see Fig.
6
c).
The DHE was originally designed for 3D models. However, a single poly-
choron (a 4D polyhedron) can be represented using the DHE without any modi-
fications, except for the use of 4D coordinates. This is done by representing the
polyhedra that lie on its boundary, in a similar manner as a 2D data structure is
commonly used to represent a single polyhedron by storing the polygons in its
boundary, cf. Baumgart (
1975
).
Lee and Zlatanova (
2008
) and Lee and Kwan (
2005
) extract from a 3D building
a graph that can be used in case of emergency, and Bogualawski et al. (
2011
) and
Bogualawski (
2011
) perform the same using a data structure, the dual half-edge
(DHE), which simultaneously represents the building (the rooms and their
boundaries) and the navigation graph. With the DHE, the construction and
manipulation operations update both representations at the same time, permitting
the simultaneous modelling and characterisation of buildings. There are several
other examples of duality in GIS: the Delaunay triangulation and the Voronoi
diagram are often used to model continuous phenomena, these two structures being
dual to each other. Dakowicz and Gold (
2003
) use them for terrain modelling, Lee
and Gahegan (
2002
) for interactive analysis, and Ledoux and Gold (
2008
) for three-
dimensional fields in geosciences. In higher dimensions, we can also store the
geometry and the topological relationships in a pair of dual subdivisions.
3 Geometric Topology
This section has been written from the excellent visual introduction topic on alge-
braic topology and geometric topology Fomenko (
1994
). Geometric topology is the
branch of mathematics that studies topology (i.e. all the properties that are invariant
by continuous mappings) through some geometric objects called manifolds (topo-
logical spaces that near each point resemble Euclidean spaces) and their mappings
and embeddings into each other. A central concept in geometric topology is the
concept of ramification (the intuitive concept of several sheets joining together,
further explained in next paragraph and illustrated by Fig.
7
). An important topo-
logical concept is the genus of a topological object i.e., the number of handles that
need to be added to a sphere, so that the result is homeomorphic (resembles) the given
topological object, i.e., there is a one-to-one continuous mapping between them
whose inverse mapping is also continuous. Another important topological concept is
the Euler-Poincaré characteristic, which is defined as v ¼ k
0
k
1
þ
k
2
k
3
þ
,
where k
i
denotes the number of cells of dimension i in the cell complex. However, the
