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.
#light
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
result
end
type TurtleGraphics =
class
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 }
t.position
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)
end
Now we need to represent the elements required for the
L-Systems. The following elements are required:
| Start point or axiom | the initial sequence of elements |
| Rules | L-system substitution rules |
| Angle | The angle used when rotating the turtle |
| Number of iterations | The number of times the rules will be applied to the axiom |
| Size of the initial segment | The 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 |
| Letter | If activated, draws a line forward |
The following code shows the implementation of this:
#light
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)
else
gettingLsystemElements str (i+1) ((Constant(str.[i].ToString()))::result)
else
List.rev result
let getLsystemElements str =
gettingLsystemElements str 0 []
end
type TurtleGraphicsLsystemProcessor =
class
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
in
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
tg.Advance(lp.seg_size)
lp.generate_points iterations rest tg (tg.Position::current) lines
else
lp.generate_points iterations rest tg current lines
| ((Constant "+")::rest) ->
tg.Rotate(lp.angle)
lp.generate_points iterations rest tg current lines
| ((Constant "-")::rest) ->
tg.Rotate(-1.0*lp.angle)
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
end
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);
this.canvas1.Children.Clear();
List<Rule> rules = GetRules();
TurtleGraphicsLsystemProcessor tgls =
new TurtleGraphicsLsystemProcessor(
LsystemFuncs.getLsystemElements(this.axiomTextBox.Text),
int.Parse(this.angleTB.Text),
Microsoft.FSharp.Collections.ListModule.of_IEnumerable<List<Rule>, Rule>(rules),
int.Parse(this.segmentSizeTB.Text),
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;
this.canvas1.Children.Add(pLine);
}
}
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.
