Simulating L-systems.pdf

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Simulating the development of
Fraxinus
pennsylvanica
shoots using L-systems
Mark Hammel and Przemyslaw Prusinkiewicz
University of Calgary
William Remphrey
University of Manitoba
Campbell Davidson
Agriculture Canada
From
Proceedings of the Sixth Western Computer Graphics
Symposium
(Banff, Alberta, 20–22 March, 1995),
pages 49–58, March 1995.
48
Simulating the development of
Fraxinus pennsylvanica
shoots using L-systems
Mark Hammel, Przemyslaw Prusinkiewicz
University of Calgary
William Remphrey
University of Manitoba
Campbell Davidson
Agriculture Canada
...in that Empire, the Cartographer’s art achieved such a degree of perfection that
the Map of a single Province occupied an entire City, and the map of the Empire,
an entire Province. In time, these vast Maps were no longer sufficient. The Guild
of Cartographers created a Map of the Empire, which perfectly coincided with the
Empire itself. But Succeeding Generations, with diminished interest in the Study of
Cartography, believed that this immense Map was of no use...
Viajes de Varones Prudentes, 1658
(quoted by Umberto Eco [2, page 95])
1 Introduction
This paper presents a methodology for creating computer models that capture the development of
plants using the formalism of L-systems and incorporating biological data. The modelling process
is divided into the following steps:
A qualitative model is constructed according to observations of plant growth and form.
Measurements of key characteristics are gathered from actual plants.
Statistical analysis is performed to convert the raw data into functions which describe growth.
The quantitative model is formed from the qualitative model and growth functions, as well
as approximated functions describing the growth of model elements for which data has not
been obtained.
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A visualisation of the model is produced.
An evaluation of the model is performed. The visualisation aids in exposing any flaws in the
qualitative or quantitative models, and helps identify any incorrectly estimated functions.
If a further iteration of this modelling process is required, this process is repeated. The
choices and assumptions made in the construction of this model are then reconsidered and
modifications are applied to improve the model’s reflection of reality.
This paper makes use of a model of shoots of
Fraxinus pennsylvanica,
also called green ash,
to illustrate the presented modelling methodology. It compliments research performed in 1994 as
a collaboration between biologists and computer scientists, which resulted in a publication in the
Canadian Journal of Botany
[11]. In the following sections, a brief introduction to L-systems is
given and then the methodology is described in greater detail.
2 L-systems
In Lindenmayer systems (L-systems) [7, 10], development is defined on the level of individual
plant components — internodes (stem segments) elongate, buds initiate new lateral branches,
leaves unfold, and so forth. To formally describe these processes, productions are used to capture
each developmental event. They state how a component or module is replaced by a subsequent
form. A string notation is employed to represent modules and productions [10].
For example, the following production defines an apex which is replaced by two new internodes,
two lateral or branching apices, and another terminal apex (an apex is a generative centre for a
plant, where new forms are initiated):
A
!
F[+A][-A]FA
The symbol before the arrow is called the
predecessor
which is the module being replaced. The
word,
or sequence of symbols, after the arrow is called the
successor
and it indicates the modules
replacing the predecessor. Apices are represented by the symbol
A
and internodes by
F.
Symbols
which define geometric aspects of the model may also be included. In this example, the square
brackets
[]
enclose the modules that compose a branch, and the
+
and
-
symbols indicate rotations.
This production is illustrated in Figure 1.
3 Qualitative Model
The modelling process begins with the specification of the qualitative model. It captures aspects
of the plant that can be obtained through observations and are deemed essential to its form and
development. These include the arrangement (topology) and the sequence of activities of various
plant modules. The main components of the plant are distinguished and their developmental stages
are identified. The connections between these components are also defined.
The qualitative portion of the green ash model consists of three main parts: the bud, the shoot
units, and the leaves.
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A
A
predecessor
A
F A
successor
F
Figure 1: A production describing the development of a plant apex.
S
S
Figure 2: An initial green ash bud.
The simulation begins with a single bud (Figure 2). This is captured by the L-system
axiom,
or
the initial sequence of modules:
!
:
f
[S][S]
g
K
A
The axiom represents the bud as a sequence of bud scale pairs, each pair enclosing the next. Each
scale
S
is represented as a branch. The “exponent”
K
is used to specify the
K
-fold repetition
of bud scale pairs. The apex
A
is initially contained within the bud. The label
!
identifies this
statement as the axiom.
After some amount of time has passed, the bud scales fall from the tree. This event is captured
by production
p
1
:
p
1
:
S
!
The symbol represents the
empty
word. As a result, each bud scale
S
is removed from the model.
The basic function of the apex is to produce
shoot units,
as shown by production
p
2
:
p
2
:
A
!
I[L][B][L][B]A
Each shoot unit consists of an internode
I,
a pair of leaves
L,
a pair of lateral buds
B,
and another
apex
A.
The appearance of the apex in the successor provides for the repeated application of this
production (Figure 3).
After a predefined number of shoot units have been initiated, production stops with the apex
forming a terminal bud
T:
p
3
:
A
!
T
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T
L
A
L
A
L
A
I
I
I
B
B
L
L
B
B
B
I
B
L
L
B
L
L
B
B
B
I
B
I
B
L
L
L
Figure 3: The development of a green ash shoot unit.
E
term
R
E
L
E
lat
R
L
E
lat
L
R
E
lat
lat
E
R
lat
E
lat
E
lat
R
E
lat
E
lat
R
E
lat
E
lat
E
lat
R
Figure 4: Development of a green ash leaf.
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