Tuesday, August 28, 2007

Exploring L-Systems with F# and C#

In this post I'm going to show a little program for displaying graphical representations of L-Systems using turtle graphics implemented in F#, C# and WPF.

First of all this program could be easily implemented using only F#, but for me is interesting to see the interaction between native F# type/structures and C#. Because of this, the code that performs the L-system rewrite is written in F# and the code that takes the result is written in C# and uses WPF.

The first thing we need is an implementation of the turtle. Since we want to use this library with several graphics toolkits, we define our own point type for the generated data.


namespace Langexplr.Lsystems

open System
open Microsoft.FSharp.Math.Vector

type Point =
{x : int; y : int }

module Funcs = begin

let my_create_vector(i,j) =
let result = (create 2 0.0)
result.[0] <- i
result.[1] <- j

type TurtleGraphics =
val mutable direction : vector
val mutable position : Point

new(iX,iY) = { position = {x = iX; y = iY};
direction = Funcs.my_create_vector(1.0,0.0)}

member t.Position
with get() = t.position and
set(v) = t.position <- {x = v.x; y = v.y }

member t.Direction
with get() = t.direction and
set(v) = t.direction <- v

member t.Advance(distance : int) =
let aX = int_of_float (t.direction.[0] * float_of_int distance)
let aY = int_of_float (t.direction.[1] * float_of_int distance)
t.position <- { x = aX+t.position.x;
y = aY+t.position.y }

member t.Rotate(angle) =
let nI = (t.direction.[0] * Math.Cos(angle)) - (t.direction.[1] * Math.Sin(angle))
let nJ = (t.direction.[0] * Math.Sin(angle)) + (t.direction.[1] * Math.Cos(angle))
t.direction <- Funcs.my_create_vector(nI,nJ)


Now we need to represent the elements required for the L-Systems. The following elements are required:

Start point or axiomthe initial sequence of elements
Rules L-system substitution rules
AngleThe angle used when rotating the turtle
Number of iterationsThe number of times the rules will be applied to the axiom
Size of the initial segmentThe size in pixels of the line that is drawn when the turtle moves forward

Also the elements inside the rule and the axiom must be translated to turtle graphics commands. The following commands are supported:

|Draws a line forward, the size of the line inversely proportional to the iteration number
+Turn left by the specified angle
-Right left by the specified angle
[Saves the position and direction of the turtle in a stack
]Restores the position and direction of the turtle from the stack
LetterIf activated, draws a line forward

The following code shows the implementation of this:


namespace Langexplr.Lsystems

open Langexplr.Lsystems

open System

type LsystemElement =
| Var of String
| Constant of String
| PipeCommand of int

type Rule =
| Rule of LsystemElement * LsystemElement list

module LsystemFuncs = begin
let rec gettingLsystemElements (str:string) i result =
if str.Length > i then
if System.Char.IsLetterOrDigit(str.[i]) then
gettingLsystemElements str (i+1) ((Var(str.[i].ToString()))::result)
gettingLsystemElements str (i+1) ((Constant(str.[i].ToString()))::result)
List.rev result
let getLsystemElements str =
gettingLsystemElements str 0 []

type TurtleGraphicsLsystemProcessor =
val start : LsystemElement list
val angle : double
val rules : Rule list
val seg_size : int
val mutable saved_positions : Point list
val mutable saved_directions : vector list
val mutable drawVariables : bool

new (a_start,a_angle,t_rules,s_size,drawVars) = {
start = a_start;
angle = (Math.PI/180.0)* a_angle;
rules = t_rules;
seg_size = s_size;
saved_positions = [];
saved_directions = [];
drawVariables = drawVars}

member lp.generate_for n current =
match n with
| 0 -> current
| o -> lp.generate_for (n - 1) (lp.apply_rules current n)

member lp.apply_rules elements iteration =
match elements with
| ((Var v)::rest) -> List.append (lp.apply_rule_for v) (lp.apply_rules rest iteration)
| ((Constant "|")::rest) -> (PipeCommand iteration)::(lp.apply_rules rest iteration)
| (e::rest) -> e::(lp.apply_rules rest iteration)
| [] -> elements

member lp.apply_rule_for v =
match (List.tryfind (fun r -> match r with
| Rule(Var vvar,_) when vvar = v -> true
| _ -> false)
lp.rules) with
| Some (Rule(_, result)) -> result
| None -> [Var v]

member lp.generate_iteration n (tg:TurtleGraphics)=
let final = lp.generate_for n lp.start
List.rev(lp.generate_points n final tg [tg.Position] [])

member lp.generate_points iterations elements (tg:TurtleGraphics) current lines =
match elements with
| ((Var _)::rest) ->
if (lp.drawVariables) then
lp.generate_points iterations rest tg (tg.Position::current) lines
lp.generate_points iterations rest tg current lines
| ((Constant "+")::rest) ->
lp.generate_points iterations rest tg current lines
| ((Constant "-")::rest) ->
lp.generate_points iterations rest tg current lines
| ((Constant "[")::rest) ->
lp.saved_directions <- tg.Direction::lp.saved_directions
lp.saved_positions <- tg.Position::lp.saved_positions
lp.generate_points iterations rest tg current lines
| ((Constant "]")::rest) ->
match (lp.saved_directions,lp.saved_positions) with
| (cdir::rest_dir,cpos::rest_pos) ->
lp.saved_directions <- rest_dir
lp.saved_positions <- rest_pos
tg.Position <- cpos
tg.Direction <- cdir
lp.generate_points iterations rest tg [tg.Position] (current::lines)
| _ -> lp.generate_points iterations rest tg current lines
| ((Constant "|")::rest) ->
tg.Advance(lp.seg_size / (iterations) )
lp.generate_points iterations rest tg (tg.Position::current) lines
| ((PipeCommand iteration)::rest) ->
tg.Advance(lp.seg_size / (iterations - iteration) )
lp.generate_points iterations rest tg (tg.Position::current) lines
| (_::rest) -> lp.generate_points iterations rest tg current lines
| [] -> current::lines


The generate_iteration method is the one that generates the line information. Its result is a list of lists of elements of type Point.

The C# code that takes this F# list of lists of Points and converts it to a group of Polyline instances is the following:

private void b_Click(object sender, RoutedEventArgs e)
string origin = this.originTB.Text;
int originX = 50;
int originY = 50;
string[] oparts = origin.Split(',');
if (oparts.Length == 2)
originX = int.Parse(oparts[0]);
originY = int.Parse(oparts[1]);

TurtleGraphics tg = new TurtleGraphics(originX, originY);
int iterations = int.Parse(this.iterationsTB.Text);


List<Rule> rules = GetRules();

TurtleGraphicsLsystemProcessor tgls =
new TurtleGraphicsLsystemProcessor(
Microsoft.FSharp.Collections.ListModule.of_IEnumerable<List<Rule>, Rule>(rules),
drawVariablesCB.IsChecked == true);

var pointCollections =
from pc in (tgls.generate_iteration(int.Parse(this.iterationsTB.Text))).Invoke(tg)
select (new PointCollection(
from p in pc
select new System.Windows.Point(p.x, p.y)));

foreach (PointCollection pcol in pointCollections)
Polyline pLine = new Polyline();

pLine.Points = pcol;
pLine.Stroke = this.colorButton.Background;

What is interesting to see is that the list generated in F# is easily manipulated using LINQ. The expression that sets the value of pointCollections takes the native list of lists of points and converts it to a list of PointCollection objects in one expression .

Another interesting thing about the interaction between F# and C# is that the definition of the generate_iteration says that it could be applied with one or two arguments (because of Currying) this is used in C# by invoking the result of calling the method with one argument: (tgls.generate_iteration(int.Parse(this.iterationsTB.Text))).Invoke(tg).

Executing the program with the following L-system (from Wikipedia):

Also with using the "|" command:

Code for this experiment can be found here.