Mathematical Models of Trees by Fractal
2. METHOD
2.1 OBJECTIVE AREA
Modeling was aimed at the trees in the Musashi-Kyuryo National Government Park located in
Namegawa, Saitama.
2.2 CALCULATION OF FRACTAL DIMENSION FOR TREES
In this research, the fractal dimension was obtained with the box counting for photographs and
models (Takayasu, 1986). First, photographs of the trees were transformed into binary digit
images. The logarithm of a pixel size (resolution) was taken along a horizontal axis, while the
logarithm of pixels which overlapped the objective figure along vertical axis. The gradient of the
regression line for these plots becomes a fractal dimension with the minimum square method.
2.3 MODEL CONSTRUCTION
In this research, the model using the fractal figure was called fractal trees. The branch
development was called a bifurcation-type tree model.
2.3.1 ALGORISM METHOD OF FRACTAL TREES
Fractal trees are generated as an internal self-similarity set and a self-similarity set combining
reduction, rotation, parallel, and symmetrical mappings (Ishimura, 1990).
2.3.2 ALGORISM OF TWO-DIMENSIONAL BRANCH TYPE TREES
(1) Number of branching: 0 to 3
The probability of occurrence of each branch was assumed as
right branch, 78 %,
left branch, 78 %,
the same direction, 52 %.
(2) Length of branch
In the case of the branch extended left and right:
where a
n is the length of the n-th branch and Rnd is random numbers with 65% to 75% of the
length before branching.
In the case of the branch extended the almost same direction as the former branch before
branching:
where Rnd is random numbers with 80% to 90% of the length before branching.
(3) Angle at a branch (direction)
In the case of the branch extended left and right:
Its direction changes 0.3 rad to 0.5 rad to the branch before branching.
In the case of the branch extended the almost same direction as the former branch before
branching:
Compared with the branch before branching, direction changes -0.05 rad to 0.05 rad.
The fractions of the above parameters were given by random numbers.
Figure 1 Two reduction mappings of the tree model generated as a self-similarity set
Figure 2 Mappings of the tree model generated as an internal self-similarity set
2.3.3 THREE-DIMENSIONAL BRANCHING
The input parameters were defined as shown in Table 1, and the first trunk was set up. Next,
from the first branch, the lengths and angles of side branching were calculated and drawn
(Fuchigami, 1992).
Table 1 Input parameters
