Environmental Engineering Reference
In-Depth Information
L-system to a given string s consists of the simultaneous replacement of all symbols
in s occurring on the left-hand side of a rule by their corresponding right-hand side.
Symbols which cannot be replaced remain unchanged. By beginning with the start
string and iteratively performing one application step to the result of the preceding
one, we get the developmental sequence of strings generated by the L-system:
Axiom ! s 1 ! s 2 ! s 3 ! ...
As a simple example, let us consider the L-system with
S ¼
{A; B}, Axiom
¼
A,
and with the two rules
A
¼¼>
B
¼¼>
B
AB
The resulting developmental sequence is:
A
! ...
From Lindenmayer's original viewpoint, A and B can be interpreted as two
different cell types of a filamentous organism. The rules express the facts that a cell
of type A can grow into a cell of type B, and a cell of type B can divide into two
cells of type A and B, respectively. The developmental sequence then reflects the
growth of the filament in discrete time steps. (Note that the number of cells in this
sequence increases according to the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,
!
B
!
AB
!
BAB
!
ABBAB
!
BABABBAB
...
, i.e.
each number is the sum of its two predecessors.)
To let L-systems produce more interesting geometries than just linear filaments
of cells, the definition of an L-system is extended by a fourth component, turtle
geometry. It provides a geometrical interpretation: With each string, particularly
with each S i from the developmental sequence above, a geometrical structure S i in
two- or three-dimensional space is associated. This is realized by letting the
alphabet
of the L-system contain the set T of all turtle commands. The strings
s i obtained from the L-system are then separately interpreted by the turtle: They are
scanned from left to right, and the geometrical structure S i is constructed by
processing the occurring commands one by one. Symbols from
S
which are not
in T are ignored by the turtle. Figure 11.2 summarizes the resulting scheme of
interpreted L-system application.
An example (from Prusinkiewicz and Lindenmayer 1990, p. 25, adapted)
demonstrates this mechanism: The rules of our L-system are:
Axiom
S
L(100) F0 and
F0 ¼¼> F0 [RU(25.7) F0] F0 [RU(-25.7) F0] F0 .
The resulting structures S 1 ,S 2 ,S 3 and S 4 are shown in Fig. 11.3 .
¼¼>
Fig. 11.2 L-system
application with geometrical
interpretation. The dotted
arrows symbolize the
interpretation of strings by the
turtle (from Kurth 2007)
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