Image Processing Reference
In-Depth Information
Fig. 11.15
Pattern produced by
PFSS
T
driven by 1D2C r-
2
CA rule 53404 selected from
Fig. 11.5
epidemiological processes on the globe have been presented in [31]. Although the
sphere is not a particularly practical shape for a building, domes have special posi-
tion in the history of architecture. More recently, geodesic domes and spheres [47]
have gained certain popularity due to their structural and aesthetic properties. A
geodesic sphere (GS) is a spherical shell structure or lattice shell based on a net-
work of great circles (geodesics) on the surface of a sphere, which intersect to form
rigid triangular elements that also distribute the stress across the structure. Icosa-
hedral geodesic sphere (IGS) is a spherical polyhedron with Euler characteristic
χ
=
2. All triangular facets of IGS have 6 triangles per vertex, except, according
to Euler's polyhedron formula [16], for 12 vertices with 5 triangles, regardless of
the recursive subdivisions of the triangles (mesh resolution). The dual polyhedron
of IGSs give Goldberg polyhedra [22], which consists of all hexagonal faces, except
for also 12 [16], which are pentagons. [57] explores GOL on such a tessellation; for
illustrative animation see [56].
The most general case of BE is a free-form surface, which cannot be constructed
with regular polygons. Irregular tessellations have been introduced to more real-
istically model natural phenomena, e.g.: social networks have been simulated on
irregular grid structures based on Voronoi-diagrams [18]. A geographic informa-
tion system (GIS)-based CA on irregularly sized and shaped land parcels, at syn-
chronous and asynchronous development have been used to model land-use change
at the land parcel scale in [53]. Incremental matching method for processing of large
structures that extend across many neighborhoods intended to enhance the data con-
tained within topographic maps has been proposed in [37]. A cell-based wildfire
simulator that uses an irregular grid that produces much faster results comparable in
accuracy with traditional fire front propagation schemes has been presented in [28].
Graph-based cellular automata for CA models on irregular lattices have been intro-
duced in [39].
For a free-form BE, however, triangular tessellation is the most suitable. Any
3D surface can be triangulated [45], that is divided into a set of (planar) triangles,
with the restriction that each triangle side is entirely shared by two adjacent trian-
gles. Analogous operation is impossible with (planar) quadrilaterals or hexagons.
The control of state of such CASS is much more limited than in the previously de-
scribed 1DCA-based systems. In the examples shown below, there is a single IC
cell located arbitrary in the mesh. Alteration of the state of that cell triggers the 2D
CA evolution, which propagates over entire mesh until the state of each cell sta-
bilizes. It is imaginable that complex patterns perpetually fluctuating on BE could
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