6. Results and discussion

The results are different systems which are summarized in Table 3. They are numbered in the first column; the second column contains the model consisting of the axiom and the production rules; and the following columns contain the parameters for each system. For this chapter, the implementation was done using Python 3 and the TK module, which is the standard graphic user interface for Python. Images of the results are presented after Table 3.

According to the classification presented in [6], systems 1, 2, 3, and 4 (which are shown in Figure 2) generate deltas that are tide-dominated. These present many distributaries that become entangled and cover a wide area; an example is the Fly River presented in part (b) of Figure 1. These systems generate skeletons that

Iteration

126

1 2 3 4

[[[[XF]

[++FFX + FFX [XF] + [FX] [XF] + [FX] + FFX] + [+FFX [XF] + [FX]]]

+ ++FFX + FFX [XF] + [FX] + +FFX + FFX [XF] + [FX] [[XF] + [FX] + FFX] +

[+FFX [XF] + [FX]]] + [+++F FX + FFX [XF] + [FX] + +FFX + FFX [XF] + [FX]

 [[XF] + [FX] + FFX] + [+FFX [XF] + [FX]] 

 [XF] + [FX]] + +FFX + FFX [XF] + [FX]] + [++FF X + FF X [XF] + [FX] [[XF]

+ [FX] + FFX] + [+FFX [XF] + [FX]]]]

> Table 2.

Resulting words after several iterations and their graphic

interpretation.

[[[XF] + [FX] + FF X] + [+FFX

 + [FX] + FFX] + [+FFX [XF] + [FX]] + +FFX + FFX [XF] + [FX]] +

[[[XF] + [FX] + FFX] + [+FFX [XF] + [FX]] + +FFX + FFX [XF] + [FX]] +

[++FFX + FFX [XF] + [FX] [XF] + [FX] + FFX] + [+FFX [XF] + [FX]]]

[[XF] + [FX] + FFX] + [+FFX [XF] + [FX]]

Resulting string

[XF] + [FX]

Graphical

interpretation

Technology, Science and Culture - A Global Vision, Volume II


#### Table 3.

Resulting L-systems.

become too cluttered and stop looking natural when there are too many iterations. A problem when using straight lines is that their entanglement may look like a grid as observed in systems 2 and 4.

> Also, system 8 has one more "+" symbol than "" symbols in the axiom, and because of this, all the branches tend to move upwards. As mentioned before, balance is needed in the number of angle-changing symbols. System 11 is another variation where the last "+" is omitted; therefore, the shape is even more curved at it would become like a spiral if more iterations are performed. This also proves how challenging the control over these systems could be, as changes are minimal but

Systems that generate tide-dominated deltas: (a) system 1, (b) system 2, (c) system 3, and (d) system 4.

Figure 5 depicts the evolution of system 5. It needs five iterations to reach the first stage shown, the second stage is reached at seven iterations, and the last one is at nine iterations. Exponential growth is easily observed. When comparing the systems, it can be observed that those systems that have longer rules need fewer iterations to reach the desired skeleton. This also applies to those that recall the axiom several times. There are important differences with L-systems that generate vegetation. In plants changes in angle occur almost exclusively when creating a new branch. This is that the symbols "+" or "" appear just after opening a new branch with the symbol "[." In the case of the rivers, the change-angle symbols surround the drawline symbols "F" and may also appear outside of branching. As an example of this, if system 9 has its change-angle symbols eliminated from the rule that rewrites each

rendered images are quite different.

An Approach for River Delta Generation Using L-Systems

DOI: http://dx.doi.org/10.5772/intechopen.90099

Figure 2.

129

Systems 5, 6, and 7 (Figure 3) generate deltas that are of the river-dominated type. These have more elongated shapes, and their branches tend to have smaller lengths; also, the main branch can be identified. The Mississippi River is the best example of this type of deltas; it is shown on part (a) of Figure 1.

When generating this type of deltas, too many iterations have not much effect; because of the fractal nature of the generation, the general shape remains the same. For this reason, it makes no sense to have too many iterations nor for this type of delta nor for the previous one; in both cases, the number of iterations in every system remains low as can be observed in Table 3. To obtain river-dominated deltas, the symbol "F" has a rule that calls the axiom every time; this means that each line will start another system in the following iteration.

Systems 8, 9, 10, and 11, depicted in Figure 4, present combinations of the previously mentioned types of deltas. System 9 is more closely related to the riverdominated type, while system 10 has more influence of the tide-dominated type. Systems 8 and 11 are variations of system 2, and the difference is recalling the axiom more times and adding more straight lines; therefore, they have more selfsimilarity. The problem with having more lines is that they tend to look less sinuous and thus less natural.

Also, system 8 has one more "+" symbol than "" symbols in the axiom, and because of this, all the branches tend to move upwards. As mentioned before, balance is needed in the number of angle-changing symbols. System 11 is another variation where the last "+" is omitted; therefore, the shape is even more curved at it would become like a spiral if more iterations are performed. This also proves how challenging the control over these systems could be, as changes are minimal but rendered images are quite different.

Figure 5 depicts the evolution of system 5. It needs five iterations to reach the first stage shown, the second stage is reached at seven iterations, and the last one is at nine iterations. Exponential growth is easily observed. When comparing the systems, it can be observed that those systems that have longer rules need fewer iterations to reach the desired skeleton. This also applies to those that recall the axiom several times.

There are important differences with L-systems that generate vegetation. In plants changes in angle occur almost exclusively when creating a new branch. This is that the symbols "+" or "" appear just after opening a new branch with the symbol "[." In the case of the rivers, the change-angle symbols surround the drawline symbols "F" and may also appear outside of branching. As an example of this, if system 9 has its change-angle symbols eliminated from the rule that rewrites each

become too cluttered and stop looking natural when there are too many iterations. A problem when using straight lines is that their entanglement may look like a grid

System Axiom Production rules Iterations β r x<sup>0</sup>

S ! F+I L ! F – I

S ! F � I L ! F+I

S ! F � I L ! F+I

S ! F � I L ! F+I

X ! +F � [R] I F ! F+I�

> S ! [�I] F ! -F + I

S ! F � I L ! F+I

> S ! FL L ! F

S ! F � I+ L ! F+I�

S ! F � I L ! F+I

1 ω =I I ! F + +F � �R[X]

3 ω =I I ! +F � F[�R]X

4 ω = I I ! FF[�R]X

5 ω = I I ! F[X]

6 ω =I I ! �F[�X] + [+X]FI

7 ω =I I ! -F[-FR] + F[+FR]FI

9 ω =I I ! �F[�RI] + F[R] � FI

10 ω = I I ! +F – F � RX

8 ω =I I ! +F � F[�RI] � F + F[RI] � FI+

11 ω =I I ! +F � F[�RI] � F + F[RI] � FI

2 ω =I I ! �F + F[�R] � F + F[+R] I

X ! FR [I] R ! +FI – [LS]

Technology, Science and Culture - A Global Vision, Volume II

R ! F – LSF � F

X ! +F � F[+R] I R ! F � L + SF

> X ! FF[+R] I R ! FLS

R ! F[�I] F [+I] F

X ! +F � F[�R] FF R ! F[+S] [�S] F

R ! F[+R � I] F [�R + I] F

R ! F – LSF � F+F

R ! F[+I] F[�I] F F ! �F+S

X ! +F � F + RI R ! F[SL] F

R ! F – LSF � F+F

, y<sup>0</sup> θ<sup>0</sup>

9 20 17 10, 30 0

8 25 15 40, 420 20

11 15 15 40, 220 20

12 25 15 140, 120 20

10 30 10 10, 130 310

6 20 10 10, 320 60

5 15 14 40, 220 44

6 15 15 40,220 60

10 20 15 110, 150 60

4 15 15 40, 220 70

F ! �FI 5 20 9 10, 150 40

Systems 5, 6, and 7 (Figure 3) generate deltas that are of the river-dominated type. These have more elongated shapes, and their branches tend to have smaller lengths; also, the main branch can be identified. The Mississippi River is the best

When generating this type of deltas, too many iterations have not much effect; because of the fractal nature of the generation, the general shape remains the same. For this reason, it makes no sense to have too many iterations nor for this type of delta nor for the previous one; in both cases, the number of iterations in every system remains low as can be observed in Table 3. To obtain river-dominated deltas, the symbol "F" has a rule that calls the axiom every time; this means that

Systems 8, 9, 10, and 11, depicted in Figure 4, present combinations of the previously mentioned types of deltas. System 9 is more closely related to the riverdominated type, while system 10 has more influence of the tide-dominated type. Systems 8 and 11 are variations of system 2, and the difference is recalling the axiom

similarity. The problem with having more lines is that they tend to look less sinuous

more times and adding more straight lines; therefore, they have more self-

example of this type of deltas; it is shown on part (a) of Figure 1.

each line will start another system in the following iteration.

as observed in systems 2 and 4.

Table 3. Resulting L-systems.

and thus less natural.

128

Figure 3. River-dominated deltas: (a) system 5, (b) system 6, and (c) system 7.

line (symbol F), most of the sinuosity is eliminated, and the resulting shape resembles more of a pine branch than a river as could be seen in Figure 6.

Figure 4.

Figure 5.

Figure 6.

131

System 9 when eliminating changes in angles.

Deltas that present combined characteristics: (a) system 8, (b) system 9, (c) system 10, and (d) system 11.

From left to right system 5 at 5, 7, and 9 iterations, respectively.

An Approach for River Delta Generation Using L-Systems

DOI: http://dx.doi.org/10.5772/intechopen.90099

Another difference is that trees use exponential growth to differentiate the trunk from the branches and leaves. This is not desirable for river deltas as the sinuosity and the presence of branching are not restricted to areas far from the start point as it is in trees.

The presented skeletons are preliminary results, and they only use straight lines. Also, the systems are deterministic; this means that each symbol only has one rule at most and each system generates the same skeleton every time when using the same parameters.

River deltas have different widths in their branches; this changes the way they look, but the underlying structure remains constant. For example, section (a) of Figure 7 shows the entire Skeidarársandur river delta, and section (b) shows a zoom in of the central part, which shows smaller entangled distributaries. The entirety of the delta has these formations, but the greater width of some of the distributaries affects the shape and hides the root-like structure. As a comparison section, (c) shows an extract of system 3 where the width of distributaries is constant.

In Figure 8 there are comparisons of real river deltas and some of the deltas generated by the systems; the shown rivers are the Lena river (a), Papua river (b), and a secondary delta of the Mississippi system (c). Systems 1 (d) and 10 (e) were rotated for this comparison; this is achieved by altering the initial angle θ<sup>0</sup> , and the final image (f) corresponds to system 6.

An Approach for River Delta Generation Using L-Systems DOI: http://dx.doi.org/10.5772/intechopen.90099

#### Figure 4.

line (symbol F), most of the sinuosity is eliminated, and the resulting shape resem-

Another difference is that trees use exponential growth to differentiate the trunk from the branches and leaves. This is not desirable for river deltas as the sinuosity and the presence of branching are not restricted to areas far from the start point as it

The presented skeletons are preliminary results, and they only use straight lines. Also, the systems are deterministic; this means that each symbol only has one rule at most and each system generates the same skeleton every time when using the same

River deltas have different widths in their branches; this changes the way they look, but the underlying structure remains constant. For example, section (a) of Figure 7 shows the entire Skeidarársandur river delta, and section (b) shows a zoom in of the central part, which shows smaller entangled distributaries. The entirety of the delta has these formations, but the greater width of some of the distributaries affects the shape and hides the root-like structure. As a comparison section, (c) shows an extract of system 3 where the width of distributaries is constant.

In Figure 8 there are comparisons of real river deltas and some of the deltas generated by the systems; the shown rivers are the Lena river (a), Papua river (b), and a secondary delta of the Mississippi system (c). Systems 1 (d) and 10 (e) were

, and the

rotated for this comparison; this is achieved by altering the initial angle θ<sup>0</sup>

final image (f) corresponds to system 6.

bles more of a pine branch than a river as could be seen in Figure 6.

River-dominated deltas: (a) system 5, (b) system 6, and (c) system 7.

Technology, Science and Culture - A Global Vision, Volume II

is in trees.

Figure 3.

parameters.

130

Deltas that present combined characteristics: (a) system 8, (b) system 9, (c) system 10, and (d) system 11.

#### Figure 5.

From left to right system 5 at 5, 7, and 9 iterations, respectively.

Figure 6. System 9 when eliminating changes in angles.

Finally, Figure 9 presents a comparison of the results of the method proposed by Seybold [9, 6], Teoh [9], and system 7 of the proposed method. It is to note that the method proposed by Seybold is intended for simulation of delta growth and not for use in virtual worlds. Therefore, it is quite accurate but is computationally expen-

In this chapter, a method for generating branching river skeletons using Lsystems was presented. It was proven that L-systems could be successfully adapted to this task as it was shown by the preliminary resulting systems and their graphic interpretations. This is an original method for generating river deltas and a new application of L-systems. The generated skeletons resemble deltas dominated by the river or by tides and some combinations in between. Also, general guidelines were provided to generate this type of branching structures. These skeletons can be used in procedural terrain generation to add more features in virtual terrains depending

As this is a work in progress, some changes will be done in the future to generate more realistic results; curved lines could be used to increase the resemblance to river deltas. By using neural networks, textures could be added automatically over these skeletons. Variability could be improved by switching to stochastic systems, and this approach could be paired with machine learning techniques to perform an

sive in comparison to the other two.

An Approach for River Delta Generation Using L-Systems

DOI: http://dx.doi.org/10.5772/intechopen.90099

7. Conclusions and future work

on the characteristics of the generated terrains.

Luis Oswaldo Valencia-Rosado\* and Oleg Starostenko\*

\*Address all correspondence to: luis.valenciaro@udlap.mx

Américas Puebla, Cholula, Puebla, Mexico

provided the original work is properly cited.

and oleg.starostenko@udlap.mx;

Department of Computing, Electronics, and Mechatronics, Universidad de las

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

automatic generation of the skeletons.

Author details

133

Figure 7.

(a) Skeidarársandur delta [8]; (b) central part zoom in [16]; and (c) extract of system 3.

#### Figure 8.

Comparison between river deltas and generated systems, (a) Lena [8], (b) Papua, (c) [8] Mississippi branch, (d) [16] system 1, (e) system 10, and (f) system 6.

Figure 9.

Results of different methods, (a) method proposed by Seybold [6], (b) method proposed by Teoh [9], and (c) system 7.

### An Approach for River Delta Generation Using L-Systems DOI: http://dx.doi.org/10.5772/intechopen.90099

Finally, Figure 9 presents a comparison of the results of the method proposed by Seybold [9, 6], Teoh [9], and system 7 of the proposed method. It is to note that the method proposed by Seybold is intended for simulation of delta growth and not for use in virtual worlds. Therefore, it is quite accurate but is computationally expensive in comparison to the other two.
