L-systems II

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Current revision (22:38, 16 September 2013) (view source)

(→Types of L-systems used in Generative Art: small edit)

* Bracketed OL-systems: these systems are deterministic and context free. The use of bracket notation allows the use of a stack that can remember and return to previous locations and headings.

* Stochastic OL-systems: these systems are also context free but not deterministic. Rather than using a single mapping from predecessor to replacement string, these systems allow multiple replacements strings each with a probability of selection.

-

* Context Sensitive L-systems: these systems are deterministic but add the use of angle-brackets to either side of the strict-predecessor. To trigger replacement the context prior to~~, or~~ after, the strict-predecessor must also match. Such systems can simulate the transmission of signals up and down the resulting tree.

+

* Context Sensitive L-systems: these systems are deterministic but add the use of angle-brackets to either side of the strict-predecessor. To trigger replacement the context prior to and after the strict-predecessor must also match. Such systems can simulate the transmission of signals up and down the resulting tree.

* Parametric OL-systems: these systems add one or more floating point quantities within parenthesis to one or more letters creating a predecessor word. In addition a test is applied to the corresponding quantity. Such systems allow the creation of non-quantized lengths for stem segments.

* Parametric 2L-systems: these systems combine parameters and context sensitivity. Such systems can simulate models that involve the diffusion of substances throughout the tree.

* Other Stochastic L-systems: as a practical matter a stochastic element can be added to any combination of context sensitive and parametric features.

-

== Modeling using L-Systems in 3 Dimensions ==

-

{{SingleImage|imageWidthPlusTen=510|imageURL=http://www-viz.tamu.edu/courses/viza658/wiki/lsys2/01.jpg|caption=}}

+

{{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/01.jpg|caption=}}

+

== How context sensitive L-systems transmit signals ==

+

Here is a simple example of signal transmission:

+

The following sample L-system makes use of context to simulate signal

+

propagation throughout a string of symbols:

+

w: baaaaaaaa

+

p1: b < a → b

+

pg: b → a

+

The first few words generated by this L-system are given below:

+

baaaaaaaa

+

abaaaaaaa

+

aabaaaaaa

+

aaabaaaaa

+

aaaabaaaa

+

For fully animated growth models see "Timed DOL-systems" in section 6.1 of The [http://algorithmicbotany.org/papers/#abop Algorithmic Beauty of Plants.]

+

== How parametric (context free) OL-systems work ==

+

Here is a simple example of a parametric (context free) OL-system:

+

ω : B(2) A(4, 4)

+

p1 : A(x, y) : y ≤ 3 → A(x × 2, x + y)

+

p2 : A(x, y) : y > 3 → B(x) A(x/y, 0)

+

p3 : B(x) : x < 1 → C

+

p4 : B(x) : x ≥ 1 → B(x - 1)

+

{{SingleImage|imageWidthPlusTen=290|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/02.jpg|caption=}}

+

Here are the extended control symbols used in a 3D parametric L-system

+

<table>

+

<tr>

+

+

<td>Move forward a step of length a > 0. The position of the

+

turtle changes to (x', y', z'), where x' = x + aHx, y' = y + aHy and z' = z + aHz. A line segment is drawn between points (x, y, z) and (x', y', z').</td>

+

</tr>

+

<tr>

+

+

<td>Move forward a step of length a without drawing a line.</td>

+

</tr>

+

<tr>

+

+

<td>Rotate around U by an angle of a degrees. If a is positive, the turtle is turned to the left and if a is negative, the turn is to the right.</td>

+

</tr>

+

<tr>

+

+

<td>Rotate around L by an angle of a degrees. If a is positive, the turtle is pitched down and if a is negative, the turtle is pitched up.</td>

+

</tr>

+

<tr>

+

+

<td>Rotate around H by an angle of a degrees. If a is positive, the turtle is rolled to the right and if a is negative, it is rolled to the left.</td>

+

</tr>

+

</table>

+

{{SingleImage|imageWidthPlusTen=702|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/03.jpg|caption=}}

+

{{SingleImage|imageWidthPlusTen=728|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/04.jpg|caption=}}

+

== How parametric (context sensitive) L-systems work ==

+

Here is a simple example of a parametric 2L-system:

+

Productions in parametric OL-systems are context-free, ie., applicable

+

regardless of the context in which the predecessor appears. A context-

+

sensitive extension is necessary to model information exchange between

+

neighboring modules. In the parametric case, each component of the

+

production predecessor (the left context, the strict predecessor and the

+

right context) is a parametric word with letters from the alphabet V

+

and formal parameters from the set E. Any formal parameters may

+

appear in the condition and the production successor.

+

A sample context-sensitive production is given below:

+

A(x) < B(y) > C(z) : x + y + z > 10 → E((x + y)/2)F((y + z)/2)

+

It can be applied to the module B(5) that appears in a parametric word

+

…A(4)B(5)C(6)…

+

Because the test succeeds symbol replacement is made as follows (assuming "A(4)" and "C(6)" are not also words requiring replacement):

+

((x+y)/2) = ((4+5)/2) = 4.5

+

((y+z)/2) = ((5+6)/2) = 5.5

+

…A(4)E(4.5)F(5.5)C(6)…

+

== Artistic use of L-Systems ==

+

One of the problems with using L-System is that they are difficult to build with a specific end design in mind. This is one reason why evolving L-Systems using genetic algorithms is so useful. It allows the artist to "steer" the result towards a desired goal.

+

{{SingleImage|imageWidthPlusTen=962|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/05.jpg|caption=}}

+

{{SingleImage|imageWidthPlusTen=846|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/06.jpg|caption=}}

+

{{SingleImage|imageWidthPlusTen=919|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/07.jpg|caption=}}

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 * Bracketed OL-systems: these systems are deterministic and context free. The use of bracket notation allows the use of a stack that can remember and return to previous locations and headings.
 * Stochastic OL-systems: these systems are also context free but not deterministic. Rather than using a single mapping from predecessor to replacement string, these systems allow multiple replacements strings each with a probability of selection.
-* Context Sensitive L-systems: these systems are deterministic but add the use of angle-brackets to either side of the strict-predecessor. To trigger replacement the context prior to, or after, the strict-predecessor must also match. Such systems can simulate the transmission of signals up and down the resulting tree.
+* Context Sensitive L-systems: these systems are deterministic but add the use of angle-brackets to either side of the strict-predecessor. To trigger replacement the context prior to and after the strict-predecessor must also match. Such systems can simulate the transmission of signals up and down the resulting tree.
 * Parametric OL-systems: these systems add one or more floating point quantities within parenthesis to one or more letters creating a predecessor word. In addition a test is applied to the corresponding quantity. Such systems allow the creation of non-quantized lengths for stem segments.
 * Parametric 2L-systems: these systems combine parameters and context sensitivity. Such systems can simulate models that involve the diffusion of substances throughout the tree.
 * Other Stochastic L-systems: as a practical matter a stochastic element can be added to any combination of context sensitive and parametric features.
 == Modeling using L-Systems in 3 Dimensions ==
-{{SingleImage|imageWidthPlusTen=510|imageURL=http://www-viz.tamu.edu/courses/viza658/wiki/lsys2/01.jpg|caption=}}
+{{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/01.jpg|caption=}}
+== How context sensitive L-systems transmit signals ==
+Here is a simple example of signal transmission:
+The following sample L-system makes use of context to simulate signal
+propagation throughout a string of symbols:
+  w:  baaaaaaaa
+  p1: b < a &rarr; b
+  pg: b &rarr; a
+The first few words generated by this L-system are given below:
+  baaaaaaaa
+  abaaaaaaa
+  aabaaaaaa
+  aaabaaaaa
+  aaaabaaaa
+For fully animated growth models see "Timed DOL-systems" in section 6.1 of The [http://algorithmicbotany.org/papers/#abop Algorithmic Beauty of Plants.]
+== How parametric (context free) OL-systems work ==
+Here is a simple example of a parametric (context free) OL-system:
+ &omega;  : B(2) A(4, 4)
+ p1 : A(x, y)  : y ≤ 3 &rarr; A(x &times; 2, x + y)
+ p2 : A(x, y)  : y > 3 &rarr; B(x) A(x/y, 0)
+ p3 : B(x)     : x < 1 &rarr; C
+ p4 : B(x)     : x ≥ 1 &rarr; B(x - 1)
+{{SingleImage|imageWidthPlusTen=290|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/02.jpg|caption=}}
+Here are the extended control symbols used in a 3D parametric L-system
+<table>
+<tr>
+<td valign="top" width="10%"><i>F</i>(<i>a</i>)</td>
+<td>Move forward a step of length <i>a</i> > 0. The position of the
+turtle changes to (<i>x', y', z'</i>), where <i>x'</i> = <i>x</i> + <i>aH<sub>x</sub></i>, <i>y'</i> = <i>y</i> + <i>aH<sub>y</sub></i> and <i>z'</i> = <i>z</i> + <i>aH<sub>z</sub></i>. A line segment is drawn between points (<i>x, y, z</i>) and (<i>x', y', z'</i>).</td>
+</tr>
+<tr>
+<td valign="top" width="10%"><i>f</i>(<i>a</i>)</td>
+<td>Move forward a step of length a without drawing a line.</td>
+</tr>
+<tr>
+<td valign="top" width="10%">+(<i>a</i>)</td>
+<td>Rotate around U by an angle of <i>a</i> degrees. If <i>a</i> is positive, the turtle is turned to the left and if a is negative, the turn is to the right.</td>
+</tr>
+<tr>
+<td valign="top" width="10%">&amp;(<i>a</i>)</td>
+<td>Rotate around <i>L</i> by an angle of <i>a</i> degrees. If <i>a</i> is positive, the turtle is pitched down and if <i>a</i> is negative, the turtle is pitched up.</td>
+</tr>
+<tr>
+<td valign="top" width="10%">/(<i>a</i>)</td>
+<td>Rotate around <i>H</i> by an angle of <i>a</i> degrees. If <i>a</i> is positive, the turtle is rolled to the right and if <i>a</i> is negative, it is rolled to the left.</td>
+</tr>
+</table>
+{{SingleImage|imageWidthPlusTen=702|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/03.jpg|caption=}}
+{{SingleImage|imageWidthPlusTen=728|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/04.jpg|caption=}}
+== How parametric (context sensitive) L-systems work ==
+Here is a simple example of a parametric 2L-system:
+Productions in parametric OL-systems are context-free, ie., applicable
+regardless of the context in which the predecessor appears. A context-
+sensitive extension is necessary to model information exchange between
+neighboring modules. In the parametric case, each component of the
+production predecessor (the left context, the strict predecessor and the
+right context) is a parametric word with letters from the alphabet V
+and formal parameters from the set E. Any formal parameters may
+appear in the condition and the production successor.
+A sample context-sensitive production is given below:
+A(<i>x</i>) < B(<i>y</i>) > C(<i>z</i>) : <i>x</i> + <i>y</i> + <i>z</i> > 10 &rarr; E((<i>x</i> + <i>y</i>)/2)F((<i>y</i> + <i>z</i>)/2)
+It can be applied to the module <i>B</i>(5) that appears in a parametric word
+ …<i>A</i>(4)<i>B</i>(5)<i>C</i>(6)…
+Because the test succeeds symbol replacement is made as follows (assuming "A(4)" and "C(6)" are not also words requiring replacement):
+ ((<i>x</i>+<i>y</i>)/2) = ((4+5)/2) = 4.5
+ ((<i>y</i>+z)/2) = ((5+6)/2) = 5.5
+ …<i>A</i>(4)<i>E</i>(4.5)<i>F</i>(5.5)<i>C</i>(6)…
+== Artistic use of L-Systems ==
+One of the problems with using L-System is that they are difficult to build with a specific end design in mind. This is one reason why evolving L-Systems using genetic algorithms is so useful. It allows the artist to "steer" the result towards a desired goal.
+{{SingleImage|imageWidthPlusTen=962|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/05.jpg|caption=}}
+{{SingleImage|imageWidthPlusTen=846|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/06.jpg|caption=}}
+{{SingleImage|imageWidthPlusTen=919|imageURL=http://www.viz.tamu.edu/courses/viza658/wiki/lsys2/07.jpg|caption=}}