Dragon curve
Create and display a dragon curve fractal.
You are encouraged to solve this task according to the task description, using any language you may know.
(You may either display the curve directly or write it to an image file.)
- Algorithms
Here are some brief notes the algorithms used and how they might suit various languages.
- Recursively a right curling dragon is a right dragon followed by a left dragon, at 90-degree angle. And a left dragon is a left followed by a right.
*---R----* expands to * *
\ /
R L
\ /
*
*
/ \
L R
/ \
*---L---* expands to * *
- The co-routines
dclanddcrin various examples do this recursively to a desired expansion level.
- The curl direction right or left can be a parameter instead of two separate routines.
- Recursively, a curl direction can be eliminated by noting the dragon consists of two copies of itself drawn towards a central point at 45-degrees.
*------->* becomes * * Recursive copies drawn
\ / from the ends towards
\ / the centre.
v v
*
- This can be seen in the SVG example. This is best suited to off-line drawing since the reversal in the second half means the drawing jumps backward and forward (in binary reflected Gray code order) which is not very good for a plotter or for drawing progressively on screen.
- Successive approximation repeatedly re-writes each straight line as two new segments at a right angle,
*
*-----* becomes / \ bend to left
/ \ if N odd
* *
* *
*-----* becomes \ / bend to right
\ / if N even
*
- Numbering from the start of the curve built so far, if the segment is at an odd position then the bend introduced is on the right side. If the segment is an even position then on the left. The process is then repeated on the new doubled list of segments. This constructs a full set of line segments before any drawing.
- The effect of the splitting is a kind of bottom-up version of the recursions. See the Asymptote example for code doing this.
- Iteratively the curve always turns 90-degrees left or right at each point. The direction of the turn is given by the bit above the lowest 1-bit of n. Some bit-twiddling can extract that efficiently.
n = 1010110000
^
bit above lowest 1-bit, turn left or right as 0 or 1
LowMask = n BITXOR (n-1) # eg. giving 0000011111
AboveMask = LowMask + 1 # eg. giving 0000100000
BitAboveLowestOne = n BITAND AboveMask
- The first turn is at n=1, so reckon the curve starting at the origin as n=0 then a straight line segment to position n=1 and turn there.
- If you prefer to reckon the first turn as n=0 then take the bit above the lowest 0-bit instead. This works because "...10000" minus 1 is "...01111" so the lowest 0 in n-1 is where the lowest 1 in n is.
- Going by turns suits turtle graphics such as Logo or a plotter drawing with a pen and current direction.
- If a language doesn't maintain a "current direction" for drawing then you can always keep that separately and apply turns by bit-above-lowest-1.
- Absolute direction to move at point n can be calculated by the number of bit-transitions in n.
n = 11 00 1111 0 1
^ ^ ^ ^ 4 places where change bit value
so direction=4*90degrees=East
- This can be calculated by counting the number of 1 bits in "n XOR (n RIGHTSHIFT 1)" since such a shift and xor leaves a single 1 bit at each position where two adjacent bits differ.
- Absolute X,Y coordinates of a point n can be calculated in complex numbers by some powers (i+1)^k and add/subtract/rotate. This is done in the gnuplot code. This might suit things similar to Gnuplot which want to calculate each point independently.
- Predicate test for whether a given X,Y point or segment is on the curve can be done. This might suit line-by-line output rather than building an entire image before printing. See M4 for an example of this.
- A predicate works by dividing out complex number i+1 until reaching the origin, so it takes roughly a bit at a time from X and Y is thus quite efficient. Why it works is slightly subtle but the calculation is not difficult. (Check segment by applying an offset to move X,Y to an "even" position before dividing i+1. Check vertex by whether the segment either East or West is on the curve.)
- The number of steps in the predicate corresponds to doublings of the curve, so stopping the check at say 8 steps can limit the curve drawn to 2^8=256 points. The offsets arising in the predicate are bits of n the segment number, so can note those bits to calculate n and limit to an arbitrary desired length or sub-section.
- As a Lindenmayer system of expansions. The simplest is two symbols F and S both straight lines, as used by the PGF code.
Axiom F, angle 90 degrees F -> F+S S -> F-S
This always has F at even positions and S at odd. Eg. after 3 levels F_S_F_S_F_S_F_S. The +/- turns in between bend to the left or right the same as the "successive approximation" method above. Read more at for instance Joel Castellanos' L-system page.
Variations are possible if you have only a single symbol for line draw, for example the Icon and Unicon and Xfractint code. The angles can also be broken into 45-degree parts to keep the expansion in a single direction rather than the endpoint rotating around.
The string rewrites can be done recursively without building the whole string, just follow its instructions at the target level. See for example C by IFS Drawing code. The effect is the same as "recursive with parameter" above but can draw other curves defined by L-systems.
Action!
Action! language does not support recursion. Therefore an iterative approach with a stack has been proposed. <lang Action!>DEFINE MAXSIZE="20"
INT ARRAY
xStack(MAXSIZE),yStack(MAXSIZE), dxStack(MAXSIZE),dyStack(MAXSIZE)
BYTE ARRAY
dirStack(MAXSIZE), depthStack(MAXSIZE),stageStack(MAXSIZE)
BYTE stacksize=[0]
BYTE FUNC IsEmpty()
IF stacksize=0 THEN RETURN (1) FI
RETURN (0)
BYTE FUNC IsFull()
IF stacksize=MAXSIZE THEN RETURN (1) FI
RETURN (0)
PROC Push(INT x,y,dx,dy BYTE dir,depth,stage)
IF IsFull() THEN Break() FI xStack(stacksize)=x yStack(stacksize)=y dxStack(stacksize)=dx dyStack(stacksize)=dy dirStack(stacksize)=dir depthStack(stacksize)=depth stageStack(stackSize)=stage stacksize==+1
RETURN
PROC Pop(INT POINTER x,y,dx,dy BYTE POINTER dir,depth,stage)
IF IsEmpty() THEN Break() FI stacksize==-1 x^=xStack(stacksize) y^=yStack(stacksize) dx^=dxStack(stacksize) dy^=dyStack(stacksize) dir^=dirStack(stacksize) depth^=depthStack(stacksize) stage^=stageStack(stacksize)
RETURN
PROC DrawDragon(INT x,y,dx,dy BYTE depth)
BYTE dir,stage INT nx,ny,dx2,dy2,tmp Push(x,y,dx,dy,1,depth,0)
WHILE IsEmpty()=0
DO
Pop(@x,@y,@dx,@dy,@dir,@depth,@stage)
IF depth=0 THEN
Plot(x,y) DrawTo(x+dx,y+dy)
ELSE
IF stage<2 THEN
Push(x,y,dx,dy,dir,depth,stage+1)
FI
nx=dx/2 ny=dy/2
dx2=nx-ny dy2=nx+ny
IF stage=0 THEN
IF dir THEN
Push(x,y,dx2,dy2,1,depth-1,0)
ELSE
Push(x,y,dy2,-dx2,1,depth-1,0)
FI
ELSEIF stage=1 THEN
IF dir THEN
tmp=-dx2 ;to avoid the compiler error
Push(x+dx2,y+dy2,dy2,tmp,0,depth-1,0)
ELSE
Push(x+dy2,y-dx2,dx2,dy2,0,depth-1,0)
FI
FI
FI
OD
RETURN
PROC Main()
BYTE CH=$02FC,COLOR1=$02C5,COLOR2=$02C6
Graphics(8+16) Color=1 COLOR1=$0C COLOR2=$02
DrawDragon(104,72,128,0,12)
DO UNTIL CH#$FF OD CH=$FF
RETURN</lang>
- Output:
ALGOL 68
File: prelude/turtle_draw.a68<lang algol68># -*- coding: utf-8 -*- #
STRUCT (REAL x, y, heading, BOOL pen down) turtle;
PROC turtle init = VOID: (
draw erase (window); turtle := (0.5, 0.5, 0, TRUE); draw move (window, x OF turtle, y OF turtle); draw colour name(window, "white")
);
PROC turtle left = (REAL left turn)VOID:
heading OF turtle +:= left turn;
PROC turtle right = (REAL right turn)VOID:
heading OF turtle -:= right turn;
PROC turtle forward = (REAL distance)VOID:(
x OF turtle +:= distance * cos(heading OF turtle) / width * height; y OF turtle +:= distance * sin(heading OF turtle); IF pen down OF turtle THEN draw line ELSE draw move FI (window, x OF turtle, y OF turtle)
);
SKIP</lang>File: prelude/exception.a68<lang algol68># -*- coding: utf-8 -*- #
COMMENT
REQUIRES : MODE EXCEPTOBJ = UNION(VOID, MODEA, MODEB, MODEC ...); OP FIRMSTR = (EXCEPTOBJ obj)STRING: ~
END COMMENT
MODE EXCEPTOBJS = [0]EXCEPTOBJ;
OP STR = (EXCEPTOBJS obj)STRING: (
STRING out := "(", fs := "";
FOR this FROM LWB obj TO UPB obj DO out +:= fs+FIRMSTR obj[this]; fs:=", " OD;
out +")"
);
MODE EXCEPTMEND = PROC(EXCEPTOBJS #obj#,STRING #msg#)BOOL;
PROC super mend = (EXCEPTOBJS obj,STRING sub exception, msg)BOOL:
( put(stand error, ("exception/",sub exception,": ", msg," - ", STR obj, new line)); break; TRUE);
PROC super break mend = (EXCEPTOBJS obj,STRING sub exception, msg)BOOL: ( super mend(obj, sub exception, msg); break; TRUE); PROC super stop mend = (EXCEPTOBJS obj,STRING sub exception, msg)BOOL: ( super mend(obj, sub exception, msg); stop; FALSE); PROC super ignore mend = (EXCEPTOBJS obj,STRING sub exception, msg)BOOL: ( #super mend(obj, sub exception, msg);# TRUE);
EXCEPTMEND on undefined mend := super break mend(,"undefined",); PROC on undefined = (EXCEPTMEND undefined mend)VOID: on undefined mend := undefined mend; PROC raise undefined = (EXCEPTOBJS obj, STRING msg)VOID: IF NOT on undefined mend(obj, msg) THEN stop FI;
EXCEPTMEND on value error mend := super break mend(,"value error",); PROC on value error = (EXCEPTMEND value error mend)VOID: on value error mend := value error mend; PROC raise value error = (EXCEPTOBJS obj, STRING msg)VOID: IF NOT on value error mend(obj, msg) THEN stop FI;
EXCEPTMEND on bounds error mend := super break mend(,"bounds error",); PROC on bounds error = (EXCEPTMEND bounds error mend)VOID: on bounds error mend := bounds error mend; PROC raise bounds error = (EXCEPTOBJS obj, STRING msg)VOID: IF NOT on bounds error mend(obj, msg) THEN stop FI;
EXCEPTMEND on tagged union error mend := super break mend(,"tagged union error",); PROC on tagged union error = (EXCEPTMEND tagged union error mend)VOID: on tagged union error mend := tagged union error mend; PROC raise tagged union error = (EXCEPTOBJS obj, STRING msg)VOID: IF NOT on tagged union error mend(obj, msg) THEN stop FI;
EXCEPTMEND on untested mend := super break mend(,"untested",); PROC on untested = (EXCEPTMEND untested mend)VOID: on untested mend := untested mend; PROC raise untested = (EXCEPTOBJS obj, STRING msg)VOID: IF NOT on untested mend(obj, msg) THEN stop FI;
EXCEPTMEND on unimplemented mend := super break mend(,"unimplemented",); PROC on unimplemented = (EXCEPTMEND unimplemented mend)VOID: on unimplemented mend := unimplemented mend; PROC raise unimplemented = (EXCEPTOBJS obj, STRING msg)VOID: IF NOT on unimplemented mend(obj, msg) THEN stop FI;
SKIP</lang>File: test/Dragon_curve.a68<lang algol68>#!/usr/bin/a68g --script #
- -*- coding: utf-8 -*- #
PR read "prelude/turtle_draw.a68" PR; MODE EXCEPTOBJ = FILE; OP FIRMSTR = (EXCEPTOBJ obj)STRING: "FILE";
PR read "prelude/exception.a68" PR;
REAL sqrt 2 = sqrt(2), degrees = pi/180;
STRUCT ( INT count, depth, current shade, upb lines, upb colours ) morph;
PROC morph init = (INT depth)VOID: (
count OF morph := 0; depth OF morph := depth; current shade OF morph := -1; upb lines OF morph := 2**depth; upb colours OF morph := 16
);
PROC morph colour = VOID: (
INT colour sectors = 3; # RGB #
INT candidate shade = ENTIER ( count OF morph / upb lines OF morph * upb colours OF morph );
IF candidate shade /= current shade OF morph THEN
current shade OF morph := candidate shade;
REAL colour sector = colour sectors * candidate shade / upb colours OF morph - 0.5;
REAL shade = colour sector - ENTIER colour sector;
CASE ENTIER colour sector + 1 # of 3 # IN
draw colour (window, 1 - shade, shade, 0),
draw colour (window, 0, 1 - shade, shade)
OUT
draw colour (window, shade, 0, 1 - shade)
ESAC
FI;
count OF morph +:= 1
);
PROC dragon init = VOID: (
pen down OF turtle := FALSE; turtle forward(23/64); turtle right(90*degrees); turtle forward (1/8); turtle right(90*degrees); pen down OF turtle := TRUE
);
PROC dragon = (REAL in step, in length, PROC(REAL)VOID zig, zag)VOID: (
IF in step <= 0 THEN morph colour; turtle forward(in length) ELSE REAL step = in step - 1; REAL length = in length / sqrt 2; zig(45*degrees); dragon(step, length, turtle right, turtle left); zag(90*degrees); dragon(step, length, turtle left, turtle right); zig(45*degrees) FI
);
PROC window init = VOID: (
STRING aspect; FILE f; associate(f, aspect); putf(f, ($g(-4)"x"g(-3)$, width, height));
CO # depricated #
IF NOT draw device (window, "X", aspect)THEN raise undefined(window, "cannot initialise X draw device") FI;
END CO
IF open (window, "Dragon Curve", stand draw channel) = 0 THEN raise undefined(window, "cannot open Dragon Curve window") FI; IF NOT make device (window, "X", aspect) THEN raise undefined(window, "cannot make device X draw device") FI
);
INT width = 800-15, height = 600-15;
FILE window; window init;
INT cycle length = 18; FOR snap shot TO cycle length DO INT depth := (snap shot - 2) MOD cycle length; turtle init; dragon init; morph init(depth);
- move to initial turtle location #
dragon(depth, 7/8, turtle right, turtle left);
draw show (window);
VOID(system("sleep 1"))
OD;
close (window)</lang>
Output:
 |
Note: each Dragon curve is composed of many smaller dragon curves (shown in a different colour).
AmigaE
Example code using mutual recursion can be found in Recursion Example of "A Beginner's Guide to Amiga E".
Applesoft BASIC
Apple IIe BASIC code can be found in Thomas Bannon, "Fractals and Transformations", Mathematics Teacher, March 1991, pages 178-185. (At JSTOR.)
Asymptote
The Asymptote source code includes an examples/dragon.asy which draws the dragon curve (four interlocking copies actually),
As of its version 2.15 it uses the successive approximation method. Vertices are represented as an array of "pairs" (complex numbers). Between each two vertices a new vertex is is introduced so as to double the segments, repeated to a desired level.
AutoHotkey
BASIC
<lang qbasic>DIM SHARED angle AS Double
SUB turn (degrees AS Double)
angle = angle + degrees*3.14159265/180
END SUB
SUB forward (length AS Double)
LINE - STEP (cos(angle)*length, sin(angle)*length), 7
END SUB
SUB dragon (length AS Double, split AS Integer, d AS Double)
IF split=0 THEN
forward length
ELSE
turn d*45 dragon length/1.4142136, split-1, 1 turn -d*90 dragon length/1.4142136, split-1, -1 turn d*45
END IF
END SUB
' Main program
SCREEN 12 angle = 0 PSET (150,180), 0 dragon 400, 12, 1 SLEEP</lang>
See also Sydney Afriat "Dragon Curves" paper for various approaches in BASIC
- http://www.econ-pol.unisi.it/~afriat/Papers.html
- http://www.econ-pol.unisi.it/~afriat/Math_Dragon.pdf
And TRS-80 BASIC code in Dan Rollins, "A Tiger Meets a Dragon: An examination of the mathematical properties of dragon curves and a program to print them on an IDS Paper Tiger", Byte Magazine, December 1983. (Based on generating a string of turns by appending middle turn and reversed copy. Options for the middle turn give the alternate paper folding curve and more too. The turns are then followed for the plot.)
IS-BASIC
<lang IS-BASIC>100 PROGRAM "Dragon.bas" 110 OPTION ANGLE DEGREES 120 LET SQ2=SQR(2) 130 GRAPHICS HIRES 2 140 SET PALETTE 0,33 150 PLOT 250,360,ANGLE 0; 160 CALL DC(580,0,11) 170 DEF DC(D,A,LEV) 180 IF LEV=0 THEN 190 PLOT FORWARD D; 200 ELSE 210 PLOT RIGHT A; 220 CALL DC(D/SQ2,45,LEV-1) 230 PLOT LEFT 2*A; 240 CALL DC(D/SQ2,-45,LEV-1) 250 PLOT RIGHT A; 260 END IF 270 END DEF</lang>
BASIC256
<lang basic256># Version without functions (for BASIC-256 ver. 0.9.6.66)
graphsize 390,270
level = 18 : insize = 247 # initial values x = 92 : y = 94 #
iters = 2^level # total number of iterations qiter = 510/iters # constant for computing colors SQ = sqrt(2) : QPI = pi/4 # constants
rotation = 0 : iter = 0 : rq = 1.0 # state variables dim rqs(level) # stack for rq (rotation coefficient)
color white fastgraphics rect 0,0,graphwidth,graphheight refresh gosub dragon refresh imgsave "Dragon_curve_BASIC-256.png", "PNG" end
dragon: if level<=0 then yn = sin(rotation)*insize + y xn = cos(rotation)*insize + x if iter*2<iters then color 0,iter*qiter,255-iter*qiter else color qiter*iter-255,(iters-iter)*qiter,0 end if line x,y,xn,yn iter = iter + 1 x = xn : y = yn return end if insize = insize/SQ rotation = rotation + rq*QPI level = level - 1 rqs[level] = rq : rq = 1 gosub dragon rotation = rotation - rqs[level]*QPI*2 rq = -1 gosub dragon rq = rqs[level] rotation = rotation + rq*QPI level = level + 1 insize = insize*SQ return</lang>
BBC BASIC
<lang bbcbasic> MODE 8
MOVE 800,400
GCOL 11
PROCdragon(512, 12, 1)
END
DEF PROCdragon(size, split%, d)
PRIVATE angle
IF split% = 0 THEN
DRAW BY -COS(angle)*size, SIN(angle)*size
ELSE
angle += d*PI/4
PROCdragon(size/SQR(2), split%-1, 1)
angle -= d*PI/2
PROCdragon(size/SQR(2), split%-1, -1)
angle += d*PI/4
ENDIF
ENDPROC</lang>
Befunge
This is loosely based on the M4 predicate algorithm, only it produces a more compact ASCII output (which is also a little easier to implement), and it lets you choose the depth of the expansion rather than having to specify the coordinates of the viewing area.
In Befunge-93 the 8-bit cell size restricts you to a maximum depth of 15, but in Befunge-98 you should be able go quite a bit deeper before other limits of the implementation come into play.
<lang befunge>" :htpeD">:#,_&>:00p:2%10p:2/:1+1>\#<1#*-#2:#\_$:1-20p510g2*-*1+610g4vv<v< | v%2\/3-1$_\#!4#:*#-\#1<\1+1:/4+1g00:\_\#$1<%2/2+1\g02\-1+%-g012\/-*<v"*/ _ >!>0$#0\#$\_-10p20p::00g4/:1+1>\#<1#*-#4:#\_$1-2*3/\2%!>0$#0\#$\_--vv|+2 v:\p06!*-1::p05<g00+1--g01g03\+-g01-p04+1:<0p03:-1_>>$$$1-\1->>:v:+1\<v~:: >:1+*!60g*!#v_!\!*50g0`*!40gg,::30g40g:2-#^_>>$>>:^:+1g02::\,+55_55+,@v":* v%2/2+*">~":< ^\-1g05-*">~"/2+*"|~"-%*"|~"\/*"|~":\-*">~"/2+%*"|~"\/*<^<: >60p\:"~>"*+2/2%60g+2%70p:"kI"*+2/2%60p\:"kI"*+2/2%60g+2%-\70g-"~|"**+"}"^</lang>
- Output:
Depth: 9
_ _
|_|_ |_|_
_ _|_|_ _|_|
|_|_| |_| |_|_|_ _ _
_| _|_|_| _ _| |_|_|
|_ |_| |_ |_|_ |_ |_ _
|_| _|_ _|_| _|_|_|
_|_|_|_|_|_ |_|_|
_|_|_|_|_|_|_| _ _ _|
|_| |_|_|_|_|_ |_|_ |_|_|_ _
_|_|_|_|_|_ _|_|_ _|_|_|_|_|
_|_|_|_| |_| |_|_|_|_|_| |_| |_|
_|_|_|_| _|_|_|_|
|_| |_|_ _ |_| |_|_ _
_|_|_|_| _|_|_|_|
|_| |_| |_| |_|
BQN
Works in: mlochbaum/BQN
Rot is a helper which rotates 90 degrees given direction in 𝕨.
Turns generates the move sequence of the fractal using the Wolfram MathWorld definition:
The curve can be constructed by representing a left turn by 1 and a right turn by 0. The first-order curve is then denoted 1. For higher order curves, append a 1 to the end, then append the string of preceding digits with its middle digit complemented.
The drawing may be skewed for some iterations since it is drawn to fit the bounding box.
<lang bqn>Rot ← ¬⊸{-⌾(𝕨⊸⊑)⌽𝕩} Turns ← {{𝕩∾1∾¬⌾((⌊2÷˜≠)⊸⊑)𝕩}⍟𝕩 ⥊1}
•Plot´ <˘⍉>+` (<0‿1)(Rot˜)`Turns 3</lang> Try It!
C
See: Dragon curve/C
C by IFS Drawing
C code that writes PNM of dragon curve. run as a.out [depth] > dragon.pnm. Sample image was with depth 9 (512 pixel length).
<lang C>#include <stdio.h>
- include <stdlib.h>
- include <string.h>
- include <math.h>
/* x, y: coordinates of current point; dx, dy: direction of movement.
* Think turtle graphics. They are divided by scale, so as to keep * very small coords/increments without losing precission. clen is * the path length travelled, which should equal to scale at the end * of the curve. */
long long x, y, dx, dy, scale, clen; typedef struct { double r, g, b; } rgb; rgb ** pix;
/* for every depth increase, rotate 45 degrees and scale up by sqrt(2)
* Note how coords can still be represented by integers. */
void sc_up() { long long tmp = dx - dy; dy = dx + dy; dx = tmp; scale *= 2; x *= 2; y *= 2; }
/* Hue changes from 0 to 360 degrees over entire length of path; Value
* oscillates along the path to give some contrast between segments * close to each other spatially. RGB derived from HSV gets *added* * to each pixel reached; they'll be dealt with later. */
void h_rgb(long long x, long long y) { rgb *p = &pix[y][x];
- define SAT 1
double h = 6.0 * clen / scale; double VAL = 1 - (cos(3.141592653579 * 64 * clen / scale) - 1) / 4; double c = SAT * VAL; double X = c * (1 - fabs(fmod(h, 2) - 1));
switch((int)h) { case 0: p->r += c; p->g += X; return; case 1: p->r += X; p->g += c; return; case 2: p->g += c; p->b += X; return; case 3: p->g += X; p->b += c; return; case 4: p->r += X; p->b += c; return; default: p->r += c; p->b += X; } }
/* string rewriting. No need to keep the string itself, just execute
* its instruction recursively. */
void iter_string(const char * str, int d) { long tmp;
- define LEFT tmp = -dy; dy = dx; dx = tmp
- define RIGHT tmp = dy; dy = -dx; dx = tmp
while (*str != '\0') { switch(*(str++)) { case 'X': if (d) iter_string("X+YF+", d - 1); continue; case 'Y': if (d) iter_string("-FX-Y", d - 1); continue; case '+': RIGHT; continue; case '-': LEFT; continue; case 'F':
/* draw: increment path length; add color; move. Here
* is why the code does not allow user to choose arbitrary
* image size: if it's not a power of two, aliasing will
* occur and grid-like bright or dark lines will result
* when normalized later. It can be gotten rid of, but that
* involves computing multiplicative order and would be a huge
* bore.
*/
clen ++; h_rgb(x/scale, y/scale); x += dx; y += dy; continue; } } }
void dragon(long leng, int depth) { long i, d = leng / 3 + 1; long h = leng + 3, w = leng + d * 3 / 2 + 2;
/* allocate pixel buffer */ rgb *buf = malloc(sizeof(rgb) * w * h); pix = malloc(sizeof(rgb *) * h); for (i = 0; i < h; i++) pix[i] = buf + w * i; memset(buf, 0, sizeof(rgb) * w * h);
/* init coords; scale up to desired; exec string */
x = y = d; dx = leng; dy = 0; scale = 1; clen = 0; for (i = 0; i < depth; i++) sc_up(); iter_string("FX", depth);
/* write color PNM file */ unsigned char *fpix = malloc(w * h * 3); double maxv = 0, *dbuf = (double*)buf;
/* find highest value among pixels; normalize image according
* to it. Highest value would be at points most travelled, so
* this ends up giving curve edge a nice fade -- it's more apparaent
* if we increase iteration depth by one or two.
*/
for (i = 3 * w * h - 1; i >= 0; i--) if (dbuf[i] > maxv) maxv = dbuf[i]; for (i = 3 * h * w - 1; i >= 0; i--) fpix[i] = 255 * dbuf[i] / maxv;
printf("P6\n%ld %ld\n255\n", w, h); fflush(stdout); /* printf and fwrite may treat buffer differently */ fwrite(fpix, h * w * 3, 1, stdout); }
int main(int c, char ** v) { int size, depth;
depth = (c > 1) ? atoi(v[1]) : 10; size = 1 << depth;
fprintf(stderr, "size: %d depth: %d\n", size, depth); dragon(size, depth * 2);
return 0; }</lang>
C#
<lang csharp>using System; using System.Collections.Generic; using System.Drawing; using System.Drawing.Drawing2D; using System.Windows.Forms;
public class DragonCurve : Form {
private List<int> turns; private double startingAngle, side;
public DragonCurve(int iter)
{
Size = new Size(800, 600);
StartPosition = FormStartPosition.CenterScreen;
DoubleBuffered = true;
BackColor = Color.White;
startingAngle = -iter * (Math.PI / 4);
side = 400 / Math.Pow(2, iter / 2.0);
turns = getSequence(iter); }
private List<int> getSequence(int iter)
{
var turnSequence = new List<int>();
for (int i = 0; i < iter; i++)
{
var copy = new List<int>(turnSequence);
copy.Reverse();
turnSequence.Add(1);
foreach (int turn in copy)
{
turnSequence.Add(-turn);
}
}
return turnSequence;
}
protected override void OnPaint(PaintEventArgs e)
{
base.OnPaint(e);
e.Graphics.SmoothingMode = SmoothingMode.AntiAlias;
double angle = startingAngle;
int x1 = 230, y1 = 350;
int x2 = x1 + (int)(Math.Cos(angle) * side);
int y2 = y1 + (int)(Math.Sin(angle) * side);
e.Graphics.DrawLine(Pens.Black, x1, y1, x2, y2);
x1 = x2;
y1 = y2;
foreach (int turn in turns)
{
angle += turn * (Math.PI / 2);
x2 = x1 + (int)(Math.Cos(angle) * side);
y2 = y1 + (int)(Math.Sin(angle) * side);
e.Graphics.DrawLine(Pens.Black, x1, y1, x2, y2);
x1 = x2;
y1 = y2;
}
}
[STAThread]
static void Main()
{
Application.Run(new DragonCurve(14));
}
}</lang>
C++
This program will generate the curve and save it to your hard drive. <lang cpp>
- include <windows.h>
- include <iostream>
//----------------------------------------------------------------------------------------- using namespace std;
//----------------------------------------------------------------------------------------- const int BMP_SIZE = 800, NORTH = 1, EAST = 2, SOUTH = 4, WEST = 8, LEN = 1;
//----------------------------------------------------------------------------------------- class myBitmap { public:
myBitmap() : pen( NULL ), brush( NULL ), clr( 0 ), wid( 1 ) {}
~myBitmap()
{
DeleteObject( pen ); DeleteObject( brush ); DeleteDC( hdc ); DeleteObject( bmp );
}
bool create( int w, int h )
{
BITMAPINFO bi; ZeroMemory( &bi, sizeof( bi ) ); bi.bmiHeader.biSize = sizeof( bi.bmiHeader ); bi.bmiHeader.biBitCount = sizeof( DWORD ) * 8; bi.bmiHeader.biCompression = BI_RGB; bi.bmiHeader.biPlanes = 1; bi.bmiHeader.biWidth = w; bi.bmiHeader.biHeight = -h;
HDC dc = GetDC( GetConsoleWindow() ); bmp = CreateDIBSection( dc, &bi, DIB_RGB_COLORS, &pBits, NULL, 0 ); if( !bmp ) return false;
hdc = CreateCompatibleDC( dc ); SelectObject( hdc, bmp ); ReleaseDC( GetConsoleWindow(), dc );
width = w; height = h; return true;
}
void clear( BYTE clr = 0 )
{
memset( pBits, clr, width * height * sizeof( DWORD ) );
}
void setBrushColor( DWORD bClr )
{
if( brush ) DeleteObject( brush ); brush = CreateSolidBrush( bClr ); SelectObject( hdc, brush );
}
void setPenColor( DWORD c )
{
clr = c; createPen();
}
void setPenWidth( int w )
{
wid = w; createPen();
}
void saveBitmap( string path )
{
BITMAPFILEHEADER fileheader; BITMAPINFO infoheader; BITMAP bitmap; DWORD wb;
GetObject( bmp, sizeof( bitmap ), &bitmap ); DWORD* dwpBits = new DWORD[bitmap.bmWidth * bitmap.bmHeight];
ZeroMemory( dwpBits, bitmap.bmWidth * bitmap.bmHeight * sizeof( DWORD ) ); ZeroMemory( &infoheader, sizeof( BITMAPINFO ) ); ZeroMemory( &fileheader, sizeof( BITMAPFILEHEADER ) );
infoheader.bmiHeader.biBitCount = sizeof( DWORD ) * 8; infoheader.bmiHeader.biCompression = BI_RGB; infoheader.bmiHeader.biPlanes = 1; infoheader.bmiHeader.biSize = sizeof( infoheader.bmiHeader ); infoheader.bmiHeader.biHeight = bitmap.bmHeight; infoheader.bmiHeader.biWidth = bitmap.bmWidth; infoheader.bmiHeader.biSizeImage = bitmap.bmWidth * bitmap.bmHeight * sizeof( DWORD );
fileheader.bfType = 0x4D42; fileheader.bfOffBits = sizeof( infoheader.bmiHeader ) + sizeof( BITMAPFILEHEADER ); fileheader.bfSize = fileheader.bfOffBits + infoheader.bmiHeader.biSizeImage;
GetDIBits( hdc, bmp, 0, height, ( LPVOID )dwpBits, &infoheader, DIB_RGB_COLORS );
HANDLE file = CreateFile( path.c_str(), GENERIC_WRITE, 0, NULL, CREATE_ALWAYS, FILE_ATTRIBUTE_NORMAL, NULL ); WriteFile( file, &fileheader, sizeof( BITMAPFILEHEADER ), &wb, NULL ); WriteFile( file, &infoheader.bmiHeader, sizeof( infoheader.bmiHeader ), &wb, NULL ); WriteFile( file, dwpBits, bitmap.bmWidth * bitmap.bmHeight * 4, &wb, NULL ); CloseHandle( file );
delete [] dwpBits;
}
HDC getDC() const { return hdc; }
int getWidth() const { return width; }
int getHeight() const { return height; }
private:
void createPen()
{
if( pen ) DeleteObject( pen ); pen = CreatePen( PS_SOLID, wid, clr ); SelectObject( hdc, pen );
}
HBITMAP bmp; HDC hdc; HPEN pen; HBRUSH brush; void *pBits; int width, height, wid; DWORD clr;
}; //----------------------------------------------------------------------------------------- class dragonC { public:
dragonC() { bmp.create( BMP_SIZE, BMP_SIZE ); dir = WEST; }
void draw( int iterations ) { generate( iterations ); draw(); }
private:
void generate( int it )
{
generator.push_back( 1 ); string temp;
for( int y = 0; y < it - 1; y++ ) { temp = generator; temp.push_back( 1 ); for( string::reverse_iterator x = generator.rbegin(); x != generator.rend(); x++ ) temp.push_back( !( *x ) );
generator = temp; }
}
void draw()
{
HDC dc = bmp.getDC(); unsigned int clr[] = { 0xff, 0xff00, 0xff0000, 0x00ffff }; int mov[] = { 0, 0, 1, -1, 1, -1, 1, 0 }; int i = 0;
for( int t = 0; t < 4; t++ ) { int a = BMP_SIZE / 2, b = a; a += mov[i++]; b += mov[i++]; MoveToEx( dc, a, b, NULL );
bmp.setPenColor( clr[t] ); for( string::iterator x = generator.begin(); x < generator.end(); x++ ) { switch( dir ) { case NORTH: if( *x ) { a += LEN; dir = EAST; } else { a -= LEN; dir = WEST; } break; case EAST: if( *x ) { b += LEN; dir = SOUTH; } else { b -= LEN; dir = NORTH; } break; case SOUTH: if( *x ) { a -= LEN; dir = WEST; } else { a += LEN; dir = EAST; } break; case WEST: if( *x ) { b -= LEN; dir = NORTH; } else { b += LEN; dir = SOUTH; } } LineTo( dc, a, b ); } } // !!! change this path !!! bmp.saveBitmap( "f:/rc/dragonCpp.bmp" );
}
int dir; myBitmap bmp; string generator;
}; //----------------------------------------------------------------------------------------- int main( int argc, char* argv[] ) {
dragonC d; d.draw( 17 ); return system( "pause" );
} //----------------------------------------------------------------------------------------- </lang>
COBOL
<lang cobol> >>SOURCE FORMAT FREE
- > This code is dedicated to the public domain
identification division. program-id. dragon. environment division. configuration section. repository. function all intrinsic. data division. working-storage section. 01 segment-length pic 9 value 2. 01 mark pic x value '.'. 01 segment-count pic 9999 value 513.
01 segment pic 9999. 01 point pic 9999 value 1. 01 point-max pic 9999. 01 point-lim pic 9999 value 8192. 01 dragon-curve.
03 filler occurs 8192.
05 ydragon pic s9999.
05 xdragon pic s9999.
01 x pic s9999 value 1. 01 y pic S9999 value 1.
01 xdelta pic s9 value 1. *> start pointing east 01 ydelta pic s9 value 0.
01 x-max pic s9999 value -9999. 01 x-min pic s9999 value 9999. 01 y-max pic s9999 value -9999. 01 y-min pic s9999 value 9999.
01 n pic 9999. 01 r pic 9.
01 xupper pic s9999. 01 yupper pic s9999.
01 window-line-number pic 99. 01 window-width pic 99 value 64. 01 window-height pic 99 value 22. 01 window.
03 window-line occurs 22.
05 window-point occurs 64 pic x.
01 direction pic x.
procedure division. start-dragon.
if segment-count * segment-length > point-lim
*> too many segments for the point-table
compute segment-count = point-lim / segment-length
end-if
perform varying segment from 1 by 1 until segment > segment-count
*>===========================================
*> segment = n * 2 ** b
*> if mod(n,4) = 3, turn left else turn right
*>===========================================
*> calculate the turn
divide 2 into segment giving n remainder r
perform until r <> 0
divide 2 into n giving n remainder r
end-perform
divide 2 into n giving n remainder r
*> perform the turn
evaluate r also xdelta also ydelta
when 0 also 1 also 0 *> turn right from east
when 1 also -1 also 0 *> turn left from west
*> turn to south
move 0 to xdelta
move 1 to ydelta
when 1 also 1 also 0 *> turn left from east
when 0 also -1 also 0 *> turn right from west
*> turn to north
move 0 to xdelta
move -1 to ydelta
when 0 also 0 also 1 *> turn right from south
when 1 also 0 also -1 *> turn left from north
*> turn to west
move 0 to ydelta
move -1 to xdelta
when 1 also 0 also 1 *> turn left from south
when 0 also 0 also -1 *> turn right from north
*> turn to east
move 0 to ydelta
move 1 to xdelta
end-evaluate
*> plot the segment points
perform segment-length times
add xdelta to x
add ydelta to y
move x to xdragon(point)
move y to ydragon(point)
add 1 to point
end-perform
*> update the limits for the display
compute x-max = max(x, x-max)
compute x-min = min(x, x-min)
compute y-max = max(y, y-max)
compute y-min = min(y, y-min)
move point to point-max
end-perform
*>========================================== *> display the curve *> hjkl corresponds to left, up, down, right *> anything else ends the program *>==========================================
move 1 to yupper xupper
perform with test after until direction <> 'h' and 'j' and 'k' and 'l'
*>==========================================
*> (yupper,xupper) maps to window-point(1,1)
*>==========================================
*> move the window
evaluate true
when direction = 'h' *> move window left
and xupper > x-min + window-width
subtract 1 from xupper
when direction = 'j' *> move window up
and yupper < y-max - window-height
add 1 to yupper
when direction = 'k' *> move window down
and yupper > y-min + window-height
subtract 1 from yupper
when direction = 'l' *> move window right
and xupper < x-max - window-width
add 1 to xupper
end-evaluate
*> plot the dragon points in the window
move spaces to window
perform varying point from 1 by 1
until point > point-max
if ydragon(point) >= yupper and < yupper + window-height
and xdragon(point) >= xupper and < xupper + window-width
*> we're in the window
compute y = ydragon(point) - yupper + 1
compute x = xdragon(point) - xupper + 1
move mark to window-point(y, x)
end-if
end-perform
*> display the window
perform varying window-line-number from 1 by 1
until window-line-number > window-height
display window-line(window-line-number)
end-perform
*> get the next window move or terminate
display 'hjkl?' with no advancing
accept direction
end-perform
stop run .
end program dragon.</lang>
- Output:
. . . . . . .
....... ... .........
. . . . . . . . .
... ...................
. . . . . . . . . . .
....................... ...
. . . . . . . . . . . . .
... ...........................
. . . . . . . . . . . . . . .
..................... ... ...
. . . . . . . . .
..... .............
. . . . . . .
..... ........... ...
. . . . . . . . .
... ...............
. . . . . . .
..... ... ...
.
... ...
. . .
....... ...
hjkl?q
Commodore BASIC
The values of SIN and COS for multiplies of PI / 4 are remembered in arrays, so that the program may run (a little) faster.
<lang basic> 10 REM DRAGON CURVE 20 REM SIN, COS IN ARRAYS FOR PI/4 MULTIPL. 30 DIM S(7),C(7) 40 QPI=ATN(1) 50 FOR I=0 TO 7 60 S(I)=SIN(I*QPI) 70 C(I)=COS(I*QPI) 80 NEXT I 90 LEVEL=15 100 INSIZE=128:REM 2^WHOLE_NUM (LOOKS BETTER) 110 X=112 120 Y=70 130 SQ=SQR(2) 140 ROTQPI=0:ITER=0:RQ=1 150 DIM R(LEVEL) 160 GRAPHIC 2,1 170 GOSUB 190 180 END 190 REM DRAGON 200 IF ROTQPI<0 THEN ROTQPI=ROTQPI+8:GOTO 220 210 IF ROTQPI>7 THEN ROTQPI=ROTQPI-8 220 IF LEVEL>1 THEN GO TO 290 230 YN=S(ROTQPI)*INSIZE+Y 240 XN=C(ROTQPI)*INSIZE+X 250 DRAW ,X,Y TO XN,YN 260 ITER=ITER+1 270 X=XN:Y=YN 280 RETURN 290 INSIZE=INSIZE*SQ/2 300 ROTQPI=ROTQPI+RQ 310 IF ROTQPI<0 THEN ROTQPI=ROTQPI+8:GOTO 330 320 IF ROTQPI>7 THEN ROTQPI=ROTQPI-8 330 LEVEL=LEVEL-1 340 R(LEVEL)=RQ:RQ=1 350 GOSUB 190 360 ROTQPI=ROTQPI-R(LEVEL)*2 370 IF ROTQPI<0 THEN ROTQPI=ROTQPI+8:GOTO 390 380 IF ROTQPI>7 THEN ROTQPI=ROTQPI-8 390 RQ=-1 400 GOSUB 190 410 RQ=R(LEVEL) 420 ROTQPI=ROTQPI+RQ 430 IF ROTQPI<0 THEN ROTQPI=ROTQPI+8:GOTO 450 440 IF ROTQPI>7 THEN ROTQPI=ROTQPI-8 450 LEVEL=LEVEL+1 460 INSIZE=INSIZE*SQ 470 RETURN </lang>
Common Lisp
This implementation uses nested transformations rather than turtle motions. with-scaling, etc. establish transformations for the drawing which occurs within them.
The recursive dragon-part function draws a curve connecting (0,0) to (1,0); if depth is 0 then the curve is a straight line. bend-direction is either 1 or -1 to specify whether the deviation from a straight line should be to the right or left. <lang lisp>(defpackage #:dragon
(:use #:clim-lisp #:clim) (:export #:dragon #:dragon-part))
(in-package #:dragon)
(defun dragon-part (depth bend-direction)
(if (zerop depth)
(draw-line* *standard-output* 0 0 1 0)
(with-scaling (t (/ (sqrt 2)))
(with-rotation (t (* pi -1/4 bend-direction))
(dragon-part (1- depth) 1)
(with-translation (t 1 0)
(with-rotation (t (* pi 1/2 bend-direction))
(dragon-part (1- depth) -1)))))))
(defun dragon (&optional (depth 7) (size 100))
(with-room-for-graphics ()
(with-scaling (t size)
(dragon-part depth 1))))</lang>
D
Text mode
A textual version of Dragon curve.
The Dragon curve drawn using an L-system.
- variables : X Y F
- constants : + −
- start : FX
- rules : (X → X+YF+),(Y → -FX-Y)
- angle : 90°
<lang d>import std.stdio, std.string;
struct Board {
enum char spc = ' '; char[][] b = ' '; // Set at least 1x1 board. int shiftx, shifty;
void clear() pure nothrow {
shiftx = shifty = 0;
b = '\0';
}
void check(in int x, in int y) pure nothrow {
while (y + shifty < 0) {
auto newr = new char[b[0].length];
newr[] = spc;
b = newr ~ b;
shifty++;
}
while (y + shifty >= b.length) {
auto newr = new char[b[0].length];
newr[] = spc;
b ~= newr;
}
while (x + shiftx < 0) {
foreach (ref c; b)
c = [spc] ~ c;
shiftx++;
}
while (x + shiftx >= b[0].length)
foreach (ref c; b)
c ~= [spc];
}
char opIndexAssign(in char value, in int x, in int y)
pure nothrow {
check(x, y);
b[y + shifty][x + shiftx] = value;
return value;
}
string toString() const pure {
return format("%-(%s\n%)", b);
}
}
struct Turtle {
static struct TState {
int[2] xy;
int heading;
}
enum int[2][] dirs = [[1, 0], [1, 1], [0, 1], [-1, 1],
[-1, 0], [-1, -1], [0, -1], [1, -1]];
enum string trace = r"-\|/-\|/";
TState t;
void reset() pure nothrow {
t = typeof(t).init;
}
void turn(in int dir) pure nothrow {
t.heading = (t.heading + 8 + dir) % 8;
}
void forward(ref Board b) pure nothrow {
with (t) {
xy[] += dirs[heading][];
b[xy[0], xy[1]] = trace[heading];
xy[] += dirs[heading][];
b[xy[0], xy[1]] = b.spc;
}
}
}
void dragonX(in int n, ref Turtle t, ref Board b) pure nothrow {
if (n >= 0) { // X -> X+YF+
dragonX(n - 1, t, b);
t.turn(2);
dragonY(n - 1, t, b);
t.forward(b);
t.turn(2);
}
}
void dragonY(in int n, ref Turtle t, ref Board b) pure nothrow {
if (n >= 0) { // Y -> -FX-Y
t.turn(-2);
t.forward(b);
dragonX(n - 1, t, b);
t.turn(-2);
dragonY(n - 1, t, b);
}
}
void main() {
Turtle t;
Board b;
// Seed : FX
t.forward(b); // <- F
dragonX(7, t, b); // <- X
writeln(b);
}</lang>
- Output:
- - - -
| | | | | | | |
- - - - - - - -
| | | | | | | |
- - - - - - - -
| | | | | | | |
- - - - - - - -
| | | | | | | |
- - - - - - - - - - - -
| | | | | | | | | | | | | | | |
- - - - - - - - - - - - - -
| | | | | | | | | | | |
- - - - - - - - - - - -
| | | | | | | | | | | |
- - - - - - - - - -
| | | | | | | |
- - - - - - - -
| | | | | | | |
- - - - - - - -
| | | | | | | |
- - - - - - - -
| | | | | | | |
- - - - - - - -
| | | | | | | |
- - - - - - - -
| | | | | | | |
- - - - - -
| | | |
- - - -
| | | |
- -
PostScript Output Version
<lang d>import std.stdio, std.string;
string drx(in size_t n) pure nothrow {
return n ? (drx(n - 1) ~ " +" ~ dry(n - 1) ~ " f +") : "";
}
string dry(in size_t n) pure nothrow {
return n ? (" - f" ~ drx(n - 1) ~ " -" ~ dry(n - 1)) : "";
}
string dragonCurvePS(in size_t n) pure nothrow {
return ["0 setlinewidth 300 400 moveto",
"/f{2 0 rlineto}def/+{90 rotate}def/-{-90 rotate}def\n",
"f", drx(n), " stroke showpage"].join();
}
void main() {
writeln(dragonCurvePS(9)); // Increase this for a bigger curve.
}</lang>
On a Bitmap
This uses the modules from the bresenhams line algorithm and Grayscale Image tasks.
First a small "turtle.d" module, useful for other tasks:
<lang d>module turtle;
import bitmap_bresenhams_line_algorithm, grayscale_image, std.math;
// Minimal turtle graphics. struct Turtle {
real x = 100, y = 100, angle = -90;
void left(in real a) pure nothrow { angle -= a; }
void right(in real a) pure nothrow { angle += a; }
void forward(Color)(Image!Color img, in real len) pure nothrow {
immutable r = angle * (PI / 180.0);
immutable dx = r.cos * len;
immutable dy = r.sin * len;
img.drawLine(cast(uint)x, cast(uint)y,
cast(uint)(x + dx), cast(uint)(y + dy),
Color.white);
x += dx;
y += dy;
}
}</lang>
Then the implementation is simple:
<lang d>import grayscale_image, turtle;
void drawDragon(Color)(Image!Color img, ref Turtle t, in uint depth,
in real dir, in uint step) {
if (depth == 0)
return t.forward(img, step);
t.right(dir);
img.drawDragon(t, depth - 1, 45.0, step);
t.left(dir * 2);
img.drawDragon(t, depth - 1, -45.0, step);
t.right(dir);
}
void main() {
auto img = new Image!Gray(500, 700);
auto t = Turtle(180, 510, -90);
img.drawDragon(t, 14, 45.0, 3);
img.savePGM("dragon_curve.pgm");
}</lang>
With QD
See: Dragon curve/D/QD
With DFL
See: Dragon curve/D/DFL
Delphi
<lang Delphi> program Dragon_curve;
{$APPTYPE CONSOLE}
uses
Winapi.Windows, System.SysUtils, System.Classes, Vcl.Graphics;
type
TDragon = class
private
p: TColor;
_sin: TArray<double>;
_cos: TArray<double>;
s: double;
b: TBitmap;
FAsBitmap: TBitmap;
const
sep = 512;
depth = 14;
procedure Dragon(n, a, t: Integer; d, x, y: Double; var b: TBitmap);
public
constructor Create;
destructor Destroy; override;
property AsBitmap: TBitmap read b;
end;
{ TDragon }
procedure TDragon.Dragon(n, a, t: Integer; d, x, y: Double; var b: TBitmap); begin
if n <= 1 then
begin
with b.Canvas do
begin
Pen.Color := p;
MoveTo(Trunc(x + 0.5), Trunc(y + 0.5));
LineTo(Trunc(x + d * _cos[a] + 0.5), Trunc(y + d * _sin[a] + 0.5));
exit;
end;
end;
d := d * s; var a1 := (a - t) and 7; var a2 := (a + t) and 7;
dragon(n - 1, a1, 1, d, x, y, b); dragon(n - 1, a2, -1, d, x + d * _cos[a1], y + d * _sin[a1], b);
end;
constructor TDragon.Create; begin
s := sqrt(2) / 2; _sin := [0, s, 1, s, 0, -s, -1, -s]; _cos := [1.0, s, 0.0, -s, -1.0, -s, 0.0, s]; p := Rgb(64, 192, 96); b := TBitmap.create;
var width := Trunc(sep * 11 / 6); var height := Trunc(sep * 4 / 3); b.SetSize(width, height); with b.Canvas do begin Brush.Color := clWhite; Pen.Width := 3; FillRect(Rect(0, 0, width, height)); end; dragon(14, 0, 1, sep, sep / 2, sep * 5 / 6, b);
end;
destructor TDragon.Destroy; begin
b.Free; inherited;
end;
var
Dragon: TDragon;
begin
Dragon := TDragon.Create;
Dragon.AsBitmap.SaveToFile('dragon.bmp');
Dragon.Free;
end.</lang>
EasyLang
<lang>color 050 linewidth 0.5 x = 25 y = 60 move x y angle = 0
func dragon size lev d . .
if lev = 0 x -= cos angle * size y += sin angle * size line x y else call dragon size / sqrt 2 lev - 1 1 angle -= d * 90 call dragon size / sqrt 2 lev - 1 -1 .
. call dragon 60 12 1</lang>
Elm
<lang elm>import Color exposing (..) import Collage exposing (..) import Element exposing (..) import Time exposing (..) import Html exposing (..) import Html.App exposing (program)
type alias Point = (Float, Float)
type alias Model =
{ points : List Point
, level : Int
, frame : Int
}
maxLevel = 12 frameCount = 100
type Msg = Tick Time
init : (Model,Cmd Msg) init = ( { points = [(-200.0, -70.0), (200.0, -70.0)]
, level = 0
, frame = 0
}
, Cmd.none )
-- New point between two existing points. Offset to left or right newPoint : Point -> Point -> Float -> Point newPoint (x0,y0) (x1,y1) offset =
let (vx, vy) = ((x1 - x0) / 2.0, (y1 - y0) / 2.0)
(dx, dy) = (-vy * offset , vx * offset )
in (x0 + vx + dx, y0 + vy + dy) --offset from midpoint
-- Insert between existing points. Offset to left or right side. newPoints : Float -> List Point -> List Point newPoints offset points =
case points of [] -> [] [p0] -> [p0] p0::p1::rest -> p0 :: newPoint p0 p1 offset :: newPoints -offset (p1::rest)
update : Msg -> Model -> (Model, Cmd Msg) update _ model =
let mo = if (model.level == maxLevel)
then model
else let nextFrame = model.frame + 1
in if (nextFrame == frameCount)
then { points = newPoints 1.0 model.points
, level = model.level+1
, frame = 0
}
else { model | frame = nextFrame
}
in (mo, Cmd.none)
-- break a list up into n equal sized lists. breakupInto : Int -> List a -> List (List a) breakupInto n ls =
let segmentCount = (List.length ls) - 1
breakup n ls = case ls of
[] -> []
_ -> List.take (n+1) ls :: breakup n (List.drop n ls)
in if n > segmentCount
then [ls]
else breakup (segmentCount // n) ls
view : Model -> Html Msg view model =
let offset = toFloat (model.frame) / toFloat frameCount
colors = [red, orange, green, blue]
in toHtml
<| layers
[ collage 700 500
(model.points
|> newPoints offset
|> breakupInto (List.length colors) -- for coloring
|> List.map path
|> List.map2 (\color path -> traced (solid color) path ) colors )
, show model.level
]
subscriptions : Model -> Sub Msg subscriptions _ =
Time.every (5*millisecond) Tick
main =
program
{ init = init
, view = view
, update = update
, subscriptions = subscriptions
}</lang>
Link to live demo: http://dc25.github.io/dragonCurveElm
Emacs Lisp
Drawing ascii art characters into a buffer using picture-mode
<lang lisp>(defun dragon-ensure-line-above ()
"If point is in the first line of the buffer then insert a new line above."
(when (= (line-beginning-position) (point-min))
(save-excursion
(goto-char (point-min))
(insert "\n"))))
(defun dragon-ensure-column-left ()
"If point is in the first column then insert a new column to the left.
This is designed for use in `picture-mode'."
(when (zerop (current-column))
(save-excursion
(goto-char (point-min))
(insert " ")
(while (= 0 (forward-line 1))
(insert " ")))
(picture-forward-column 1)))
(defun dragon-insert-char (char len)
"Insert CHAR repeated LEN many times.
After each CHAR point move in the current `picture-mode' direction (per `picture-set-motion' etc).
This is the same as `picture-insert' except in column 0 or row 0 a new row or column is inserted to make room, with existing buffer contents shifted down or right."
(dotimes (i len) (dragon-ensure-line-above) (dragon-ensure-column-left) (picture-insert char 1)))
(defun dragon-bit-above-lowest-0bit (n)
"Return the bit above the lowest 0-bit in N.
For example N=43 binary \"101011\" has lowest 0-bit at \"...0..\" and the bit above that is \"..1...\" so return 8 which is that bit."
(logand n (1+ (logxor n (1+ n)))))
(defun dragon-next-turn-right-p (n)
"Return non-nil if the dragon curve should turn right after segment N.
Segments are numbered from N=0 for the first, so calling with N=0 is whether to turn right after drawing that N=0 segment."
(zerop (dragon-bit-above-lowest-0bit n)))
(defun dragon-picture (len step)
"Draw the dragon curve in a *dragon* buffer.
LEN is the number of segments of the curve to draw. STEP is the length of each segment, in characters.
Any LEN can be given but a power-of-2 such as 256 shows the self-similar nature of the curve.
If STEP >= 2 then the segments are lines using \"-\" or \"|\" characters (`picture-rectangle-h' and `picture-rectangle-v'). If STEP=1 then only \"+\" corners.
There's a `sit-for' delay in the drawing loop to draw the curve progressively on screen."
(interactive (list (read-number "Length of curve " 256)
(read-number "Each step size " 3)))
(unless (>= step 1)
(error "Step length must be >= 1"))
(switch-to-buffer "*dragon*") (erase-buffer) (ignore-errors ;; if already in picture-mode (picture-mode))
(dotimes (n len) ;; n=0 to len-1, inclusive
(dragon-insert-char ?+ 1) ;; corner char
(dragon-insert-char (if (zerop picture-vertical-step)
picture-rectangle-h picture-rectangle-v)
(1- step)) ;; line chars
(if (dragon-next-turn-right-p n)
;; turn right
(picture-set-motion (- picture-horizontal-step) picture-vertical-step)
;; turn left
(picture-set-motion picture-horizontal-step (- picture-vertical-step)))
;; delay to display the drawing progressively (sit-for .01))
(picture-insert ?+ 1) ;; endpoint (picture-mode-exit) (goto-char (point-min)))
(dragon-picture 128 2)</lang>
- Output:
+-+ +-+
| | | |
+-+-+ +-+
| |
+-+ +-+ +-+
| | |
+-+-+-+
| | |
+-+-+
|
+-+ +-+ +-+ +-+
| | | | | | |
+-+ +-+-+-+ +-+-+ +-+-+
| | | | | | | | |
+-+-+-+-+-+ +-+-+-+ +-+-+-+ +-+
| | | | | | | | | | | | | | |
+-+ +-+ +-+-+-+-+-+ +-+ +-+-+
| | | | |
+-+-+-+-+ +-+
| | | | |
+-+ +-+-+ +-+ + +-+
| | | | | |
+-+-+-+-+ +-+
| | | |
+-+ +-+
ERRE
Graphic solution with PC.LIB library <lang ERRE> PROGRAM DRAGON
! ! for rosettacode.org !
!$DYNAMIC DIM RQS[0]
!$INCLUDE="PC.LIB"
PROCEDURE DRAGON
IF LEVEL<=0 THEN
YN=SIN(ROTATION)*INSIZE+Y
XN=COS(ROTATION)*INSIZE+X
LINE(X,Y,XN,YN,12,FALSE)
ITER=ITER+1
X=XN Y=YN
EXIT PROCEDURE
END IF
INSIZE=INSIZE/SQ
ROTATION=ROTATION+RQ*QPI
LEVEL=LEVEL-1
RQS[LEVEL]=RQ
RQ=1 DRAGON
ROTATION=ROTATION-RQS[LEVEL]*QPI*2
RQ=-1 DRAGON
RQ=RQS[LEVEL]
ROTATION=ROTATION+RQ*QPI
LEVEL=LEVEL+1
INSIZE=INSIZE*SQ
END PROCEDURE
BEGIN
SCREEN(9)
LEVEL=12 INSIZE=287 ! initial values
X=200 Y=120 !
SQ=SQR(2) QPI=ATN(1) ! constants
ROTATION=0 ITER=0 RQ=1 ! state variables
!$DIM RQS[LEVEL]
! stack for RQ (ROTATION coefficient)
LINE(0,0,639,349,14,TRUE)
DRAGON
GET(A$)
END PROGRAM </lang>
F#
Using
for visualization:
<lang fsharp>open System.Windows open System.Windows.Media
let m = Matrix(0.0, 0.5, -0.5, 0.0, 0.0, 0.0)
let step segs =
seq { for a: Point, b: Point in segs do
let x = a + 0.5 * (b - a) + (b - a) * m
yield! [a, x; b, x] }
let rec nest n f x =
if n=0 then x else nest (n-1) f (f x)
[<System.STAThread>] do
let path = Shapes.Path(Stroke=Brushes.Black, StrokeThickness=0.001)
path.Data <-
PathGeometry
[ for a, b in nest 13 step (seq [Point(0.0, 0.0), Point(1.0, 0.0)]) ->
PathFigure(a, [(LineSegment(b, true) :> PathSegment)], false) ]
(Application()).Run(Window(Content=Controls.Viewbox(Child=path))) |> ignore</lang>
Factor
A translation of the BASIC example, using OpenGL, drawing with HSV coloring similar to the C example.
<lang Factor> USING: accessors colors colors.hsv fry kernel locals math math.constants math.functions opengl.gl typed ui ui.gadgets ui.gadgets.canvas ui.render ;
IN: dragon
CONSTANT: depth 12
TUPLE: turtle
{ angle fixnum }
{ color float }
{ x float }
{ y float } ;
TYPED: nxt-color ( turtle: turtle -- turtle )
[ [ 360 2 depth ^ /f + ] keep
1.0 1.0 1.0 <hsva> >rgba-components glColor4d
] change-color ; inline
TYPED: draw-fwd ( x1: float y1: float x2: float y2: float -- )
GL_LINES glBegin glVertex2d glVertex2d glEnd ; inline
TYPED:: fwd ( turtle: turtle l: float -- )
turtle x>> turtle y>> turtle angle>> pi * 180 / :> ( x y angle ) l angle [ cos * x + ] [ sin * y + ] 2bi :> ( dx dy ) turtle x y dx dy [ draw-fwd ] 2keep [ >>x ] [ >>y ] bi* drop ; inline
TYPED: trn ( turtle: turtle d: fixnum -- turtle )
'[ _ + ] change-angle ; inline
TYPED:: dragon' ( turtle: turtle l: float s: fixnum d: fixnum -- )
s zero? [
turtle nxt-color l fwd ! don't like this drop
] [
turtle d 45 * trn l 2 sqrt / s 1 - 1 dragon'
turtle d -90 * trn l 2 sqrt / s 1 - -1 dragon'
turtle d 45 * trn drop
] if ;
- dragon ( -- )
0 0 150 180 turtle boa 400 depth 1 dragon' ;
TUPLE: dragon-canvas < canvas ;
M: dragon-canvas draw-gadget* [ drop dragon ] draw-canvas ; M: dragon-canvas pref-dim* drop { 640 480 } ;
MAIN-WINDOW: dragon-window { { title "Dragon Curve" } }
dragon-canvas new-canvas >>gadgets ;
MAIN: dragon-window </lang>
Forth
<lang forth>include turtle.fs
2 value dragon-step
- dragon ( depth dir -- )
over 0= if dragon-step fd 2drop exit then dup rt over 1- 45 recurse dup 2* lt over 1- -45 recurse rt drop ;
home clear 10 45 dragon</lang>
Basically the same code as the BigForth version.
<lang forth>include lib/graphics.4th include lib/gturtle.4th
2 constant dragon-step
- dragon ( depth dir -- )
over 0= if dragon-step forward 2drop exit then dup right over 1- 45 recurse dup 2* left over 1- -45 recurse right drop ;
150 pic_width ! 210 pic_height ! color_image
clear-screen 50 95 turtle! xpendown 13 45 dragon s" 4tHdragon.ppm" save_image</lang>
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.
In this page you can see the program(s) related to this task and their results.
FreeBASIC
<lang freebasic>Const pi As Double = 4 * Atn(1) Dim Shared As Double angulo = 0
Sub giro (grados As Double)
angulo += grados*pi/180
End Sub
Sub dragon (longitud As Double, division As Integer, d As Double)
If division = 0 Then
Line - Step (Cos(angulo)*longitud, Sin(angulo)*longitud), Int(Rnd * 7)
Else
giro d*45
dragon longitud/1.4142136, division-1, 1
giro -d*90
dragon longitud/1.4142136, division-1, -1
giro d*45
End If
End Sub
'--- Programa Principal --- Screen 12 Pset (150,180), 0 dragon 400, 12, 1 Bsave "Dragon_curve_FreeBASIC.bmp",0 Sleep</lang>
Gnuplot
Version #1.
Implemented by "parametric" mode running an index t through the desired number of curve segments with X,Y position calculated for each. The "lines" plot joins them up.
<lang gnuplot># Return the position of the highest 1-bit in n.
- The least significant bit is position 0.
- For example n=13 is binary "1101" and the high bit is pos=3.
- If n==0 then the return is 0.
- Arranging the test as n>=2 avoids infinite recursion if n==NaN (any
- comparison involving NaN is always false).
high_bit_pos(n) = (n>=2 ? 1+high_bit_pos(int(n/2)) : 0)
- Return 0 or 1 for the bit at position "pos" in n.
- pos==0 is the least significant bit.
bit(n,pos) = int(n / 2**pos) & 1
- dragon(n) returns a complex number which is the position of the
- dragon curve at integer point "n". n=0 is the first point and is at
- the origin {0,0}. Then n=1 is at {1,0} which is x=1,y=0, etc. If n
- is not an integer then the point returned is for int(n).
- The calculation goes by bits of n from high to low. Gnuplot doesn't
- have iteration in functions, but can go recursively from
- pos=high_bit_pos(n) down to pos=0, inclusive.
- mul() rotates by +90 degrees (complex "i") at bit transitions 0->1
- or 1->0. add() is a vector (i+1)**pos for each 1-bit, but turned by
- factor "i" when in a "reversed" section of curve, which is when the
- bit above is also a 1-bit.
dragon(n) = dragon_by_bits(n, high_bit_pos(n)) dragon_by_bits(n,pos) \
= (pos>=0 ? add(n,pos) + mul(n,pos)*dragon_by_bits(n,pos-1) : 0)
add(n,pos) = (bit(n,pos) ? (bit(n,pos+1) ? {0,1} * {1,1}**pos \
: {1,1}**pos) \
: 0)
mul(n,pos) = (bit(n,pos) == bit(n,pos+1) ? 1 : {0,1})
- Plot the dragon curve from 0 to "length" with line segments.
- "trange" and "samples" are set so the parameter t runs through
- integers t=0 to t=length inclusive.
- Any trange works, it doesn't have to start at 0. But must have
- enough "samples" that all integers t in the range are visited,
- otherwise vertices in the curve would be missed.
length=256 set trange [0:length] set samples length+1 set parametric set key off plot real(dragon(t)),imag(dragon(t)) with lines</lang>
Version #2.
- Note
- plotdcf.gp file-functions for the load command is the only possible imitation of the fine functions in the gnuplot.
- plotdcf.gp
<lang gnuplot>
- plotdcf.gp 1/11/17 aev
- Plotting a Dragon curve fractal to the png-file.
- Note: assign variables: ord (order), clr (color), filename and ttl (before using load command).
- ord (order) # a.k.a. level - defines size of fractal (also number of mini-curves).
reset set style arrow 1 nohead linewidth 1 lc rgb @clr set term png size 1024,1024 ofn=filename.ord."gp.png" # Output file name set output ofn ttl="Dragon curve fractal: order ".ord set title ttl font "Arial:Bold,12" unset border; unset xtics; unset ytics; unset key; set xrange [0:1.0]; set yrange [0:1.0]; dragon(n, x, y, dx, dy) = n >= ord ? \
sprintf("set arrow from %f,%f to %f,%f as 1;", x, y, x + dx, y + dy) : \
dragon(n + 1, x, y, (dx - dy) / 2, (dy + dx) / 2) . \
dragon(n + 1, x + dx, y + dy, - (dx + dy) / 2, (dx - dy) / 2);
eval(dragon(0, 0.2, 0.4, 0.7, 0.0)) plot -100 set output </lang>
- Plotting 3 Dragon curve fractals
<lang gnuplot>
- pDCF.gp 1/11/17 aev
- Plotting 3 Dragon curve fractals.
- Note: assign variables: ord (order), clr (color), filename and ttl (before using load command).
- ord (order) # a.k.a. level - defines size of fractal (also number of dots).
- cd 'C:\gnupData'
- DCF11
ord=11; clr = '"red"'; filename = "DCF"; ttl = "Dragon curve fractal, order ".ord; load "plotdcf.gp"
- DCF13
ord=13; clr = '"brown"'; filename = "DCF"; ttl = "Dragon curve fractal, order ".ord; load "plotdcf.gp"
- DCF15
ord=15; clr = '"navy"'; filename = "DCF"; ttl = "Dragon curve fractal, order ".ord; load "plotdcf.gp" </lang>
- Output:
1. All pDCF.gp file commands. 2. 3 plotted png-files: DCF11gp, DCF13gp and DCF15gp
Go
Version using standard image libriary is an adaptation of the version below using the Bitmap task. The only major change is that line drawing code was needed. See comments in code. <lang go>package main
import (
"fmt" "image" "image/color" "image/draw" "image/png" "math" "os"
)
// separation of the the two endpoints // make this a power of 2 for prettiest output const sep = 512 // depth of recursion. adjust as desired for different visual effects. const depth = 14
var s = math.Sqrt2 / 2 var sin = []float64{0, s, 1, s, 0, -s, -1, -s} var cos = []float64{1, s, 0, -s, -1, -s, 0, s} var p = color.NRGBA{64, 192, 96, 255} var b *image.NRGBA
func main() {
width := sep * 11 / 6
height := sep * 4 / 3
bounds := image.Rect(0, 0, width, height)
b = image.NewNRGBA(bounds)
draw.Draw(b, bounds, image.NewUniform(color.White), image.ZP, draw.Src)
dragon(14, 0, 1, sep, sep/2, sep*5/6)
f, err := os.Create("dragon.png")
if err != nil {
fmt.Println(err)
return
}
if err = png.Encode(f, b); err != nil {
fmt.Println(err)
}
if err = f.Close(); err != nil {
fmt.Println(err)
}
}
func dragon(n, a, t int, d, x, y float64) {
if n <= 1 {
// Go packages used here do not have line drawing functions
// so we implement a very simple line drawing algorithm here.
// We take advantage of knowledge that we are always drawing
// 45 degree diagonal lines.
x1 := int(x + .5)
y1 := int(y + .5)
x2 := int(x + d*cos[a] + .5)
y2 := int(y + d*sin[a] + .5)
xInc := 1
if x1 > x2 {
xInc = -1
}
yInc := 1
if y1 > y2 {
yInc = -1
}
for x, y := x1, y1; ; x, y = x+xInc, y+yInc {
b.Set(x, y, p)
if x == x2 {
break
}
}
return
}
d *= s
a1 := (a - t) & 7
a2 := (a + t) & 7
dragon(n-1, a1, 1, d, x, y)
dragon(n-1, a2, -1, d, x+d*cos[a1], y+d*sin[a1])
}</lang> Original version written to Bitmap task: <lang go>package main
// Files required to build supporting package raster are found in: // * Bitmap // * Write a PPM file
import (
"math" "raster"
)
// separation of the the two endpoints // make this a power of 2 for prettiest output const sep = 512 // depth of recursion. adjust as desired for different visual effects. const depth = 14
var s = math.Sqrt2 / 2 var sin = []float64{0, s, 1, s, 0, -s, -1, -s} var cos = []float64{1, s, 0, -s, -1, -s, 0, s} var p = raster.Pixel{64, 192, 96} var b *raster.Bitmap
func main() {
width := sep * 11 / 6
height := sep * 4 / 3
b = raster.NewBitmap(width, height)
b.Fill(raster.Pixel{255, 255, 255})
dragon(14, 0, 1, sep, sep/2, sep*5/6)
b.WritePpmFile("dragon.ppm")
}
func dragon(n, a, t int, d, x, y float64) {
if n <= 1 {
b.Line(int(x+.5), int(y+.5), int(x+d*cos[a]+.5), int(y+d*sin[a]+.5), p)
return
}
d *= s
a1 := (a - t) & 7
a2 := (a + t) & 7
dragon(n-1, a1, 1, d, x, y)
dragon(n-1, a2, -1, d, x+d*cos[a1], y+d*sin[a1])
}</lang>
Gri
Recursively by a dragon curve comprising two smaller dragons drawn towards a midpoint.
<lang Gri>`Draw Dragon [ from .x1. .y1. to .x2. .y2. [level .level.] ]' Draw a dragon curve going from .x1. .y1. to .x2. .y2. with recursion depth .level.
The total number of line segments for the recursion is 2^level. level=0 is a straight line from x1,y1 to x2,y2.
The default for x1,y1 and x2,y2 is to draw horizontally from 0,0 to 1,0. {
new .x1. .y1. .x2. .y2. .level.
.x1. = \.word3.
.y1. = \.word4.
.x2. = \.word6.
.y2. = \.word7.
.level. = \.word9.
if {rpn \.words. 5 >=}
.x2. = 1
.y2. = 0
end if
if {rpn \.words. 7 >=}
.level. = 6
end if
if {rpn 0 .level. <=}
draw line from .x1. .y1. to .x2. .y2.
else
.level. = {rpn .level. 1 -}
# xmid,ymid is half way between x1,y1 and x2,y2 and up at
# right angles away.
#
# xmid,ymid xmid = (x1+x2 + y2-y1)/2
# ^ ^ ymid = (x1-x2 + y1+y2)/2
# / . \
# / . \
# x1,y1 ........... x2,y2
#
new .xmid. .ymid.
.xmid. = {rpn .x1. .x2. + .y2. .y1. - + 2 /}
.ymid. = {rpn .x1. .x2. - .y1. .y2. + + 2 /}
# The recursion is a level-1 dragon from x1,y1 to the midpoint
# and the same from x2,y2 to the midpoint (the latter
# effectively being a revered dragon.)
#
Draw Dragon from .x1. .y1. to .xmid. .ymid. level .level.
Draw Dragon from .x2. .y2. to .xmid. .ymid. level .level.
delete .xmid. .ymid.
end if
delete .x1. .y1. .x2. .y2. .level.
}
- Dragon curve from 0,0 to 1,0 extends out by 1/3 at the ends, so
- extents -0.5 to +1.5 for a bit of margin. The Y extent is the same
- size 2 to make the graph square.
set x axis -0.5 1.5 .25 set y axis -1 1 .25
Draw Dragon</lang>
Haskell
<lang haskell>import Data.List import Graphics.Gnuplot.Simple
-- diamonds -- pl = [[0,1],[1,0]]
pl = [[0,0],[0,1]] r_90 = [[0,1],[-1,0]]
ip :: [Int] -> [Int] -> Int ip xs = sum . zipWith (*) xs matmul xss yss = map (\xs -> map (ip xs ). transpose $ yss) xss
vmoot xs = (xs++).map (zipWith (+) lxs). flip matmul r_90.
map (flip (zipWith (-)) lxs) .reverse . init $ xs where lxs = last xs
dragoncurve = iterate vmoot pl</lang> For plotting I use the gnuplot interface module from hackageDB
Use:
plotPath [] . map (\[x,y] -> (x,y)) $ dragoncurve!!13
String rewrite, and outputs a postscript: <lang haskell>x 0 = "" x n = (x$n-1)++" +"++(y$n-1)++" f +" y 0 = "" y n = " - f"++(x$n-1)++" -"++(y$n-1)
dragon n = concat ["0 setlinewidth 300 400 moveto", "/f{2 0 rlineto}def/+{90 rotate}def/-{-90 rotate}def\n", "f", x n, " stroke showpage"]
main = putStrLn $ dragon 14</lang>
HicEst
A straightforward approach, since HicEst does not know recursion (rarely needed in daily work) <lang hicest> CHARACTER dragon
1 DLG(NameEdit=orders,DNum, Button='&OK', TItle=dragon) ! input orders WINDOW(WINdowhandle=wh, Height=1, X=1, TItle='Dragon curves up to order '//orders)
IF( LEN(dragon) < 2^orders) ALLOCATE(dragon, 2^orders)
AXIS(WINdowhandle=wh, Xaxis=2048, Yaxis=2048) ! 2048: black, linear, noGrid, noScales dragon = ' ' NorthEastSouthWest = 0 x = 0 y = 1 LINE(PenUp, Color=1, x=0, y=0, x=x, y=y) last = 1
DO order = 1, orders
changeRtoL = LEN_TRIM(dragon) + 1 + (LEN_TRIM(dragon) + 1)/2
dragon = TRIM(dragon) // 'R' // TRIM(dragon)
IF(changeRtoL > 2) dragon(changeRtoL) = 'L'
DO last = last, LEN_TRIM(dragon)
NorthEastSouthWest = MOD( NorthEastSouthWest-2*(dragon(last)=='L')+5, 4 )
x = x + (NorthEastSouthWest==1) - (NorthEastSouthWest==3)
y = y + (NorthEastSouthWest==0) - (NorthEastSouthWest==2)
LINE(Color=order, X=x, Y=y)
ENDDO
ENDDO
GOTO 1 ! this is to stimulate a discussion
END</lang>
Icon and Unicon
The following implements a Heighway Dragon using the Lindenmayer system. It's based on the linden program in the Icon Programming Library. <lang Icon>link linddraw,wopen
procedure main()
gener := 12 # generations w := h := 800 # window size rewrite := table() # L rewrite rules rewrite["X"] := "X+YF+" rewrite["Y"] := "-FX-Y" every (C := ) ++:= !!rewrite every /rewrite[c := !C] := c # map all rule characters
WOpen("size=" || w || "," || h, "dx=" || (w / 2), "dy=" || (h / 2)) | stop("*** cannot open window")
WAttrib("fg=blue")
linddraw(0, 0, "FX", rewrite, 5, 90.0, gener, 0)
# x,y, axiom, rules, length, angle, generations, delay
WriteImage("dragon-unicon" || ".gif") # save the image
WDone()
end</lang>
J
<lang j>require 'plot' start=: 0 0,: 1 0 step=: ],{: +"1 (0 _1,: 1 0) +/ .*~ |.@}: -"1 {: plot <"1 |: step^:13 start</lang> In English: Start with a line segment. For each step of iteration, retrace that geometry backwards, but oriented 90 degrees about its original end point. To show the curve you need to pick some arbitrary number of iterations.
Any line segment is suitable for start. (For example, -start+123 works just fine though of course the resulting orientation and coordinates for the curve will be different from those obtained using start for the line segment.)
For a more colorful display, with a different color for the geometry introduced at each iteration, replace that last line of code with: <lang j>([:pd[:<"1|:)every'reset';|.'show';step&.>^:(i.17)<start</lang>
The curve can also be represented as a limiting set of the iterated function system
Giving the code <lang j>require 'plot' f1=.*&(-:1j1) f2=.[: -. *&(-:1j_1) plot (f1,}.@|.@f2)^:12 ]0 1</lang>
Where both functions are applied successively to starting complex values of 0 and 1.
Note the formatting of f2 as }.@|.@f2 . This allows the plotted path to go in the right order and removes redundant points, paralleling similar operations in the previous solution.
Java
<lang java>import java.awt.Color; import java.awt.Graphics; import java.util.*; import javax.swing.JFrame;
public class DragonCurve extends JFrame {
private List<Integer> turns; private double startingAngle, side;
public DragonCurve(int iter) {
super("Dragon Curve");
setBounds(100, 100, 800, 600);
setDefaultCloseOperation(EXIT_ON_CLOSE);
turns = getSequence(iter);
startingAngle = -iter * (Math.PI / 4);
side = 400 / Math.pow(2, iter / 2.);
}
public List<Integer> getSequence(int iterations) {
List<Integer> turnSequence = new ArrayList<Integer>();
for (int i = 0; i < iterations; i++) {
List<Integer> copy = new ArrayList<Integer>(turnSequence);
Collections.reverse(copy);
turnSequence.add(1);
for (Integer turn : copy) {
turnSequence.add(-turn);
}
}
return turnSequence;
}
@Override
public void paint(Graphics g) {
g.setColor(Color.BLACK);
double angle = startingAngle;
int x1 = 230, y1 = 350;
int x2 = x1 + (int) (Math.cos(angle) * side);
int y2 = y1 + (int) (Math.sin(angle) * side);
g.drawLine(x1, y1, x2, y2);
x1 = x2;
y1 = y2;
for (Integer turn : turns) {
angle += turn * (Math.PI / 2);
x2 = x1 + (int) (Math.cos(angle) * side);
y2 = y1 + (int) (Math.sin(angle) * side);
g.drawLine(x1, y1, x2, y2);
x1 = x2;
y1 = y2;
}
}
public static void main(String[] args) {
new DragonCurve(14).setVisible(true);
}
}</lang>
JavaScript
ES5 plus HTML DOM
Version #1.
I'm sure this can be simplified further, but I have this working here!
Though there is an impressive SVG example further below, this uses JavaScript to recurse through the expansion and simply displays each line with SVG. It is invoked as a method DRAGON.fractal(...) as described.
<lang javascript>var DRAGON = (function () {
// MATRIX MATH // -----------
var matrix = {
mult: function ( m, v ) {
return [ m[0][0] * v[0] + m[0][1] * v[1],
m[1][0] * v[0] + m[1][1] * v[1] ];
},
minus: function ( a, b ) {
return [ a[0]-b[0], a[1]-b[1] ];
},
plus: function ( a, b ) {
return [ a[0]+b[0], a[1]+b[1] ];
}
};
// SVG STUFF // ---------
// Turn a pair of points into an SVG path like "M1 1L2 2".
var toSVGpath = function (a, b) { // type system fail
return "M" + a[0] + " " + a[1] + "L" + b[0] + " " + b[1];
};
// DRAGON MAKING // -------------
// Make a dragon with a better fractal algorithm
var fractalMakeDragon = function (svgid, ptA, ptC, state, lr, interval) {
// make a new <path>
var path = document.createElementNS('http://www.w3.org/2000/svg', 'path');
path.setAttribute( "class", "dragon");
path.setAttribute( "d", toSVGpath(ptA, ptC) );
// append the new path to the existing <svg>
var svg = document.getElementById(svgid); // call could be eliminated
svg.appendChild(path);
// if we have more iterations to go...
if (state > 1) {
// make a new point, either to the left or right
var growNewPoint = function (ptA, ptC, lr) {
var left = [[ 1/2,-1/2 ],
[ 1/2, 1/2 ]];
var right = [[ 1/2, 1/2 ],
[-1/2, 1/2 ]];
return matrix.plus(ptA, matrix.mult( lr ? left : right,
matrix.minus(ptC, ptA) ));
};
var ptB = growNewPoint(ptA, ptC, lr, state);
// then recurse using each new line, one left, one right
var recurse = function () {
// when recursing deeper, delete this svg path
svg.removeChild(path);
// then invoke again for new pair, decrementing the state
fractalMakeDragon(svgid, ptB, ptA, state-1, lr, interval);
fractalMakeDragon(svgid, ptB, ptC, state-1, lr, interval);
};
window.setTimeout(recurse, interval);
}
};
// Export these functions
// ----------------------
return {
fractal: fractalMakeDragon
// ARGUMENTS
// ---------
// svgid id of <svg> element
// ptA first point [x,y] (from top left)
// ptC second point [x,y]
// state number indicating how many steps to recurse
// lr true/false to make new point on left or right
// CONFIG
// ------
// CSS rules should be made for the following
// svg#fractal
// svg path.dragon
};
}());</lang>
My current demo page includes the following to invoke this: <lang html>... <script src='./dragon.js'></script> ...
<svg xmlns='http://www.w3.org/2000/svg' id='fractal'></svg>
<script>
DRAGON.fractal('fractal', [100,300], [500,300], 15, false, 700);
</script> ...</lang>
Version #2.
<lang html> <html> <head> <script type='text/javascript'> function pDragon(cId) {
// Plotting Dragon curves. 2/25/17 aev
var n=document.getElementById('ord').value;
var sc=document.getElementById('sci').value;
var hsh=document.getElementById('hshi').value;
var vsh=document.getElementById('vshi').value;
var clr=document.getElementById('cli').value;
var c=c1=c2=c2x=c2y=x=y=0, d=1, n=1<<n;
var cvs=document.getElementById(cId);
var ctx=cvs.getContext("2d");
hsh=Number(hsh); vsh=Number(vsh);
x=y=cvs.width/2;
// Cleaning canvas, init plotting
ctx.fillStyle="white"; ctx.fillRect(0,0,cvs.width,cvs.height);
ctx.beginPath();
for(i=0; i<=n;) {
ctx.lineTo((x+hsh)*sc,(y+vsh)*sc);
c1=c&1; c2=c&2;
c2x=1*d; if(c2>0) {c2x=(-1)*d}; c2y=(-1)*c2x;
if(c1>0) {y+=c2y} else {x+=c2x}
i++; c+=i/(i&-i);
}
ctx.strokeStyle = clr; ctx.stroke();
} </script> </head> <body>
Please input order, scale, x-shift, y-shift, color:</>
<input id=ord value=11 type="number" min="7" max="25" size="2"> <input id=sci value=7.0 type="number" min="0.001" max="10" size="5"> <input id=hshi value=-265 type="number" min="-50000" max="50000" size="6"> <input id=vshi value=-260 type="number" min="-50000" max="50000" size="6"> <input id=cli value="red" type="text" size="14"> <button onclick="pDragon('canvId')">Plot it!</button>
Dragon curve
<canvas id="canvId" width=640 height=640 style="border: 2px inset;"></canvas> </body> </html> </lang> Testing cases:
Input parameters: ord scale x-shift y-shift color [File name to save] ------------------------------------------- 11 7. -265 -260 red DC11.png 15 2. -205 -230 brown DC15.png 17 1. -135 70 green DC17.png 19 0.6 380 440 navy DC19.png 21 0.22 1600 800 blue DC21.png 23 0.15 1100 800 violet DC23.png 25 0.07 2100 5400 darkgreen DC25.png ===========================================
- Output:
Page with different plotted Dragon curves. Right-clicking on the canvas you can save each of them as a png-file.
ES6
Declarative definition of an SVG file, in terms of functional primitives. (To test, generate and save SVG as file, and open in a browser or graphics application).
(Pure JS, without HTML or DOM) <lang javascript>(() => {
'use strict';
// ------------------ DRAGON CURVE -------------------
// dragonCurve :: Int -> Int const dragonCurve = xs => { const pivot = op => map( zipWith(op)(last(xs)) ), r90 = [ [0, 1], [-1, 0] ]; return compose( append(xs), pivot(add), flip(matrixMultiply)(r90), pivot(subtract), reverse, init )(xs); };
// ---------------------- TEST -----------------------
// main :: IO ()
const main = () =>
// SVG of 12th iteration.
console.log(
svgFromPointLists(512)(512)(
index(iterate(dragonCurve)([
[0, 0],
[0, -1]
]))(12)
)
);
// ----------------------- SVG -----------------------
// svgFromPointLists :: Int -> Int -> // (Int, Int) -> String const svgFromPointLists = cw => ch => xyss => { const polyline = xs => `<polyline points="${unwords(concat(xs).map(showJSON))}"/>`, [x, y, mx, my] = ap([minimum, maximum])( Array.from(unzip(concat(xyss))) ), [wd, hd] = map(x => Math.floor(x / 10))([ mx - x, my - y ]); return unlines([ '<?xml version="1.0" encoding="UTF-8"?>', unwords([ '<svg', `width="${cw}" height="${ch}"`, `viewBox="${x - wd} ${y - hd} ${12 * wd} ${12 * hd}"`, 'xmlns="http://www.w3.org/2000/svg">' ]), '<g stroke-width="0.2" stroke="red" fill="none">', unlines(map(polyline)(xyss)), '</g>', '</svg>' ]); };
// ---------------- GENERIC FUNCTIONS ----------------
// Just :: a -> Maybe a
const Just = x => ({
type: 'Maybe',
Nothing: false,
Just: x
});
// Nothing :: Maybe a
const Nothing = () => ({
type: 'Maybe',
Nothing: true,
});
// Tuple (,) :: a -> b -> (a, b)
const Tuple = a =>
b => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});
// add (+) :: Num a => a -> a -> a
const add = a =>
// Curried addition.
b => a + b;
// ap (<*>) :: [(a -> b)] -> [a] -> [b]
const ap = fs =>
// The sequential application of each of a list
// of functions to each of a list of values.
xs => fs.flatMap(
f => xs.map(f)
);
// append (++) :: [a] -> [a] -> [a]
// append (++) :: String -> String -> String
const append = xs =>
// A list or string composed by
// the concatenation of two others.
ys => xs.concat(ys);
// compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
const compose = (...fs) =>
fs.reduce(
(f, g) => x => f(g(x)),
x => x
);
// concat :: a -> [a] const concat = xs => [].concat(...xs);
// dotProduct :: Num a => a -> a -> a const dotProduct = xs => compose(sum, zipWith(mul)(xs));
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = m =>
n => Array.from({
length: 1 + n - m
}, (_, i) => m + i);
// flip :: (a -> b -> c) -> b -> a -> c
const flip = f =>
x => y => f(y)(x);
// index (!!) :: [a] -> Int -> Maybe a
// index (!!) :: Generator (a) -> Int -> Maybe a
// index (!!) :: String -> Int -> Maybe Char
const index = xs =>
i => (
drop(i)(xs),
take(1)(xs)
);
// drop :: Int -> [a] -> [a]
// drop :: Int -> Generator [a] -> Generator [a]
// drop :: Int -> String -> String
const drop = n =>
xs => Infinity > length(xs) ? (
xs.slice(n)
) : (take(n)(xs), xs);
// init :: [a] -> [a]
const init = xs =>
// All elements of a list except the last.
0 < xs.length ? (
xs.slice(0, -1)
) : undefined;
// iterate :: (a -> a) -> a -> Gen [a]
const iterate = f =>
function* (x) {
let v = x;
while (true) {
yield(v);
v = f(v);
}
};
// last :: [a] -> a
const last = xs =>
// The last item of a list.
0 < xs.length ? xs.slice(-1)[0] : undefined;
// length :: [a] -> Int
const length = xs =>
// Returns Infinity over objects without finite
// length. This enables zip and zipWith to choose
// the shorter argument when one is non-finite,
// like cycle, repeat etc
(Array.isArray(xs) || 'string' === typeof xs) ? (
xs.length
) : Infinity;
// map :: (a -> b) -> [a] -> [b]
const map = f =>
// The list obtained by applying f
// to each element of xs.
// (The image of xs under f).
xs => xs.map(f);
// matrixMultiply :: Num a => a -> a -> a const matrixMultiply = a => b => { const cols = transpose(b); return map( compose( flip(map)(cols), dotProduct ) )(a); };
// minimum :: Ord a => [a] -> a
const minimum = xs =>
0 < xs.length ? (
xs.slice(1)
.reduce((a, x) => x < a ? x : a, xs[0])
) : undefined;
// maximum :: Ord a => [a] -> a
const maximum = xs =>
// The largest value in a non-empty list.
0 < xs.length ? (
xs.slice(1).reduce(
(a, x) => x > a ? (
x
) : a, xs[0]
)
) : undefined;
// mul (*) :: Num a => a -> a -> a
const mul = a =>
b => a * b;
// reverse :: [a] -> [a]
const reverse = xs =>
xs.slice(0).reverse();
// showJSON :: a -> String
const showJSON = x =>
// Indented JSON representation of the value x.
JSON.stringify(x, null, 2);
// subtract :: Num -> Num -> Num
const subtract = x =>
y => y - x;
// sum :: [Num] -> Num
const sum = xs =>
// The numeric sum of all values in xs.
xs.reduce((a, x) => a + x, 0);
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = n =>
// The first n elements of a list,
// string of characters, or stream.
xs => 'GeneratorFunction' !== xs
.constructor.constructor.name ? (
xs.slice(0, n)
) : [].concat.apply([], Array.from({
length: n
}, () => {
const x = xs.next();
return x.done ? [] : [x.value];
}));
// transpose :: a -> a const transpose = rows => // The columns of the input transposed // into new rows. // Simpler version of transpose, assuming input // rows of even length. 0 < rows.length ? rows[0].map( (x, i) => rows.flatMap( x => x[i] ) ) : [];
// unlines :: [String] -> String
const unlines = xs =>
// A single string formed by the intercalation
// of a list of strings with the newline character.
xs.join('\n');
// until :: (a -> Bool) -> (a -> a) -> a -> a
const until = p => f => x => {
let v = x;
while (!p(v)) v = f(v);
return v;
};
// unwords :: [String] -> String
const unwords = xs =>
// A space-separated string derived
// from a list of words.
xs.join(' ');
// unzip :: [(a,b)] -> ([a],[b])
const unzip = xys =>
xys.reduce(
(ab, xy) => Tuple(ab[0].concat(xy[0]))(
ab[1].concat(xy[1])
),
Tuple([])([])
);
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = f =>
// A list constructed by zipping with a
// custom function, rather than with the
// default tuple constructor.
xs => ys => {
const
lng = Math.min(length(xs), length(ys)),
vs = take(lng)(ys);
return take(lng)(xs)
.map((x, i) => f(x)(vs[i]));
};
// MAIN --- return main();
})();</lang>
jq
Works with gojq, the Go implementation of jq
The programs given here generate SVG code that can be viewed directly in a browser, at least if the file suffix is .svg.
The first program uses simple turtle graphics and an L-system; the second is based on the fractalMakeDragon example in Javascript.
L-System
See Simple Turtle Graphics for the simple-turtle.jq module used in this entry. The `include` statement assumes the file is in the pwd. <lang jq>include "simple-turtle" {search: "."};
def rules:
{ F: "F+S",
S: "F-S" };
def dragon($count):
rules as $rules
| def p($count):
if $count <= 0 then .
else gsub("S"; "s") | gsub("F"; $rules["F"]) | gsub("s"; $rules["S"])
| p($count-1)
end;
"F" | p($count) ;
def interpret($x):
if $x == "+" then turtleRotate(90) elif $x == "-" then turtleRotate(-90) elif $x == "F" then turtleForward(4) elif $x == "S" then turtleForward(4) else . end;
def dragon_curve($n):
dragon($n)
| split("")
| reduce .[] as $action (turtle([200,300]) | turtleDown;
interpret($action) ) ;
dragon_curve(15) | path("none"; "red"; "0.1") | svg(1700)</lang>
fractalMakeDragon
The following is based on the JavaScript example, with some variations, notably:
- the last argument of the main function allows CSS style elements to be specified
- the output is a single SVG element that can, for example, be viewed in a web browser such as Chrome, Firefox, or Safari
- only one "path" element is emitted.
The main function is fractalMakeDragon(svgid; ptA; ptC; steps; left; style) where:
# svgid id of <svg> element
# ptA first point [x,y] (from top left)
# ptC second point [x,y]
# steps number indicating how many steps to recurse
# left if true, make new point on left; if false, then on right
# css a JSON object optionally specifying "stroke" and "stroke-width"
<lang jq># MATRIX MATH
def mult(m; v):
[ m[0][0] * v[0] + m[0][1] * v[1],
m[1][0] * v[0] + m[1][1] * v[1] ];
def minus(a; b): [ a[0]-b[0], a[1]-b[1] ];
def plus(a; b): [ a[0]+b[0], a[1]+b[1] ];
- SVG STUFF
# default values of stroke and stroke-width are provided
def style(obj):
{ "stroke": "rgb(255, 15, 131)", "stroke-width": "2px" } as $default
| ($default + obj) as $s
| "<style type='text/css' media='all'>
.dragon { stroke:\($s.stroke); stroke-width:\($s["stroke-width"]); }
</style>";
def svg(id; width; height):
"<svg width='\(width // "100%")' height='\(height // "100%") '
id='\(id)'
xmlns='http://www.w3.org/2000/svg'>";
# Turn a pair of points into an SVG path like "M1 1L2 2" (M=move to; L=line to).
def toSVGpath(a; b):
"M\(a[0]) \(a[1])L\(b[0]) \(b[1])";
- DRAGON MAKING
def fractalMakeDragon(svgid; ptA; ptC; steps; left; css):
# Make a new point, either to the left or right
def growNewPoint(ptA; ptC; left):
[[ 1/2,-1/2 ], [ 1/2, 1/2 ]] as $left
| [[ 1/2, 1/2 ], [-1/2, 1/2 ]] as $right
| plus(ptA;
mult(if left then $left else $right end;
minus(ptC; ptA)));
def grow(ptA; ptC; steps; left):
# if we have more iterations to go...
if steps > 1 then
growNewPoint(ptA; ptC; left) as $ptB
# ... then recurse using each new line, one left, one right
| grow($ptB; ptA; steps-1; left),
grow($ptB; ptC; steps-1; left)
else
toSVGpath(ptA; ptC)
end;
svg(svgid; "100%"; "100%"),
style(css),
"<path class='dragon' d='",
grow(ptA; ptC; steps; left),
"'/>",
"</svg>";</lang>
Example: <lang jq># Default values are provided for the last argument fractalMakeDragon("roar"; [100,300]; [500,300]; 15; false; {})</lang>
- Output:
The command to generate the SVG and the first few lines of output are as follows: <lang sh>$ jq -n -r -f dragon.jq <svg width='100%' height='100% '
id='roar'
xmlns='http://www.w3.org/2000/svg'>
<style type='text/css' media='all'>
.dragon { stroke:rgb(255, 15, 131); stroke-width:2px; }
</style>
<path class='dragon' d=' M259.375 218.75L259.375 221.875 M259.375 218.75L262.5 218.75 ... </lang>
Julia
Code uses Luxor library[1].
<lang julia> using Luxor function dragon(turtle::Turtle, level=4, size=200, direction=45)
if level != 0
Turn(turtle, -direction)
dragon(turtle, level-1, size/sqrt(2), 45)
Turn(turtle, direction*2)
dragon(turtle, level-1, size/sqrt(2), -45)
Turn(turtle, -direction)
else
Forward(turtle, size)
end
end
Drawing(900, 500, "./Dragon.png") t = Turtle(300, 300, true, 0, (0., 0.0, 0.0)); dragon(t, 10,400) finish() preview() </lang>
Kotlin
<lang scala>// version 1.0.6
import java.awt.Color import java.awt.Graphics import javax.swing.JFrame
class DragonCurve(iter: Int) : JFrame("Dragon Curve") {
private val turns: MutableList<Int> private val startingAngle: Double private val side: Double
init {
setBounds(100, 100, 800, 600)
defaultCloseOperation = EXIT_ON_CLOSE
turns = getSequence(iter)
startingAngle = -iter * Math.PI / 4
side = 400.0 / Math.pow(2.0, iter / 2.0)
}
fun getSequence(iterations: Int): MutableList<Int> {
val turnSequence = mutableListOf<Int>()
for (i in 0 until iterations) {
val copy = mutableListOf<Int>()
copy.addAll(turnSequence)
copy.reverse()
turnSequence.add(1)
copy.mapTo(turnSequence) { -it }
}
return turnSequence
}
override fun paint(g: Graphics) {
g.color = Color.BLUE
var angle = startingAngle
var x1 = 230
var y1 = 350
var x2 = x1 + (Math.cos(angle) * side).toInt()
var y2 = y1 + (Math.sin(angle) * side).toInt()
g.drawLine(x1, y1, x2, y2)
x1 = x2
y1 = y2
for (turn in turns) {
angle += turn * Math.PI / 2.0
x2 = x1 + (Math.cos(angle) * side).toInt()
y2 = y1 + (Math.sin(angle) * side).toInt()
g.drawLine(x1, y1, x2, y2)
x1 = x2
y1 = y2
}
}
}
fun main(args: Array<String>) {
DragonCurve(14).isVisible = true
}</lang>
Lambdatalk
1) two twinned recursive functions
<lang scheme> {def dcr
{lambda {:step :length}
{let { {:step {- :step 1}}
{:length {/ :length 1.41421}}
} {if {> :step 0}
then T45
{dcr :step :length}
T-90
{dcl :step :length}
T45
else T45
M:length
T-90
M:length
T45}
}}} -> dcr
{def dcl
{lambda {:step :length}
{let { {:step {- :step 1}}
{:length {/ :length 1.41421}}
} {if {> :step 0}
then T-45
{dcr :step :length}
T90
{dcl :step :length}
T-45
else T-45
M:length
T90
M:length
T-45}
}}} -> dcl </lang> The word Tθ rotates the drawing direction of the pen from θ degrees and the word Md moves it on d pixels. Writing {dcr 10 360} outputs 4093 words begining with T45 T45 T45 T45 T45 T45 T45 T45 T45 T45 M11.250283388970585 T-90 M11.250283388970585 T45 T-90 T-45 M11.250283388970585 T90 M11.250283388970585 ...
2) the SVG context
Lambdatalk comes with a primitive, turtle, translating the previous sequence of words into a sequence of SVG points [x0 y0 x1 y2 ... xn yn] feeding the "d" attribute of a SVG path.
3) drawing three dragon curves [2,6,10] of decreasing width: <lang scheme> {svg
{@ width="580px" height="580px"
style="box-shadow:0 0 8px #888;"}
{path {@ d="M {turtle 130 130 0 {dcr 2 360}}" {stroke 20 #ccc}}}
{path {@ d="M {turtle 130 130 0 {dcr 6 360}}" {stroke 10 #888}}}
{path {@ d="M {turtle 130 130 0 {dcr 10 360}}" {stroke 1 #000}}}
}
where {def stroke
{lambda {:w :c}
fill="transparent" stroke=":c" stroke-width=":w"}}
-> stroke </lang> The output can be seen in http://lambdaway.free.fr/lambdawalks/?view=dragon
Liberty BASIC
<lang lb>nomainwin
mainwin 50 20
WindowHeight =620 WindowWidth =690
open "Graphics library" for graphics as #a
#a, "trapclose [quit]"
#a "down"
Turn$ ="R" Pace =100 s = 16
[again]
print Turn$
#a "cls ; home ; north ; down ; fill black"
for i =1 to len( Turn$)
v =255 *i /len( Turn$)
#a "color "; v; " 120 "; 255 -v
#a "go "; Pace
if mid$( Turn$, i, 1) ="R" then #a "turn 90" else #a "turn -90"
next i
#a "color 255 120 0" #a "go "; Pace #a "flush"
FlippedTurn$ =""
for i =len( Turn$) to 1 step -1
if mid$( Turn$, i, 1) ="R" then FlippedTurn$ =FlippedTurn$ +"L" else FlippedTurn$ =FlippedTurn$ +"R"
next i
Turn$ =Turn$ +"R" +FlippedTurn$
Pace =Pace /1.35
scan
timer 1000, [j] wait
[j]
timer 0
if len( Turn$) <40000 then goto [again]
wait
[quit]
close #a end</lang>
Logo
Recursive
<lang logo>to dcr :step :length
make "step :step - 1 make "length :length / 1.41421 if :step > 0 [rt 45 dcr :step :length lt 90 dcl :step :length rt 45] if :step = 0 [rt 45 fd :length lt 90 fd :length rt 45]
end
to dcl :step :length
make "step :step - 1 make "length :length / 1.41421 if :step > 0 [lt 45 dcr :step :length rt 90 dcl :step :length lt 45] if :step = 0 [lt 45 fd :length rt 90 fd :length lt 45]
end</lang> The program can be started using dcr 4 300 or dcl 4 300.
Or removing duplication: <lang logo>to dc :step :length :dir
if :step = 0 [fd :length stop] rt :dir dc :step-1 :length/1.41421 45 lt :dir lt :dir dc :step-1 :length/1.41421 -45 rt :dir
end to dragon :step :length
dc :step :length 45
end</lang> An alternative approach by using sentence-like grammar using four productions o->on, n->wn, w->ws, s->os. O, S, N and W mean cardinal points. <lang logo>to O :step :length
if :step=1 [Rt 90 fd :length Lt 90] [O (:step - 1) (:length / 1.41421) N (:step - 1) (:length / 1.41421)]
end
to N :step :length
if :step=1 [fd :length] [W (:step - 1) (:length / 1.41421) N (:step - 1) (:length / 1.41421)]
end
to W :step :length
if :step=1 [Lt 90 fd :length Rt 90] [W (:step - 1) (:length / 1.41421) S (:step - 1) (:length / 1.41421)]
end
to S :step :length
if :step=1 [Rt 180 fd :length Lt 180] [O (:step - 1) (:length / 1.41421) S (:step - 1) (:length / 1.41421)]
end</lang>
Iterative
Or drawing iteratively by making a turn left or right at each point calculated by bit-twiddling. This allows any length to be drawn, not just powers-of-2.
<lang logo>; Return the bit above the lowest 1-bit in :n.
- If
- n = binary "...z100..00" then the return is "z000..00".
- Eg. n=22 is binary 10110 the lowest 1-bit is the "...1." and the return is
- bit above that "..1.," which is 4.
to bit.above.lowest.1bit :n
output bitand :n (1 + (bitxor :n (:n - 1)))
end
- Return angle +90 or -90 for dragon curve turn at point
- n.
- The curve is reckoned as starting from n=0 so the first turn is at n=1.
to dragon.turn.angle :n
output ifelse (bit.above.lowest.1bit :n) = 0 [90] [-90]
end
- Draw
- steps many segments of the dragon curve.
to dragon :steps
localmake "step.len 12 ; length of each step repeat :steps [ forward :step.len left dragon.turn.angle repcount ; repcount = 1 to :steps inclusive ]
end
dragon 256</lang>
<lang logo>; Draw :steps many segments of the dragon curve, with corners chamfered
- off with little 45-degree diagonals.
- Done this way the vertices don't touch.
to dragon.chamfer :steps
localmake "step.len 12 ; length of each step localmake "straight.frac 0.5 ; fraction of the step to go straight
localmake "straight.len :step.len * :straight.frac localmake "diagonal.len (:step.len - :straight.len) * sqrt(1/2)
repeat :steps [
localmake "turn (dragon.turn.angle repcount)/2 ; +45 or -45
forward :straight.len
left :turn
forward :diagonal.len
left :turn
]
end
dragon.chamfer 256</lang>
Lua
Could be made much more compact, but this was written for speed. It has two rendering modes, one which renders the curve in text mode (default,) and one which just dumps all the coordinates for use by an external rendering application. <lang Lua>function dragon()
local l = "l"
local r = "r"
local inverse = {l = r, r = l}
local field = {r}
local num = 1
local loop_limit = 6 --increase this number to render a bigger curve
for discard=1,loop_limit do
field[num+1] = r
for i=1,num do
field[i+num+1] = inverse[field[num-i+1]]
end
num = num*2+1
end
return field
end
function render(field, w, h, l)
local x = 0
local y = 0
local points = {}
local highest_x = 0
local highest_y = 0
local lowest_x = 0
local lowest_y = 0
local l = "l"
local r = "r"
local u = "u"
local d = "d"
local heading = u
local turn = {r = {r = d, d = l, l = u, u = r}, l = {r = u, u = l, l = d, d = r}}
for k, v in ipairs(field) do
heading = turn[v][heading]
for i=1,3 do
points[#points+1] = {x, y}
if heading == l then
x = x-w
elseif heading == r then
x = x+w
elseif heading == u then
y = y-h
elseif heading == d then
y = y+h
end
if x > highest_x then
highest_x = x
elseif x < lowest_x then
lowest_x = x
end
if y > highest_y then
highest_y = y
elseif y < lowest_y then
lowest_y = y
end
end
end
points[#points+1] = {x, y}
highest_x = highest_x - lowest_x + 1
highest_y = highest_y - lowest_y + 1
for k, v in ipairs(points) do
v[1] = v[1] - lowest_x + 1
v[2] = v[2] - lowest_y + 1
end
return highest_x, highest_y, points
end
function render_text_mode()
local width, height, points = render(dragon(), 1, 1, 1)
local rows = {}
for i=1,height do
rows[i] = {}
for j=1,width do
rows[i][j] = ' '
end
end
for k, v in ipairs(points) do
rows[v[2]][v[1]] = "*"
end
for i=1,height do
print(table.concat(rows[i], ""))
end
end
function dump_points()
local width, height, points = render(dragon(), 4, 4, 1)
for k, v in ipairs(points) do
print(unpack(v))
end
end
--replace this line with dump_points() to output a list of coordinates: render_text_mode() </lang> Output:
**** ****
* * * *
* * * *
**** *******
* *
* *
**** **** ****
* * *
* * *
**********
* * *
* * *
*******
*
*
**** ****
* * *
* * *
********** ****
* * * * *
* * * * *
**** ****************
* * * * * * *
* * * * * * *
*******************
* * * * *
* * * * *
******* ******* ****
* * * *
* * * *
******* **** **** ****
* * * * * *
* * * * * *
**** ********** ****
* * * *
* * * *
********** **** *******
* * * * * * * *
* * * * * * * *
******* ********** ****
* * * *
* * * *
******* *******
* * * *
* * * *
**** ****
M2000 Interpreter
<lang M2000 Interpreter> Module Checkit {
def double angle, d45, d90, change=5000
const sr2 as double= .70710676237
Cls 0
Pen 14
\\ move console full screen to second monitor
Window 12, 1
\\ reduce size (tv as second monitor cut pixels from edges)
Window 12, scale.x*.9, scale.y*.9;
\\ opacity 100%, but for 0 (black is 100%, and we can hit anything under console window)
Desktop 255, 0
\\ M2000 console can divide screen to characters/lines with automatic line space
Form 60, 30
\\ cut the border from window
Form
\\ scale.x and scale.y in twips
\\ all graphic/console commands works for printer also (except for Input)
Move scale.x/2,scale.y/10
\\ outline graphics, here outline text
\\ legend text$, font, size, angle, justify(2 for center), quality (non zero for antialiasing, works for angle 0), letter spacing.
Color {
Legend "DRAGON CURVE", "Courier",SCALE.Y/200,0,2, 1, SCALE.X/50
}
angle=0
d45=pi/4
d90=pi/2
Move scale.x/3, scale.y*2/3
bck=point
\\ twipsx is width in twips of pixel. twipsy are height in twips of a pixel
\\ so we use length:twips.x*scale.x/40 or scale.x/40 pixels.
\\ use % for integer - we can omit these, and we get integer by automatic conversion (overflow raise error)
dragon(twipsx*scale.x/40,14%, 1)
Pen 14
a$=key$
Cls 5
\\ set opacity to 100%
Desktop 255
End
\\ Subs are private to this module
\\ Subs have same scope as module
Sub turn(rand as double)
angle+=rand
End Sub
\\ angle is absolute, length is relative
Sub forward(length as double)
Draw Angle angle, length
End Sub
Sub dragon(length as double, split as integer, d as double)
If split=0 then {
forward(length)
} else {
Gosub turn(d*d45)
\\ we can omit Gosub
dragon(length*sr2,split-1,1)
turn(-d*d90)
dragon(length*sr2,split-1,-1)
turn(d*d45)
change--
If change else {
push 0: do {drop: push random(11,15) : over } until number<>pen: pen number
change=5000
}
}
End Sub
} Checkit
</lang>
M4
This code uses the "predicate" approach. A given x,y position is tested by a predicate as to whether it's on the curve or not and printed as a character or a space accordingly. The output goes row by row and column by column with no image storage or buffering.
<lang># The macros which return a pair of values x,y expand to an unquoted 123,456
- which is suitable as arguments to a further macro. The quoting is slack
- because the values are always integers and so won't suffer unwanted macro
- expansion.
- 0,1 Vertex and segment x,y numbering.
- |
- | Segments are numbered as if a
- |s=0,1 square grid turned anti-clockwise
- | by 45 degrees.
- |
- -1,0 -------- 0,0 -------- 1,0 vertex_to_seg_east(x,y) returns
- s=-1,1 | s=0,0 the segment x,y to the East,
- | so vertex_to_seg_east(0,0) is 0,0
- |
- |s=-1,0 vertex_to_seg_west(x,y) returns
- | the segment x,y to the West,
- 0,-1 so vertex_to_seg_west(0,0) is -1,1
define(`vertex_to_seg_east', `eval($1 + $2), eval($2 - $1)') define(`vertex_to_seg_west', `eval($1 + $2 - 1), eval($2 - $1 + 1)') define(`vertex_to_seg_south', `eval($1 + $2 - 1), eval($2 - $1)')
- Some past BSD m4 didn't have "&" operator, so mod2(n) using % instead.
- mod2() returns 0,1 even if "%" gives -1 for negative odds.
define(`mod2', `ifelse(eval($1 % 2),0,0,1)')
- seg_to_even(x,y) returns x,y moved to an "even" position by subtracting an
- offset in a way which suits the segment predicate test.
- seg_offset_y(x,y) is a repeating pattern
- | 1,1,0,0
- | 1,1,0,0
- | 0,0,1,1
- | 0,0,1,1
- +---------
- seg_offset_x(x,y) is the same but offset by 1 in x,y
- | 0,1,1,0
- | 1,0,0,1
- | 1,0,0,1
- | 0,1,1,0
- +---------
- Incidentally these offset values also give n which is the segment number
- along the curve. "x_offset XOR y_offset" is 0,1 and is a bit of n from
- low to high.
define(`seg_offset_y', `mod2(eval(($1 >> 1) + ($2 >> 1)))') define(`seg_offset_x', `seg_offset_y(eval($1+1), eval($2+1))') define(`seg_to_even', `eval($1 - seg_offset_x($1,$2)),
eval($2 - seg_offset_y($1,$2))');
- xy_div_iplus1(x,y) returns x,y divided by complex number i+1.
- So (x+i*y)/(i+1) which means newx = (x+y)/2, newy = (y-x)/2.
- Must have x,y "even", meaning x+y even, so newx and newy are integers.
define(`xy_div_iplus1', `eval(($1 + $2)/2), eval(($2 - $1)/2)')
- seg_is_final(x,y) returns 1 if x,y is one of the final four points.
- On these four points xy_div_iplus1(seg_to_even(x,y)) returns x,y
- unchanged, so the seg_pred() recursion does not reduce any further.
- .. | ..
- final | final y=+1
- final | final y=0
- -------+--------
- .. | ..
- x=-1 x=0
define(`seg_is_final', `eval(($1==-1 || $1==0) && ($2==1 || $2==0))')
- seg_pred(x,y) returns 1 if segment x,y is on the dragon curve.
- If the final point reached is 0,0 then the original x,y was on the curve.
- (If a different final point then x,y was one of four rotated copies of the
- curve.)
define(`seg_pred', `ifelse(seg_is_final($1,$2), 1,
`eval($1==0 && $2==0)',
`seg_pred(xy_div_iplus1(seg_to_even($1,$2)))')')
- vertex_pred(x,y) returns 1 if point x,y is on the dragon curve.
- The curve always turns left or right at a vertex, it never crosses itself,
- so if a vertex is visited then either the segment to the east or to the
- west must have been traversed. Prefer ifelse() for the two checks since
- eval() || operator is not a short-circuit.
define(`vertex_pred', `ifelse(seg_pred(vertex_to_seg_east($1,$2)),1,1,
`seg_pred(vertex_to_seg_west($1,$2))')')
- forloop(varname, start,end, body)
- Expand body with varname successively define()ed to integers "start" to
- "end" inclusive. "start" to "end" can go either increasing or decreasing.
define(`forloop', `define(`$1',$2)$4`'dnl ifelse($2,$3,,`forloop(`$1',eval($2 + 2*($2 < $3) - 1), $3, `$4')')')
- ----------------------------------------------------------------------------
- dragon01(xmin,xmax, ymin,ymax) prints an array of 0s and 1s which are the
- vertex_pred() values. `y' runs from ymax down to ymin so that y
- coordinate increases up the screen.
define(`dragon01', `forloop(`y',$4,$3, `forloop(`x',$1,$2, `vertex_pred(x,y)') ')')
- dragon_ascii(xmin,xmax, ymin,ymax) prints an ascii art dragon curve.
- Each y value results in two output lines. The first has "+" vertices and
- "--" horizontals. The second has "|" verticals.
define(`dragon_ascii', `forloop(`y',$4,$3, `forloop(`x',$1,$2, `ifelse(vertex_pred(x,y),1, `+', ` ')dnl ifelse(seg_pred(vertex_to_seg_east(x,y)), 1, `--', ` ')') forloop(`x',$1,$2, `ifelse(seg_pred(vertex_to_seg_south(x,y)), 1, `| ', ` ')') ')')
- --------------------------------------------------------------------------
divert`'dnl
- 0s and 1s directly from vertex_pred().
dragon01(-7,23, dnl X range
-11,10) dnl Y range
- ASCII art lines.
dragon_ascii(-6,5, dnl X range
-10,2) dnl Y range</lang>
- Output
# 0s and 1s directly from vertex_pred().
#
0000000000000000011111110000000
0000000000000011011111111000000
0000000000000111011111111000000
0000000000000111111111100000000
0000000000000111111111111111000
0000000000000111111111111111100
0000000000000001111111111111100
0000000000000001111111111110000
0000111100000000011111111111000
0000111110000011011110001111100
0011110110000111011110111111100
0011110000000111111000111110000
0001110000000111111100011110000
0000111100110111111110000000000
0011111101110111111110000000000
0011111111111111111000000000000
0001111111111111111100000000000
0000000011111000111110000000000
0000001111111011111110000000000
0000001111100011111000000000000
0000000111100001111000000000000
0000000000000000000000000000000
# ASCII art lines.
#
+--+ +--+
| | | |
+--+--+ +--+
| |
+--+ +--+ +--+
| | |
+--+--+--+
| | |
+--+--+
|
+--+ +--+ +--+
| | | | |
+--+ +--+--+--+ +--+--+
| | | | | | |
+--+--+--+--+--+ +--+--+--+ +--
| | | | | | | | | |
+--+ +--+ +--+--+--+--+--+
| | | |
+--+--+--+--+
| | | |
+--+ +--+--+ +--+
| | | |
+--+--+--+--+
| | | |
+--+ +--+Mathematica / Wolfram Language
Two functions: one that makes 2 lines from 1 line. And another that applies this function to all existing lines: <lang Mathematica>FoldOutLine[{a_,b_}]:=Template:A,&[a+0.5(b-a)+{{0.,0.5},{-0.5,0.}}.(b-a)] NextStep[in_]:=Flatten[FoldOutLine/@in,1] lines={{{0.,0.},{1.,0.}}}; Graphics[Line/@Nest[NextStep,lines,11]]</lang>
Metafont
Metafont is a language to create fonts; since fonts normally are not too big, Metafont has hard encoded limits which makes it difficult to produce large images. This is one of the reasons why Metapost came into being.
The following code produces a single character font, 25 points wide and tall (0 points in depth), and store it in the position where one could expect to find the character D.
<lang metafont>mode_setup; dragoniter := 8; beginchar("D", 25pt#, 25pt#, 0pt#);
pickup pencircle scaled .5pt;
x1 = 0; x2 = w; y1 = y2 = .5h;
mstep := .5; sg := -1;
for i = 1 upto dragoniter:
for v = 1 step mstep until (2-mstep):
if unknown z[v+mstep]:
pair t; t := .7071[ z[v], z[v+2mstep] ]; z[v+mstep] = t rotatedaround(z[v], 45sg); sg := -1*sg;
fi endfor mstep := mstep/2; endfor draw for v:=1 step 2mstep until (2-2mstep): z[v] -- endfor z[2];
endchar; end</lang>
The resulting character, magnified by 2, looks like:
Nim
The program is an adaptation of the second version of Go solution with some changes.
Rather than producing a big PPM file, we output a PNG file.
<lang Nim>import math import imageman
const
## Separation of the two endpoints. ## Make this a power of 2 for prettier output. Sep = 512 ## Depth of recursion. Adjust as desired for different visual effects. Depth = 18
S = sqrt(2.0) / 2 Sin = [float 0, S, 1, S, 0, -S, -1, -S] Cos = [float 1, S, 0, -S, -1, -S, 0, S]
LineColor = ColorRGBU [byte 64, 192, 96] Width = Sep * 11 div 6 Height = Sep * 4 div 3
Output = "dragon.png"
- ---------------------------------------------------------------------------------------------------
func dragon(img: var Image; n, a, t: int; d, x, y: float) =
if n <= 1: img.drawLine((x.toInt, y.toInt), ((x + d * Cos[a]).toInt, (y + d * Sin[a]).toInt), LineColor) return let d = d * S let a1 = (a - t) and 7 let a2 = (a + t) and 7 img.dragon(n - 1, a1, 1, d, x, y) img.dragon(n - 1, a2, -1, d, x + d * Cos[a1], y + d * Sin[a1])
- ---------------------------------------------------------------------------------------------------
var image = initImage[ColorRGBU](Width, Height) image.fill(ColorRGBU [byte 0, 0, 0]) image.dragon(Depth, 0, 1, Sep, Sep / 2, Sep * 5 / 6)
- Save into a PNG file.
image.savePNG(Output, compression = 9)</lang>
OCaml
Example solution, using an OCaml class and displaying the result in a Tk canvas, mostly inspired by the Tcl solution. <lang ocaml>(* This constant does not seem to be defined anywhere in the standard modules *) let pi = acos (-1.0);
(*
- CLASS dragon_curve_computer:
- ----------------------------
- Computes the coordinates for the line drawing the curve.
- - initial_x initial_y: coordinates for starting point for curve
- - total_length: total length for the curve
- - total_splits: total number of splits to perform
- )
class dragon_curve_computer initial_x initial_y total_length total_splits =
object(self)
val mutable current_x = (float_of_int initial_x) (* current x coordinate in curve *)
val mutable current_y = (float_of_int initial_y) (* current y coordinate in curve *)
val mutable current_angle = 0.0 (* current angle *)
(*
** METHOD compute_coords:
** ----------------------
** Actually computes the coordinates in the line for the curve
** - length: length for current iteration
** - nb_splits: number of splits to perform for current iteration
** - direction: direction for current line (-1.0 or 1.0)
** Returns: the list of coordinates for the line in this iteration
*)
method compute_coords length nb_splits direction =
(* If all splits have been done *)
if nb_splits = 0
then
begin
(* Draw line segment, updating current coordinates *)
current_x <- current_x +. length *. cos current_angle;
current_y <- current_y +. length *. sin current_angle;
[(int_of_float current_x, int_of_float current_y)]
end
(* If there are still splits to perform *)
else
begin
(* Compute length for next iteration *)
let sub_length = length /. sqrt 2.0 in
(* Turn 45 degrees to left or right depending on current direction and draw part
of curve in this direction *)
current_angle <- current_angle +. direction *. pi /. 4.0;
let coords1 = self#compute_coords sub_length (nb_splits - 1) 1.0 in
(* Turn 90 degrees in the other direction and draw part of curve in that direction *)
current_angle <- current_angle -. direction *. pi /. 2.0;
let coords2 = self#compute_coords sub_length (nb_splits - 1) (-1.0) in
(* Turn back 45 degrees to set head in the initial direction again *)
current_angle <- current_angle +. direction *. pi /. 4.0;
(* Concatenate both sub-curves to get the full curve for this iteration *)
coords1 @ coords2
end
(*
** METHOD get_coords:
** ------------------
** Returns the coordinates for the curve with the parameters set in the object initializer
*)
method get_coords = self#compute_coords total_length total_splits 1.0
end;;
(*
- MAIN PROGRAM:
- =============
- )
let () =
(* Curve is displayed in a Tk canvas *) let top=Tk.openTk() in let c = Canvas.create ~width:400 ~height:400 top in Tk.pack [c]; (* Create instance computing the curve coordinates *) let dcc = new dragon_curve_computer 100 200 200.0 16 in (* Create line with these coordinates in canvas *) ignore (Canvas.create_line ~xys: dcc#get_coords c); Tk.mainLoop ();
- </lang>
A functional version
Here is another OCaml solution, in a functional rather than OO style: <lang OCaml>let zig (x1,y1) (x2,y2) = (x1+x2+y1-y2)/2, (x2-x1+y1+y2)/2 let zag (x1,y1) (x2,y2) = (x1+x2-y1+y2)/2, (x1-x2+y1+y2)/2
let rec dragon p1 p2 p3 n =
if n = 0 then [p1;p2] else (dragon p1 (zig p1 p2) p2 (n-1)) @ (dragon p2 (zag p2 p3) p3 (n-1))
let _ =
let top = Tk.openTk() in let c = Canvas.create ~width:430 ~height:300 top in Tk.pack [c]; let p1, p2 = (100, 100), (356,100) in let points = dragon p1 (zig p1 p2) p2 15 in ignore (Canvas.create_line ~xys: points c); Tk.mainLoop ()</lang>
producing:
Run an example with:
ocaml -I +labltk labltk.cma dragon.ml
Openscad
Using the two sub-curves inward approach. The sub-curves are rotated and shifted explicitly. That could be combined into a multmatrix() each if desired. Lines segments are drawn as elongated cuboids.
<lang SCAD>level = 8; linewidth = .1; // fraction of segment length sqrt2 = pow(2, .5);
// Draw a dragon curve "level" going from [0,0] to [1,0] module dragon(level) {
if (level <= 0) {
translate([.5,0]) cube([1+linewidth,linewidth,linewidth],center=true);
} else {
rotate(-45) scale(1/sqrt2) dragon(level-1);
translate([1,0]) rotate(-135) scale(1/sqrt2) dragon(level-1);
}
}
scale(40) { // scale to nicely visible in the default GUI
sphere(1.5*linewidth / pow(2,level/2)); // mark the start of the curve dragon(level);
} </lang>
PARI/GP
Version #1.
Using the "high level" plothraw with real and imaginary parts of vertex points as X and Y coordinates. Change plothraw() to psplothraw() to write a PostScript file "pari.ps" instead of drawing on-screen.
<lang parigp>level = 13 p = [0, 1]; \\ complex number points, initially 0 to 1
\\ "unfold" at the current endpoint p[#p]. \\ p[^-1] so as not to duplicate that endpoint. \\ \\ * end \\ --> | \\ / | \\ v \\ *------->* \\ 0,0 p[#p] \\ for(i=1,level, my(end = (1+I)*p[#p]); \
p = concat(p, apply(z->(end - I*z), Vecrev(p[^-1]))))
plothraw(apply(real,p),apply(imag,p), 1); \\ flag=1 join points</lang>
Version #2.
Using the "low level" plotting functions to draw to a GUI window (X etc). <lang parigp>len=256;
bit_above_low_1(n) = bittest(n, valuation(n,2)+1);
plotinit(0); plotscale(0, -32,32, 32,-32); \\ Y increasing up the screen plotmove(0, 0,0); plotstring(0, "start", 8+32); \\ flags 8=top + 32=gap
dx=1; dy=0; turn_right()= [dx,dy]=[-dy,dx]; turn_left() = [dx,dy]=[dy,-dx];
for(i=1,len, plotrline(0,dx,dy); \
if(bit_above_low_1(i), turn_right(), turn_left()));
plotdraw([0,100,100]);</lang>
Version #3.
This is actualy Version #1 upgraded to the reusable function.
<lang parigp> \\ Dragon curve \\ 4/8/16 aev Dragon(level)={my(p=[0,1],end); print(" *** Dragon curve, level ",level); for(i=1,level, end=(1+I)*p[#p];
p=concat(p,apply(z->(end-I*z),Vecrev(p[^-1]))) );
plothraw(apply(real,p),apply(imag,p), 1); }
{\\ Executing/Testing:
Dragon(13); \\ Dragon13.png
Dragon(17); \\ Dragon17.png
Dragon(21); \\ Dragon21.png
Dragon(23); \\ No result }
</lang>
- Output:
*** Dragon curve, level 13 *** last result computed in 282 ms. *** Dragon curve, level 17 *** last result computed in 453 ms. *** Dragon curve, level 21 *** last result computed in 7,266 ms. *** Dragon curve, level 23 *** concat: the PARI stack overflows ! *** last result computed in 0 ms.
Pascal
using Compas (Pascal with Logo-expansion): <lang pascal>procedure dcr(step,dir:integer;length:real);
begin;
step:=step -1;
length:= length/sqrt(2);
if dir > 0 then
begin
if step > 0 then
begin
turnright(45);
dcr(step,1,length);
turnleft(90);
dcr(step,0,length);
turnright(45);
end
else
begin
turnright(45);
forward(length);
turnleft(90);
forward(length);
turnright(45);
end;
end
else
begin
if step > 0 then
begin
turnleft(45);
dcr(step,1,length);
turnright(90);
dcr(step,0,length);
turnleft(45);
end
else
begin
turnleft(45);
forward(length);
turnright(90);
forward(length);
turnleft(45);
end;
end;
end;</lang> main program: <lang pascal>begin
init; penup; back(100); pendown; dcr(step,direction,length); close;
end.</lang>
Perl
As in the Raku solution, we'll use a Lindenmayer system and draw the dragon in SVG. <lang perl>use SVG; use List::Util qw(max min);
use constant pi => 2 * atan2(1, 0);
- Compute the curve with a Lindemayer-system
my %rules = (
X => 'X+YF+', Y => '-FX-Y'
); my $dragon = 'FX'; $dragon =~ s/([XY])/$rules{$1}/eg for 1..10;
- Draw the curve in SVG
($x, $y) = (0, 0); $theta = 0; $r = 6;
for (split //, $dragon) {
if (/F/) {
push @X, sprintf "%.0f", $x;
push @Y, sprintf "%.0f", $y;
$x += $r * cos($theta);
$y += $r * sin($theta);
}
elsif (/\+/) { $theta += pi/2; }
elsif (/\-/) { $theta -= pi/2; }
}
$xrng = max(@X) - min(@X); $yrng = max(@Y) - min(@Y); $xt = -min(@X)+10; $yt = -min(@Y)+10; $svg = SVG->new(width=>$xrng+20, height=>$yrng+20); $points = $svg->get_path(x=>\@X, y=>\@Y, -type=>'polyline'); $svg->rect(width=>"100%", height=>"100%", style=>{'fill'=>'black'}); $svg->polyline(%$points, style=>{'stroke'=>'orange', 'stroke-width'=>1}, transform=>"translate($xt,$yt)");
open $fh, '>', 'dragon_curve.svg'; print $fh $svg->xmlify(-namespace=>'svg'); close $fh;</lang> Dragon curve (offsite image)
Phix
You can run this online here. Changing the colour and depth give some mildly interesting results.
-- -- demo\rosetta\DragonCurve.exw -- ============================ -- with javascript_semantics include pGUI.e Ihandle dlg, canvas cdCanvas cddbuffer, cdcanvas integer colour = 0 procedure Dragon(integer depth, atom x1, y1, x2, y2) depth -= 1 if depth<=0 then cdCanvasSetForeground(cddbuffer, colour) cdCanvasLine(cddbuffer, x1, y1, x2, y2) -- (some interesting colour patterns emerge) colour += 2 -- colour += 2000 -- colour += #100 else atom dx = x2-x1, dy = y2-y1, nx = x1+(dx-dy)/2, ny = y1+(dx+dy)/2 Dragon(depth,x1,y1,nx,ny) Dragon(depth,x2,y2,nx,ny) end if end procedure function redraw_cb(Ihandle /*ih*/, integer /*posx*/, /*posy*/) cdCanvasActivate(cddbuffer) cdCanvasClear(cddbuffer) -- (note: depths over 21 take a long time to draw, -- depths <= 16 look a little washed out) Dragon(17,100,100,100+256,100) cdCanvasFlush(cddbuffer) return IUP_DEFAULT end function function map_cb(Ihandle ih) cdcanvas = cdCreateCanvas(CD_IUP, ih) cddbuffer = cdCreateCanvas(CD_DBUFFER, cdcanvas) cdCanvasSetBackground(cddbuffer, CD_PARCHMENT) return IUP_DEFAULT end function procedure main() IupOpen() canvas = IupCanvas(NULL) IupSetAttribute(canvas, "RASTERSIZE", "420x290") IupSetCallback(canvas, "MAP_CB", Icallback("map_cb")) IupSetCallback(canvas, "ACTION", Icallback("redraw_cb")) dlg = IupDialog(canvas,"RESIZE=NO") IupSetAttribute(dlg, "TITLE", "Dragon Curve") IupShow(dlg) if platform()!=JS then IupMainLoop() IupClose() end if end procedure main()
PicoLisp
This uses the 'brez' line drawing function from Bitmap/Bresenham's line algorithm#PicoLisp. <lang PicoLisp># Need some turtle graphics (load "@lib/math.l")
(setq
*TurtleX 100 # X position *TurtleY 75 # Y position *TurtleA 0.0 ) # Angle
(de fd (Img Len) # Forward
(let (R (*/ *TurtleA pi 180.0) DX (*/ (cos R) Len 1.0) DY (*/ (sin R) Len 1.0))
(brez Img *TurtleX *TurtleY DX DY)
(inc '*TurtleX DX)
(inc '*TurtleY DY) ) )
(de rt (A) # Right turn
(inc '*TurtleA A) )
(de lt (A) # Left turn
(dec '*TurtleA A) )
- Dragon curve stuff
(de *DragonStep . 4)
(de dragon (Img Depth Dir)
(if (=0 Depth)
(fd Img *DragonStep)
(rt Dir)
(dragon Img (dec Depth) 45.0)
(lt (* 2 Dir))
(dragon Img (dec Depth) -45.0)
(rt Dir) ) )
- Run it
(let Img (make (do 200 (link (need 300 0)))) # Create image 300 x 200
(dragon Img 10 45.0) # Build dragon curve
(out "img.pbm" # Write to bitmap file
(prinl "P1")
(prinl 300 " " 200)
(mapc prinl Img) ) )</lang>
PL/I
This was written for the Ministry of Works IBM390 system running MVS/XA. Odd results when linking from a library of previously-compiled procedures led to the preference for employing libraries via including source files. That way, all of the prog. would be compiled with the same settings: optimisation, bound checking, etc. and the odd behaviour vanished. As complexity grew, these libraries tended to take advantage of each other, so small ad-hoc progs. still ended up needing many inclusions. GOODIES for example defined INTEGER to be FIXED BINARY(16,0), BOOLEAN as FIXED BIT(1) ALIGNED, etc. and so was nearly always wanted. RUNFILE offered an interface to the special assembler routines (written by the MOW) that enabled run-time file allocation and also helped with error messages. CARDINAL and ORDINAL are for presenting numbers as texts. And PSTUFF supplied my notions of an interface to the local plotting routines that allowed output to an IBM3268 screen or a CalComp pen plotter and a few others. These routines are alas no longer available, but I do have an order 19 Dragoncurve that was plotted on a sheet of 32" by 56" by the Calcomp shortly before it was retired, still in excellent order: the + plotted at the start and the x at the end were perfectly aligned. To the 119 secs of cpu time to generate the plot file (the Calcomp format was used, in units of a thousandth of an inch), a further 350 seconds was needed to present the results to the plotter. The charge rate was a dollar a second...
The source file was used to test plotting opportunities, and I have removed the code to draw the likes of a snowflake, pursuit curves, Lissajou curves, and a few others. If the dragon curve order was less than twelve, then all up to that order would be drawn, otherwise only the specified order for the larger jobs. The odd layout (especially of the documentation for DRAGONCURVE) was grist to the "prettyprint" process of PLIST that would list pl/i source files with whole-line comments textflowed into a lineprinter width of 132 columns and end-of-line comments were aligned to the right, away from the source on the left. Each printer line began with the line sequence number, normally in columns 73-80, though they have been removed here. Display screens only had a width of 72 for the source and six for the line sequence: with the ISPF editor, each field control code occupied one space on the display.
The method uses a bit string to represent the turn direction, and each "fold" to construct the next dragon curve involved appending an inverted and reversed copy of the current bit string to the end of the current string after a "1" bit representing the fold. That is, source 1 ecruos where "ecruos" is inverted via not - this scheme was described to me by an acquaintance at Auckland University in 1970. The dragon curve was not drawn by straight lines, because that meant that the dragon curve would intersect with itself at many corners. So, instead of showing each bend as two lines at right angles, a quarter-turn of a circle was used with the same orientation. No collisions, and no bewildering areas of simple squares huddled together. There cannot be any intersections, because the original involves a sheet of paper and no matter how folded it never passes through itself.
A restriction of the pl/i compiler in the 1980s was that array indices could not exceed 32767, thus the escalation to a two-dimensional array, as in DECLARE FOLD(0:31,0:32767) BOOLEAN; /*Oh for (0:1000000) or so..*/ This made the array indexing rather messy.
<lang PLI>
- PROCESS GONUMBER, MARGINS(1,72), NOINTERRUPT, MACRO;
TEST:PROCEDURE OPTIONS(MAIN);
DECLARE SYSIN FILE STREAM INPUT, DRAGON FILE STREAM OUTPUT PRINT, SYSPRINT FILE STREAM OUTPUT PRINT; DECLARE (MIN,MAX,MOD,INDEX,LENGTH,SUBSTR,VERIFY,TRANSLATE) BUILTIN; DECLARE (COMPLEX,SQRT,REAL,IMAG,ATAN,SIN,EXP,COS,ABS) BUILTIN; %INCLUDE PLILIB(GOODIES); %INCLUDE PLILIB(SCAN); %INCLUDE PLILIB(GRAMMAR); %INCLUDE PLILIB(CARDINAL); %INCLUDE PLILIB(ORDINAL); %INCLUDE PLILIB(ANSWAROD); %INCLUDE PLILIB(RUNFILE); %INCLUDE PLILIB(PSTUFF);
DECLARE (TWOPI,TORAD) REAL; DECLARE RANGE(4) REAL; DECLARE TRACERANGE BOOLEAN INITIAL(FALSE); DECLARE FRESHRANGE BOOLEAN INITIAL(TRUE);
BOUND:PROCEDURE(Z);
DECLARE Z COMPLEX;
DECLARE (ZX,ZY) REAL;
ZX = REAL(Z); ZY = IMAG(Z);
IF FRESHRANGE THEN
DO;
RANGE(1),RANGE(2) = ZX;
RANGE(3),RANGE(4) = ZY;
END;
ELSE
DO;
RANGE(1) = MIN(RANGE(1),ZX);
RANGE(2) = MAX(RANGE(2),ZX);
RANGE(3) = MIN(RANGE(3),ZY);
RANGE(4) = MAX(RANGE(4),ZY);
END;
FRESHRANGE = FALSE;
END BOUND;
PLOTZ:PROCEDURE(Z,PEN); DECLARE Z COMPLEX; DECLARE PEN INTEGER; IF TRACERANGE THEN CALL BOUND(Z); CALL PLOT(REAL(Z),IMAG(Z),PEN); END PLOTZ;
%PAGE; DRAGONCURVE:PROCEDURE(ORDER,HOP); /*Folding paper in two...*/
/*Some statistics on runs with x = 56.25", y = 32.6" &(the calcomp plotter).*/ /*The actual size of the picture determines the number of steps &to each quarter-turn.*/ /* n turns x y secs dx dy &*/ /* 20 1,048,575 -2389:681 -682:1364 180+ 3070 2046 &*/ /* 19 524,287 -1365:681 -340:1364 119 2046 1704 &*/ /* 18 262,143 -341:681 -340:1194 71 1022 1554 &*/ /* 17 131,071 -171:681 -340:682 35 852 1022 &*/
DECLARE ORDER BIGINT; /*So how many folds.*/
DECLARE HOP BOOLEAN;
DECLARE FOLD(0:31,0:32767) BOOLEAN; /*Oh for (0:1000000) or so..*/
DECLARE (TURN,N,IT,I,I1,I2,J1,J2,L,LL) BIGINT;
DECLARE (XMIN,XMAX,YMIN,YMAX,XMID,YMID) REAL;
DECLARE (IXMIN,IXMAX,IYMIN,IYMAX) BIGINT;
DECLARE (S,H,TORAD) REAL;
DECLARE (ZMID,Z,Z2,DZ,ZL) COMPLEX;
DECLARE (FULLTURN,ABOUTTURN,QUARTERTURN) INTEGER;
DECLARE (WAY,DIRECTION,ND,LD,LD1,LD2) INTEGER;
DECLARE LEAF(0:3,0:360) COMPLEX; /*Corner turning.*/
DECLARE SWAPXY BOOLEAN; /*Try to align rectangles.*/
DECLARE (T1,T2) CHARACTER(200) VARYING;
IF ¬PLOTCHOICE() THEN RETURN; /*Ascertain the plot device.*/
N = 0;
FOR TURN = 1 TO ORDER;
IT = N + 1;
I1 = IT/32768; I2 = MOD(IT,32768);
FOLD(I1,I2) = TRUE;
FOR I = 1 TO N;
I1 = (IT + I)/32768; I2 = MOD(IT + I,32768);
J1 = (IT - I)/32768; J2 = MOD(IT - I,32768);
FOLD(I1,I2) = ¬FOLD(J1,J2);
END;
N = N*2 + 1;
IF HOP & TURN < ORDER THEN GO TO XX;
XMIN,XMAX,YMIN,YMAX = 0;
Z = 0; /*Start at the origin.*/
DZ = 1; /*Step out unilaterally.*/
FOR I = 1 TO N;
Z = Z + DZ; /*Take the step before the kink.*/
I1 = I/32768; I2 = MOD(I,32768);
IF FOLD(I1,I2) THEN DZ = DZ*(0 + 1I); ELSE DZ = DZ*(0 - 1I);
Z = Z + DZ; /*The step after the kink.*/
XMIN = MIN(XMIN,REAL(Z)); XMAX = MAX(XMAX,REAL(Z));
YMIN = MIN(YMIN,IMAG(Z)); YMAX = MAX(YMAX,IMAG(Z));
END;
SWAPXY = ((XMAX - XMIN) >= (YMAX - YMIN)) /*Contemplate */
¬= (PLOTSTUFF.XSIZE >= PLOTSTUFF.YSIZE); /* rectangularities.*/
IF SWAPXY THEN
DO;
H = XMIN;
XMIN = YMIN;
YMIN = -XMAX;
XMAX = YMAX;
YMAX = -H;
END;
IXMAX = XMAX; IYMAX = YMAX; IXMIN = XMIN; IYMIN = YMIN;
XMID = (XMAX + XMIN)/2; YMID = (YMAX + YMIN)/2;
ZMID = COMPLEX(XMID,YMID);
XMAX = XMAX - XMID; YMAX = YMAX - YMID;
XMIN = XMIN - XMID; YMIN = YMIN - YMID;
T1 = 'Order ' || IFMT(TURN) || ' Dragoncurve, '
|| SAYNUM(0,N,'turn') || '.';
IF SWAPXY THEN T2 = 'y range ' || IFMT(IYMIN) || ':' || IFMT(IYMAX)
|| ', x range ' || IFMT(IXMIN) || ':' || IFMT(IXMAX);
ELSE T2 = 'x range ' || IFMT(IXMIN) || ':' || IFMT(IXMAX)
|| ', y range ' || IFMT(IYMIN) || ':' || IFMT(IYMAX);
S = MIN(PLOTSTUFF.XSIZE/(XMAX - XMIN), /*Rectangularity */
(PLOTSTUFF.YSIZE - 4*H)/(YMAX - YMIN)); /* matching?*/
H = MIN(PLOTSTUFF.XSIZE,S*(XMAX - XMIN)); /*X-width for text.*/
H = MIN(PLOTCHAR,H/(MAX(LENGTH(T1),LENGTH(T2)) + 6));
IF ¬NEWRANGE(XMIN*S,XMAX*S,YMIN*S-2*H,YMAX*S+2*H) THEN STOP('Urp!');
CALL PLOTTEXT(-LENGTH(T1)*H/2,YMAX*S + 2*PLOTTICK,H,T1,0);
CALL PLOTTEXT(-LENGTH(T2)*H/2,YMIN*S - 2*H + 2*PLOTTICK,H,T2,0);
QUARTERTURN = MIN(MAX(3,12*SQRT(S)),90); /*Angle refinement.*/
ABOUTTURN = QUARTERTURN*2;
FULLTURN = QUARTERTURN*4; /*Ensures divisibility.*/
TORAD = TWOPI/FULLTURN; /*Imagine if FULLTURN was 360.*/
ZL = 1; /*Start with 0 degrees.*/
FOR L = 0 TO 3; /*The four directions.*/
FOR I = 0 TO FULLTURN; /*Fill out the petals in the corner.*/
LEAF(L,I) = ZL + EXP((0 + 1I)*I*TORAD); /*Poke!*/
END; /*Fill out the full circle for each for simplicity.*/
ZL = ZL*(0 + 1I); /*Rotate to the next axis.*/
END; /*Four circles, centred one unit along each axial direction.*/
Z = -ZMID; /*The start point. Was 0, before shift by ZMID.*/
CALL PLOTZ(S*Z,3); /*Position the pen.*/
DIRECTION = 0; /*The way ahead is along the x-axis.*/
DZ = 1; /*The step before the kink.*/
IF SWAPXY THEN DIRECTION = -QUARTERTURN; /*Or maybe y.*/
IF SWAPXY THEN DZ = (0 - 1I); /*An x-y swap.*/
FRESHRANGE = TRUE; /*A sniffing.*/
FOR I = 1 TO N; /*The deviationism begins.*/
I1 = I/32768; I2 = MOD(I,32768);
IF FOLD(I1,I2) THEN WAY = +1; ELSE WAY = -1;
ND = DIRECTION + QUARTERTURN*WAY;
IF ND >= FULLTURN THEN ND = ND - FULLTURN;
IF ND < 0 THEN ND = ND + FULLTURN;
LD = ND/QUARTERTURN; /*Select a leaf.*/
LD1 = MOD(ND + ABOUTTURN,FULLTURN);
LD2 = LD1 + WAY*QUARTERTURN; /*No mod, see the FOR loop below.*/
FOR L = LD1 TO LD2 BY WAY; /*Round the kink.*/
LL = L; /*A copy to wrap into range.*/
IF LL < 0 THEN LL = LL + FULLTURN;
IF LL >= FULLTURN THEN LL = LL - FULLTURN;
ZL = Z + LEAF(LD,LL); /*Work along the curve.*/
CALL PLOTZ(S*ZL,2); /*Move a bit.*/
END; /*On to the next step.*/
DIRECTION = ND; /*The new direction.*/
Z = Z + DZ; /*The first half of the step that has been rounded.*/
DZ = DZ*(0 + 1I)*WAY; /*A right-angle, one way or the other.*/
Z = Z + DZ; /*Avoid the roundoff of hordes of fractional moves.*/
END; /*On to the next fold.*/
CALL PLOT(0,0,998);
IF TRACERANGE THEN PUT SKIP(3) FILE(DRAGON) LIST('Dragoncurve: ');
IF TRACERANGE THEN PUT FILE(DRAGON) DATA(RANGE,ORDER,S,ZMID);
XX:END;
END DRAGONCURVE; %PAGE; %PAGE; %PAGE; RANDOM:PROCEDURE(SEED) RETURNS(REAL); DECLARE SEED INTEGER; SEED = SEED*497 + 4032; IF SEED <= 0 THEN SEED = SEED + 32767; IF SEED > 32767 THEN SEED = MOD(SEED,32767); RETURN(SEED/32767.0); END RANDOM;
%PAGE; TRACE:PROCEDURE(O,R,A,N,G); DECLARE (I,N,G) INTEGER; DECLARE (O,R,A(*),X0,X1,X2) COMPLEX; X1 = O + R*A(1); X0 = X1; CALL PLOT(REAL(X1),IMAG(X1),3); FOR I = 2 TO N; X2 = O + R*A(I); CALL PLOT(REAL(X2),IMAG(X2),2); X1 = X2; END; CALL PLOT(REAL(X0),IMAG(X0),2); END TRACE;
CENTREZ:PROCEDURE(A,N); DECLARE (A(*),T) COMPLEX; DECLARE (I,N) INTEGER; T = 0; FOR I = 1 TO N; T = T + A(I); END; T = T/N; FOR I = 1 TO N; A(I) = A(I) - T; END; END CENTREZ; %PAGE; %PAGE; DECLARE (BELCH,ORDER,CHASE,TWIRL) INTEGER; DECLARE HOP BOOLEAN;
TWOPI = 8*ATAN(1);
TORAD = TWOPI/360;
BELCH = REPLYN('How many dragoncurves (max 20)');
IF BELCH < 12 THEN HOP = FALSE;
ELSE HOP = YEA('Go directly to order ' || IFMT(BELCH));
/*ORDER = REPLYN('The depth of recursion (eg 4)');
CHASE = REPLYN('How many pursuits');
TWIRL = REPLYN('How many twirls');
TRACERANGE = YEA('Trace the ranges');*/
CALL DRAGONCURVE(BELCH,HOP);
/*CALL TRIANGLEPLEX(ORDER);
CALL SQUAREBASH(ORDER,+1); CALL SQUAREBASH(ORDER,-1); CALL SNOWFLAKE(ORDER); CALL SNOWFLAKE3(ORDER); CALL PURSUE(CHASE); CALL LISSAJOU(TWIRL); CALL CARDIOD; CALL HEART;*/ CALL PLOT(0,0,-3); CALL PLOT(0,0,999);
END TEST; </lang>
PostScript
<lang postscript>%!PS %%BoundingBox: 0 0 550 400 /ifpendown false def /rotation 0 def /srootii 2 sqrt def /turn {
rotation add /rotation exch def
} def
/forward {
dup rotation cos mul
exch rotation sin mul
ifpendown
{ rlineto }
{ rmoveto }
ifelse
} def
/penup {
/ifpendown false def
} def
/pendown {
/ifpendown true def
} def
/dragon { % [ length, split, d ]
dup
dup 1 get 0 eq
{ 0 get forward }
{ dup 2 get 45 mul turn
dup aload pop pop
1 sub exch srootii div exch
1 3 array astore dragon pop
dup 2 get 90 mul neg turn
dup aload pop pop
1 sub exch srootii div exch
-1 3 array astore dragon
dup 2 get 45 mul turn
}
ifelse
pop
} def
150 150 moveto pendown [ 300 12 1 ] dragon stroke % 0 0 moveto 550 0 rlineto 0 400 rlineto -550 0 rlineto closepath stroke showpage %%END</lang>
Or (almost) verbatim string rewrite: (this is a 20 page document, and don't try to print it, or you might have a very angry printer). <lang postscript>%!PS-Adobe-3.0 %%BoundingBox 0 0 300 300
/+ { 90 rotate } def /- {-90 rotate } def /!1 { dup 1 sub dup 0 eq not } def
/F { 180 0 rlineto } def /X { !1 { X + Y F + } if pop } def /Y { !1 { - F X - Y } if pop } def
/dragon {
gsave
70 180 moveto
dup 1 sub { 1 2 div sqrt dup scale -45 rotate } repeat
F X stroke
grestore
} def
1 1 20 { dragon showpage } for
%%EOF</lang>
- See also
POV-Ray
Example code recursive and iterative can be found at Courbe du Dragon.
PowerShell
WinForms
<lang powershell># handy constants with script-wide scope [Single]$script:QUARTER_PI=0.7853982 [Single]$script:HALF_PI=1.570796
- leverage GDI (Forms and Drawing)
Add-Type -AssemblyName System.Drawing Add-Type -AssemblyName System.Windows.Forms $script:TurtlePen = New-Object Drawing.Pen darkGreen $script:Form = New-Object Windows.Forms.Form $script:Canvas = $Form.CreateGraphics()
- implement a turtle graphics model
Class Turtle { # relies on the script-scoped $TurtlePen and $Canvas
# member properties for turtle's position and orientation [Single]$TurtleX [Single]$TurtleY [Single]$TurtleAngle
# constructors
Turtle() {
$this.TurtleX, $this.TurtleY = 44.0, 88.0
$this.TurtleAngle = 0.0
}
Turtle([Single]$InitX, [Single]$InitY, [Single]$InitAngle) {
$this.TurtleX, $this.TurtleY = $InitX, $InitY
$this.TurtleAngle = $InitAngle
}
# methods for turning and drawing
[Void]Turn([Single]$Angle) { # $Angle measured in radians
$this.TurtleAngle += $Angle # use positive $Angle for right turn, negative for left turn
}
[Void]Forward([Single]$Distance) {
# draw line segment
$TargetX = $this.TurtleX + $Distance * [Math]::Cos($this.TurtleAngle)
$TargetY = $this.TurtleY + $Distance * [Math]::Sin($this.TurtleAngle)
$script:Canvas.DrawLine($script:TurtlePen, $this.TurtleX, $this.TurtleY, $TargetX, $TargetY)
# relocate turtle to other end of segment
$this.TurtleX = $TargetX
$this.TurtleY = $TargetY
}
} # end of Turtle class definition
- Implement dragon curve drawing methods in a subclass that inherits Turtle
Class DragonTurtle : Turtle {
# DCL: recursive dragon curve, starting with left turns
[Void]DCL([Byte]$Step, [Single]$Length) {
$AdjustedStep = $Step - 1
$AdjustedLength = $Length / [Math]::Sqrt(2.0)
$this.Turn(-$script:QUARTER_PI)
if ($AdjustedStep -gt 0) {
$this.DCR($AdjustedStep, $AdjustedLength)
} else {
$this.Forward($AdjustedLength)
}
$this.Turn($script:HALF_PI)
if ($AdjustedStep -gt 0) {
$this.DCL($AdjustedStep, $AdjustedLength)
} else {
$this.Forward($AdjustedLength)
}
$this.Turn(-$script:QUARTER_PI)
}
# DCR: recursive dragon curve, starting with right turns
[Void]DCR([Byte]$Step, [Single]$Length) {
$AdjustedStep = $Step - 1
$AdjustedLength = $Length / [Math]::Sqrt(2.0)
$this.Turn($script:QUARTER_PI)
if ($AdjustedStep -gt 0) {
$this.DCR($AdjustedStep, $AdjustedLength)
} else {
$this.Forward($AdjustedLength)
}
$this.Turn(-$script:HALF_PI)
if ($AdjustedStep -gt 0) {
$this.DCL($AdjustedStep, $AdjustedLength)
} else {
$this.Forward($AdjustedLength)
}
$this.Turn($script:QUARTER_PI)
}
} # end of DragonTurtle subclass definition
- prepare anonymous dragon-curve painting function for WinForms dialog
$Form.add_paint({
[DragonTurtle]$Dragon = [DragonTurtle]::new() $Dragon.DCR(14,128)
})
- display the GDI Form window, which triggers its prepared anonymous drawing function
$Form.ShowDialog() </lang>
Processing
<lang java>float l = 3; int ints = 13;
void setup() {
size(700, 600); background(0, 0, 255); translate(150, 100); stroke(255); turn_left(l, ints); turn_right(l, ints);
}
void turn_right(float l, int ints) {
if (ints == 0) {
line(0, 0, 0, -l);
translate(0, -l);
} else {
turn_left(l, ints-1);
rotate(radians(90));
turn_right(l, ints-1);
}
}
void turn_left(float l, int ints) {
if (ints == 0) {
line(0, 0, 0, -l);
translate(0, -l);
} else {
turn_left(l, ints-1);
rotate(radians(-90));
turn_right(l, ints-1);
}
}</lang>The sketch can be run online :
here.
Processing Python mode
<lang python>l = 3 ints = 13
def setup():
size(700, 600) background(0, 0, 255) translate(150, 100) stroke(255) turn_left(l, ints) turn_right(l, ints)
def turn_right(l, ints):
if ints == 0:
line(0, 0, 0, -l)
translate(0, -l)
else:
turn_left(l, ints - 1)
rotate(radians(90))
turn_right(l, ints - 1)
def turn_left(l, ints):
if ints == 0:
line(0, 0, 0, -l)
translate(0, -l)
else:
turn_left(l, ints - 1)
rotate(radians(-90))
turn_right(l, ints - 1)</lang>
Prolog
Works with SWI-Prolog which has a Graphic interface XPCE.
DCG are used to compute the list of "turns" of the Dragon Curve and the list of points.
<lang Prolog>dragonCurve(N) :-
dcg_dg(N, [left], DCL, []),
Side = 4,
Angle is -N * (pi/4),
dcg_computePath(Side, Angle, DCL, point(180,400), P, []),
new(D, window('Dragon Curve')),
send(D, size, size(800,600)),
new(Path, path(poly)),
send_list(Path, append, P),
send(D, display, Path),
send(D, open).
% compute the list of points of the Dragon Curve
dcg_computePath(Side, Angle, [left | DCT], point(X1, Y1)) -->
[point(X1, Y1)],
{ X2 is X1 + Side * cos(Angle),
Y2 is Y1 + Side * sin(Angle),
Angle1 is Angle + pi / 2
},
dcg_computePath(Side, Angle1, DCT, point(X2, Y2)).
dcg_computePath(Side, Angle, [right | DCT], point(X1, Y1)) --> [point(X1, Y1)], { X2 is X1 + Side * cos(Angle), Y2 is Y1 + Side * sin(Angle), Angle1 is Angle - pi / 2 }, dcg_computePath(Side, Angle1, DCT, point(X2, Y2)).
dcg_computePath(_Side, _Angle, [], point(X1, Y1)) -->
[ point(X1, Y1)].
% compute the list of the "turns" of the Dragon Curve
dcg_dg(1, L) --> L.
dcg_dg(N, L) --> {dcg_dg(L, L1, []), N1 is N - 1}, dcg_dg(N1, L1).
% one interation of the process dcg_dg(L) --> L, [left], inverse(L).
inverse([H | T]) --> inverse(T), inverse(H).
inverse([]) --> [].
inverse(left) --> [right].
inverse(right) --> [left].</lang> Output :
1 ?- dragonCurve(13). true
PureBasic
<lang PureBasic>#SqRt2 = 1.4142136
- SizeH = 800: #SizeV = 550
Global angle.d, px, py, imageNum
Procedure turn(degrees.d)
angle + degrees * #PI / 180
EndProcedure
Procedure forward(length.d)
Protected w = Cos(angle) * length Protected h = Sin(angle) * length LineXY(px, py, px + w, py + h, RGB(255,255,255)) px + w: py + h
EndProcedure
Procedure dragon(length.d, split, d.d)
If split = 0 forward(length) Else turn(d * 45) dragon(length / #SqRt2, split - 1, 1) turn(-d * 90) dragon(length / #SqRt2, split - 1, -1) turn(d * 45) EndIf
EndProcedure
OpenWindow(0, 0, 0, #SizeH, #SizeV, "DragonCurve", #PB_Window_SystemMenu) imageNum = CreateImage(#PB_Any, #SizeH, #SizeV, 32) ImageGadget(0, 0, 0, 0, 0, ImageID(imageNum))
angle = 0: px = 185: py = 190 If StartDrawing(ImageOutput(imageNum))
dragon(400, 15, 1) StopDrawing() SetGadgetState(0, ImageID(imageNum))
EndIf
Repeat: Until WaitWindowEvent(10) = #PB_Event_CloseWindow</lang>
Python
<lang python>from turtle import *
def dragon(step, length):
dcr(step, length)
def dcr(step, length):
step -= 1
length /= 1.41421
if step > 0:
right(45)
dcr(step, length)
left(90)
dcl(step, length)
right(45)
else:
right(45)
forward(length)
left(90)
forward(length)
right(45)
def dcl(step, length):
step -= 1 length /= 1.41421
if step > 0:
left(45)
dcr(step, length)
right(90)
dcl(step, length)
left(45)
else:
left(45)
forward(length)
right(90)
forward(length)
left(45)</lang>
A more pythonic version: <lang python>from turtle import right, left, forward, speed, exitonclick, hideturtle
def dragon(level=4, size=200, zig=right, zag=left):
if level <= 0:
forward(size)
return
size /= 1.41421 zig(45) dragon(level-1, size, right, left) zag(90) dragon(level-1, size, left, right) zig(45)
speed(0) hideturtle() dragon(6) exitonclick() # click to exit</lang> Other version: <lang python>from turtle import right, left, forward, speed, exitonclick, hideturtle
def dragon(level=4, size=200, direction=45):
if level:
right(direction)
dragon(level-1, size/1.41421356237, 45)
left(direction * 2)
dragon(level-1, size/1.41421356237, -45)
right(direction)
else:
forward(size)
speed(0) hideturtle() dragon(6) exitonclick() # click to exit</lang>
Quackery
<lang Quackery> [ $ "turtleduck.qky" loadfile ] now!
[ 2 * 2dup turn 4 1 walk turn ] is corner ( n/d --> )
forward is right ( n --> )
forward is left ( n --> )
[ dup 0 = iff
[ drop 8 1 walk ] done
1 - dup
left
1 4 corner
right ] resolves right ( n --> )
[ dup 0 = iff
[ drop 8 1 walk ] done
1 - dup
left
-1 4 corner
right ] resolves left ( n --> )
turtle -260 1 fly 3 4 turn 100 1 fly 5 8 turn 11 left</lang>
- Output:
R
Version #1.
<lang R> Dragon<-function(Iters){
Rotation<-matrix(c(0,-1,1,0),ncol=2,byrow=T) ########Rotation multiplication matrix Iteration<-list() ###################################Set up list for segment matrices for 1st Iteration1 <- matrix(rep(0,16), ncol = 4) Iteration1[1,]<-c(0,0,1,0) Iteration1[2,]<-c(1,0,1,-1) Moveposition<-rep(0,Iters) ##########################Which point should be shifted to origin Moveposition[1]<-4 if(Iters > 1){#########################################where to move to get to origin for(l in 2:Iters){#####################################only if >1, because 1 set before for loop Moveposition[l]<-(Moveposition[l-1]*2)-2#############sets vector of all positions in matrix where last point is }} Move<-list() ########################################vector to add to all points to shift start at origin
for (i in 1:Iters){ half<-dim(Iterationi)[1]/2 half<-1:half for(j in half){########################################Rotate all points 90 degrees clockwise
Iterationi[j+length(half),]<-c(Iterationi[j,1:2]%*%Rotation,Iterationi[j,3:4]%*%Rotation)
} Movei<-matrix(rep(0,4),ncol=4) Movei[1,1:2]<-Movei[1,3:4]<-(Iterationi[Moveposition[i],c(3,4)]*-1) Iterationi+1<-matrix(rep(0,2*dim(Iterationi)[1]*4),ncol=4)##########move the dragon, set next Iteration's matrix for(k in 1:dim(Iterationi)[1]){#########################################move dragon by shifting all previous iterations point
Iterationi+1[k,]<-Iterationi[k,]+Movei###so the start is at the origin
} xlimits<-c(min(Iterationi[,3])-2,max(Iterationi[,3]+2))#Plot ylimits<-c(min(Iterationi[,4])-2,max(Iterationi[,4]+2)) plot(0,0,type='n',axes=FALSE,xlab="",ylab="",xlim=xlimits,ylim=ylimits) s<-dim(Iterationi)[1] s<-1:s segments(Iterationi[s,1], Iterationi[s,2], Iterationi[s,3], Iterationi[s,4], col= 'red') }}######################################################################### </lang> [2]
![[2]](https://commons.wikimedia.org/wiki/File:Dragon_Curve_16_Iterations_R_programming_language.png#mediaviewer/File:Dragon_Curve_16_Iterations_R_programming_language.png){kind=link}
Version #2.
Note: This algorithm in R works only for orders <= 16. For bigger values it returns error
in bitwAnd() [bit-wise AND].
It means: 32-bit integer is not long enough. This is true even on 64-bit computer.
See samples using the same algorithm in JavaScript version #2 (order is up to 25, may be even greater).
<lang r>
- Generate and plot Dragon curve.
- translation of JavaScript v.#2: http://rosettacode.org/wiki/Dragon_curve#JavaScript
- 2/27/16 aev
- gpDragonCurve(ord, clr, fn, d, as, xsh, ysh)
- Where: ord - order (defines the number of line segments);
- clr - color, fn - file name (.ext will be added), d - segment length,
- as - axis scale, xsh - x-shift, ysh - y-shift
gpDragonCurve <- function(ord, clr, fn, d, as, xsh, ysh) {
cat(" *** START:", date(), "order=",ord, "color=",clr, "\n");
d=10; m=640; ms=as*m; n=bitwShiftL(1, ord);
c=c1=c2=c2x=c2y=i1=0; x=y=x1=y1=0;
if(fn=="") {fn="DCR"}
pf=paste0(fn, ord, ".png");
ttl=paste0("Dragon curve, ord=",ord);
cat(" *** Plot file -", pf, "title:", ttl, "n=",n, "\n");
plot(NA, xlim=c(-ms,ms), ylim=c(-ms,ms), xlab="", ylab="", main=ttl);
for (i in 0:n) {
segments(x1+xsh, y1+ysh, x+xsh, y+ysh, col=clr); x1=x; y1=y;
c1=bitwAnd(c, 1); c2=bitwAnd(c, 2);
c2x=d; if(c2>0) {c2x=(-1)*d}; c2y=(-1)*c2x;
if(c1>0) {y=y+c2y} else {x=x+c2x}
i1=i+1; ii=bitwAnd(i1, -i1); c=c+i1/ii;
}
dev.copy(png, filename=pf, width=m, height=m); # plot to png-file
dev.off(); graphics.off(); # Cleaning
cat(" *** END:",date(),"\n");
}
- Testing samples:
gpDragonCurve(7, "red", "", 20, 0.2, -30, -30)
- gpDragonCurve(11, "red", "", 10, 0.6, 100, 200)
gpDragonCurve(13, "navy", "", 10, 1, 300, -200)
- gpDragonCurve(15, "darkgreen", "", 10, 2, -450, -500)
gpDragonCurve(16, "darkgreen", "", 10, 3, -1050, -500) </lang>
- Output:
> gpDragonCurve(7, "red", "", 20, 0.2, -30, -30) *** START: Mon Feb 27 12:53:57 2017 order= 7 color= red *** Plot file - DCR7.png title: Dragon curve, ord=7 n= 128 *** END: Mon Feb 27 12:53:57 2017 > gpDragonCurve(13, "navy", "", 10, 1, 300, -200) *** START: Mon Feb 27 12:44:04 2017 order= 13 color= navy *** Plot file - DCR13.png title: Dragon curve, ord=13 n= 8192 *** END: Mon Feb 27 12:44:06 2017 > gpDragonCurve(16, "darkgreen", "", 10, 3, -1050, -500) *** START: Mon Feb 27 12:18:56 2017 order= 16 color= darkgreen *** Plot file - DCR16.png title: Dragon curve, ord=16 n= 65536 *** END: Mon Feb 27 12:19:03 2017
Racket
<lang racket>#lang racket
(require plot)
(define (dragon-turn n)
(if (> (bitwise-and (arithmetic-shift (bitwise-and n (- n)) 1) n) 0)
'L
'R))
(define (rotate heading dir)
(cond
[(eq? dir 'R) (cond [(eq? heading 'N) 'E]
[(eq? heading 'E) 'S]
[(eq? heading 'S) 'W]
[(eq? heading 'W) 'N])]
[(eq? dir 'L) (cond [(eq? heading 'N) 'W]
[(eq? heading 'E) 'N]
[(eq? heading 'S) 'E]
[(eq? heading 'W) 'S])]))
(define (step pos heading)
(cond [(eq? heading 'N) (list (car pos) (add1 (cadr pos)))] [(eq? heading 'E) (list (add1 (car pos)) (cadr pos))] [(eq? heading 'S) (list (car pos) (sub1 (cadr pos)))] [(eq? heading 'W) (list (sub1 (car pos)) (cadr pos))] ))
(let-values ([(dir pos trail)
(for/fold ([dir 'N]
[pos (list 0 0)]
[trail '((0 0))])
([n (in-range 0 50000)])
(let* ([new-dir (rotate dir (dragon-turn n))]
[new-pos (step pos new-dir)])
(values new-dir
new-pos
(cons new-pos trail))))])
(plot-file (lines trail) "dragon.png" 'png))</lang>
Raku
(formerly Perl 6) We'll use a L-System role, and draw the dragon in SVG. <lang perl6>use SVG;
role Lindenmayer {
has %.rules;
method succ {
self.comb.map( { %!rules{$^c} // $c } ).join but Lindenmayer(%!rules)
}
}
my $dragon = "FX" but Lindenmayer( { X => 'X+YF+', Y => '-FX-Y' } );
$dragon++ xx ^15;
my @points = 215, 350;
for $dragon.comb {
state ($x, $y) = @points[0,1];
state $d = 2 + 0i;
if /'F'/ { @points.append: ($x += $d.re).round(.1), ($y += $d.im).round(.1) }
elsif /< + - >/ { $d *= "{$_}1i" }
}
say SVG.serialize(
svg => [
:600width, :450height, :style<stroke:rgb(0,0,255)>,
:rect[:width<100%>, :height<100%>, :fill<white>],
:polyline[ :points(@points.join: ','), :fill<white> ],
],
);</lang>
RapidQ
This implementation displays the Dragon Curve fractal in a GUI window. <lang rapidq>DIM angle AS Double DIM x AS Double, y AS Double DECLARE SUB PaintCanvas
CREATE form AS QForm
Width = 800
Height = 600
CREATE canvas AS QCanvas
Height = form.ClientHeight
Width = form.ClientWidth
OnPaint = PaintCanvas
END CREATE
END CREATE
SUB turn (degrees AS Double)
angle = angle + degrees*3.14159265/180
END SUB
SUB forward (length AS Double)
x2 = x + cos(angle)*length y2 = y + sin(angle)*length canvas.Line(x, y, x2, y2, &Haaffff) x = x2: y = y2
END SUB
SUB dragon (length AS Double, split AS Integer, d AS Double)
IF split=0 THEN
forward length
ELSE
turn d*45
dragon length/1.4142136, split-1, 1
turn -d*90
dragon length/1.4142136, split-1, -1
turn d*45
END IF
END SUB
SUB PaintCanvas
canvas.FillRect(0, 0, canvas.Width, canvas.Height, &H102800) x = 220: y = 220: angle = 0 dragon 384, 12, 1
END SUB
form.ShowModal</lang>
REXX
This REXX version uses a unique plot character to indicate which part of the dragon curve is being shown; the
number of "parts" of the dragon curve can be specified (the 1st argument).
The initial (facing) direction may be specified (North, East, South, or West) (the 2nd argument).
A specific plot character can be specified instead for all curve parts (the 3rd argument).
This, in effect, allows the dragon curve to be plotted/displayed with a different (starting) orientation. <lang rexx>/*REXX program creates & draws an ASCII Dragon Curve (or Harter-Heighway dragon curve).*/ d.= 1; d.L= -d.; @.= ' '; x= 0; x2= x; y= 0; y2= y; z= d.; @.x.y= "∙" plot_pts = '123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZΘ' /*plot chars*/ loX= 0; hiX= 0; loY= 0; hiY= 0 /*assign various constants & variables.*/ parse arg # p c . /*#: number of iterations; P=init dir.*/ if #== | #=="," then #= 11 /*Not specified? Then use the default.*/ if p== | p=="," then p= 'north'; upper p /* " " " " " " */ if c== then c= plot_pts /* " " " " " " */ if length(c)==2 then c= x2c(c) /*was a hexadecimal code specified? */ if length(c)==3 then c= d2c(c) /* " " decimal " " */ p= translate( left(p, 1), 0123, 'NESW') /*get the orientation for dragon curve.*/ $= /*initialize the dragon curve to a null*/
do # /*create the start of a dragon curve. */ $= $'R'reverse( translate($, "RL", 'LR') ) /*append the rest of dragon curve. */ end /*#*/ /* [↑] append char, flip, and reverse.*/
do j=1 for length($); _= substr($, j, 1) /*get next cardinal direction for curve*/
p= (p + d._) // 4 /*move dragon curve in a new direction.*/
if p< 0 then p= p + 4 /*Negative? Then use a new direction. */
if p==0 then do; y= y + 1; y2= y + 1; end /*curve is going east cartologically.*/
if p==1 then do; x= x + 1; x2= x + 1; end /* " " south " */
if p==2 then do; y= y - 1; y2= y - 1; end /* " " west " */
if p==3 then do; x= x - 1; x2= x - 1; end /* " " north " */
if j>2**z then z= z + 1 /*identify a part of curve being built.*/
!= substr(c, z, 1) /*choose plot point character (glyph). */
if !==' ' then != right(c, 1) /*Plot point a blank? Then use a glyph*/
@.x.y= !; @.x2.y2= ! /*draw part of the dragon curve. */
loX= min(loX,x,x2); hiX= max(hiX,x,x2); x= x2 /*define the min & max X graph limits*/
loY= min(loY,y,y2); hiY= max(hiY,y,y2); y= y2 /* " " " " " Y " " */
end /*j*/ /* [↑] process all of $ char string.*/
do r=loX to hiX; a= /*nullify the line that will be drawn. */
do c=loY to hiY; a= a || @.r.c /*create a line (row) of curve points. */
end /*c*/ /* [↑] append a single column of a row.*/
if a\== then say strip(a, "T") /*display a line (row) of curve points.*/
end /*r*/ /*stick a fork in it, we're all done. */</lang>
Choosing a high visibility glyph can really help make the dragon much more viewable; the
solid fill ASCII character (█ or hexadecimal db in code page 437) is quite good for this.
output when using the following input: 12 south db
(Shown at 1/6 size)
███ ███ ███ ███ ███ ███ ███ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████ █████████ █████████ █████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ █████ ███ ███ █████ ███ ███ █████ ███ ███ █████ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████ █████████ █████████ █████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ ███ ███████████ ███ █████████ ███ ███ ███████████ ███ █████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████████████████████████████ █████████████████████████████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ █████████████████████████████ ███ ███ █████████████████████████████ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████████████████████████████ █████████████████████████████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████ ███ ███████████████████ ███ ███ █████████ ███ ███████████████████ ███ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████ █████████████████ █████████ █████████ █████████████████ █████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ █████ ███ ███ █████████████ ███ ███ █████ ███ ███ █████ ███ ███ █████████████ ███ ███ █████ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████ █████████████████ █████████ █████████ █████████████████ █████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ ███ ███████████████████ ███ █████████ ███ ███ ███████████████████ ███ █████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████████████████████████████ █████████████████████████████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ █████████████████████████████ ███ ███ █████████████████████████████ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████████████████████████████ █████████████████████████████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ ███ ███ ███ ███ ███ █████████████████████████████████ ███ ███ ███ ███ █████████████████████████ ███ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████ █████████ █████████████████████████████████████████ █████████ █████████████████████████████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ █████ ███ ███ █████ ███ ███ █████████████████████████████████████ ███ ███ █████ ███ ███ █████████████████████████████ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
█████████ █████████ █████████████████████████████████████████ █████████ █████████████████████████████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ ███ ███████████ ███ ███████████ ███ ███████████████████████████████████████████ ███ ███████████ ███ █████████████████████████ ███ ███
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███████████████████████████████████████████ ███████ ███████████████████████ ███████ █████████████████████████████████████████████
█ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █
███ ███████████████████████████████████████ █████ █████████████████████ █████ ███████████████████████████████████████████ ███
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Ruby
<lang ruby>Point = Struct.new(:x, :y) Line = Struct.new(:start, :stop)
Shoes.app(:width => 800, :height => 600, :resizable => false) do
def split_segments(n)
dir = 1
@segments = @segments.inject([]) do |new, l|
a, b, c, d = l.start.x, l.start.y, l.stop.x, l.stop.y
mid_x = a + (c-a)/2.0 - (d-b)/2.0*dir
mid_y = b + (d-b)/2.0 + (c-a)/2.0*dir
mid_p = Point.new(mid_x, mid_y)
dir *= -1
new << Line.new(l.start, mid_p)
new << Line.new(mid_p, l.stop)
end
end
@segments = [Line.new(Point.new(200,200), Point.new(600,200))]
15.times do |n|
info "calculating frame #{n}"
split_segments(n)
end
stack do
@segments.each do |l|
line l.start.x, l.start.y, l.stop.x, l.stop.y
end
end
end</lang>
<lang ruby>LEN = 3 GEN = 14 attr_reader :angle
def setup
sketch_title 'Heighway Dragon' background(0, 0, 255) translate(170, 170) stroke(255) @angle = 90.radians turn_left(GEN)
end
def draw_line
line(0, 0, 0, -LEN) translate(0, -LEN)
end
def turn_right(gen)
return draw_line if gen.zero?
turn_left(gen - 1) rotate(angle) turn_right(gen - 1)
end
def turn_left(gen)
return draw_line if gen.zero?
turn_left(gen - 1) rotate(-angle) turn_right(gen - 1)
end
def settings
size(700, 600)
end </lang>
Context Free Art version <lang ruby>require 'cf3'
INV_SQRT = 1 / Math.sqrt(2)
def setup_the_dragon
@dragon = ContextFree.define do
shape :start do
dragon alpha: 1
end
shape :dragon do
square hue: 0, brightness: 0, saturation: 1, alpha: 0.02
split do
dragon size: INV_SQRT, rotation: -45, x: 0.25, y: 0.25
rewind
dragon size: INV_SQRT, rotation: 135, x: 0.25, y: 0.25
rewind
end
end
end
end
def settings
size 800, 500
end
def setup
sketch_title 'Heighway Dragon' setup_the_dragon draw_it
end
def draw_it
background 255
@dragon.render :start, size: width * 0.8, stop_size: 2,
start_x: width / 3, start_y: height / 3.5
end </lang>
Run BASIC
<lang runbasic>graphic #g, 600,600 RL$ = "R" loc = 90 pass = 0
[loop]
- g "cls ; home ; north ; down ; fill black"
for i =1 to len(RL$)
v = 255 * i /len(RL$) #g "color "; v; " 120 "; 255 -v #g "go "; loc if mid$(RL$,i,1) ="R" then #g "turn 90" else #g "turn -90"
next i
- g "color 255 120 0"
- g "go "; loc
LR$ ="" for i =len( RL$) to 1 step -1
if mid$( RL$, i, 1) ="R" then LR$ =LR$ +"L" else LR$ =LR$ +"R"
next i
RL$ = RL$ + "R" + LR$ loc = loc / 1.35 pass = pass + 1 render #g input xxx cls
if pass < 16 then goto [loop] end</lang>
Rust
<lang rust> use ggez::{
conf::{WindowMode, WindowSetup},
error::GameResult,
event,
graphics::{clear, draw, present, Color, MeshBuilder},
nalgebra::Point2,
Context,
}; use std::time::Duration;
// L-System to create the sequence needed for a Dragon Curve. // This function creates the next generation given the current one // L-System from https://www.cs.unm.edu/~joel/PaperFoldingFractal/L-system-rules.html // fn l_system_next_generation(current_generation: &str) -> String {
let f_rule = "f-h";
let h_rule = "f+h";
let mut next_gen = String::new();
for char in current_generation.chars() {
match char {
'f' => next_gen.push_str(f_rule),
'h' => next_gen.push_str(h_rule),
'-' | '+' => next_gen.push(char),
_ => panic!("Unknown char {}", char),
}
}
next_gen
}
// The rest of the code is for drawing the output and is specific to using the // ggez 2d game library: https://ggez.rs/
const WINDOW_WIDTH: f32 = 700.0; const WINDOW_HEIGHT: f32 = 700.0; const START_X: f32 = WINDOW_WIDTH / 6.0; const START_Y: f32 = WINDOW_HEIGHT / 6.0; const MAX_DEPTH: i32 = 15; const LINE_LENGTH: f32 = 20.0;
struct MainState {
start_gen: String, next_gen: String, line_length: f32, max_depth: i32, current_depth: i32,
}
impl MainState {
fn new() -> GameResult<MainState> {
let start_gen = "f";
let next_gen = String::new();
let line_length = LINE_LENGTH;
let max_depth = MAX_DEPTH;
let current_depth = 0;
Ok(MainState {
start_gen: start_gen.to_string(),
next_gen,
line_length,
max_depth,
current_depth,
})
}
}
impl event::EventHandler for MainState {
// In each repetition of the event loop a new generation of the L-System
// is generated and drawn, until the maximum depth is reached.
// Each time the line length is reduced so that the overall dragon curve
// can be seen in the window as it spirals and gets bigger.
// The update sleeps for 0.5 seconds just so that its pogression can be watched.
//
fn update(&mut self, _ctx: &mut Context) -> GameResult {
if self.current_depth < self.max_depth {
self.next_gen = l_system_next_generation(&self.start_gen);
self.start_gen = self.next_gen.clone();
self.line_length -= (self.line_length / self.max_depth as f32) * 1.9;
self.current_depth += 1;
}
ggez::timer::sleep(Duration::from_millis(500));
Ok(())
}
fn draw(&mut self, ctx: &mut Context) -> GameResult {
let grey = Color::from_rgb(77, 77, 77);
let blue = Color::from_rgb(51, 153, 255);
let initial_point_blue = Point2::new(START_X, START_Y);
clear(ctx, grey);
draw_lines(
&self.next_gen,
ctx,
self.line_length,
blue,
initial_point_blue,
)?;
present(ctx)?;
Ok(())
}
}
fn next_point(current_point: Point2<f32>, heading: f32, line_length: f32) -> Point2<f32> {
let next_point = (
(current_point.x + (line_length * heading.to_radians().cos().trunc() as f32)),
(current_point.y + (line_length * heading.to_radians().sin().trunc() as f32)),
);
Point2::new(next_point.0, next_point.1)
}
fn draw_lines(
instructions: &str, ctx: &mut Context, line_length: f32, colour: Color, initial_point: Point2<f32>,
) -> GameResult {
let line_width = 2.0;
let mut heading = 0.0;
let turn_angle = 90.0;
let mut start_point = initial_point;
let mut line_builder = MeshBuilder::new();
for char in instructions.chars() {
let end_point = next_point(start_point, heading, line_length);
match char {
'f' | 'h' => {
line_builder.line(&[start_point, end_point], line_width, colour)?;
start_point = end_point;
}
'+' => heading += turn_angle,
'-' => heading -= turn_angle,
_ => panic!("Unknown char {}", char),
}
}
let lines = line_builder.build(ctx)?;
draw(ctx, &lines, (initial_point,))?;
Ok(())
}
fn main() -> GameResult {
let cb = ggez::ContextBuilder::new("dragon curve", "huw")
.window_setup(WindowSetup::default().title("Dragon curve"))
.window_mode(WindowMode::default().dimensions(WINDOW_WIDTH, WINDOW_HEIGHT));
let (ctx, event_loop) = &mut cb.build()?;
let state = &mut MainState::new()?;
event::run(ctx, event_loop, state)
} </lang>
Scala
<lang scala>import javax.swing.JFrame import java.awt.Graphics
class DragonCurve(depth: Int) extends JFrame(s"Dragon Curve (depth $depth)") {
setBounds(100, 100, 800, 600);
setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
val len = 400 / Math.pow(2, depth / 2.0);
val startingAngle = -depth * (Math.PI / 4);
val steps = getSteps(depth).filterNot(c => c == 'X' || c == 'Y')
def getSteps(depth: Int): Stream[Char] = {
if (depth == 0) {
"FX".toStream
} else {
getSteps(depth - 1).flatMap{
case 'X' => "XRYFR"
case 'Y' => "LFXLY"
case c => c.toString
}
}
}
override def paint(g: Graphics): Unit = {
var (x, y) = (230, 350)
var (dx, dy) = ((Math.cos(startingAngle) * len).toInt, (Math.sin(startingAngle) * len).toInt)
for (c <- steps) c match {
case 'F' => {
g.drawLine(x, y, x + dx, y + dy)
x = x + dx
y = y + dy
}
case 'L' => {
val temp = dx
dx = dy
dy = -temp
}
case 'R' => {
val temp = dx
dx = -dy
dy = temp
}
}
}
}
object DragonCurve extends App {
new DragonCurve(14).setVisible(true);
}</lang>
Scilab
It uses complex numbers and treats them as vectors to perform rotations of the edges of the curve around one of its ends. The output is a shown in a graphic window. <lang>n_folds=10
folds=[]; folds=[0 1];
old_folds=[]; vectors=[];
i=[];
for i=2:n_folds+1
curve_length=length(folds);
vectors=folds(1:curve_length-1)-folds(curve_length);
vectors=vectors.*exp(90/180*%i*%pi);
new_folds=folds(curve_length)+vectors;
j=curve_length;
while j>1
folds=[folds new_folds(j-1)]
j=j-1;
end
end
scf(0); clf(); xname("Dragon curve: "+string(n_folds)+" folds")
plot2d(real(folds),imag(folds),5);
set(gca(),"isoview","on"); set(gca(),"axes_visible",["off","off","off"]);</lang>
Seed7
<lang seed7>$ include "seed7_05.s7i";
include "float.s7i"; include "math.s7i"; include "draw.s7i"; include "keybd.s7i";
var float: angle is 0.0; var integer: x is 220; var integer: y is 220;
const proc: turn (in integer: degrees) is func
begin angle +:= flt(degrees) * PI / 180.0 end func;
const proc: forward (in float: length) is func
local var integer: x2 is 0; var integer: y2 is 0; begin x2 := x + trunc(cos(angle) * length); y2 := y + trunc(sin(angle) * length); lineTo(x, y, x2, y2, black); x := x2; y := y2; end func;
const proc: dragon (in float: length, in integer: split, in integer: direct) is func
begin
if split = 0 then
forward(length);
else
turn(direct * 45);
dragon(length/1.4142136, pred(split), 1);
turn(-direct * 90);
dragon(length/1.4142136, pred(split), -1);
turn(direct * 45);
end if;
end func;
const proc: main is func
begin screen(976, 654); clear(curr_win, white); KEYBOARD := GRAPH_KEYBOARD; dragon(768.0, 14, 1); ignore(getc(KEYBOARD)); end func;</lang>
Original source: [3]
SequenceL
Tail-Recursive SequenceL Code:
<lang sequencel>import <Utilities/Math.sl>;
import <Utilities/Conversion.sl>;
initPoints := [[0,0],[1,0]];
f1(point(1)) := let matrix := [[cos(45 * (pi/180)), -sin(45 * (pi/180))], [sin(45 * (pi/180)), cos(45 * (pi/180))]]; in head(transpose((1/sqrt(2)) * matmul(matrix, transpose([point]))));
f2(point(1)) := let matrix := [[cos(135 * (pi/180)), -sin(135 * (pi/180))], [sin(135 * (pi/180)), cos(135 * (pi/180))]]; in head(transpose((1/sqrt(2)) * matmul(matrix, transpose([point])))) + initPoints[2];
matmul(X(2),Y(2))[i,j] := sum(X[i,all]*Y[all,j]);
entry(steps(0), maxX(0), maxY(0)) := let scaleX := maxX / 1.5; scaleY := maxY;
shiftX := maxX / 3.0 / 1.5; shiftY := maxY / 3.0; in round(run(steps, initPoints) * [scaleX, scaleY] + [shiftX, shiftY]);
run(steps(0), result(2)) := let next := f1(result) ++ f2(result); in result when steps <= 0 else run(steps - 1, next);</lang>
C++ Driver Code:
<lang c>#include <iostream>
- include <vector>
- include "SL_Generated.h"
- include "Cimg.h"
using namespace cimg_library; using namespace std;
int main(int argc, char** argv) { int threads = 0; if(argc > 1) threads = atoi(argv[1]); Sequence< Sequence<int> > result;
sl_init(threads);
int width = 500; if(argc > 2) width = atoi(argv[2]); int height = width; if(argc > 3) height = atoi(argv[3]);
CImg<unsigned char> visu(width, height, 1, 3, 0); CImgDisplay draw_disp(visu);
SLTimer compTimer; SLTimer drawTimer;
int steps = 0; int maxSteps = 18; if(argc > 4) maxSteps = atoi(argv[4]); int waitTime = 200; if(argc > 5) waitTime = atoi(argv[5]); bool adding = true; while(!draw_disp.is_closed()) { compTimer.start(); sl_entry(steps, width, height, threads, result); compTimer.stop();
drawTimer.start(); visu.fill(0);
double thirdSize = ((result.size() / 2.0) / 3.0); thirdSize = (int)thirdSize == 0 ? 1 : thirdSize;
for(int i = 1; i <= result.size(); i+=2) { unsigned char shade = (unsigned char)(255 * ((((i / 2) % (int)thirdSize) / thirdSize)) + 0.5);
unsigned char r = i / 2 <= thirdSize ? shade : 255/2; unsigned char g = thirdSize < i / 2 && i / 2 <= thirdSize * 2 ? shade : 255/2; unsigned char b = thirdSize * 2 < i / 2 && i / 2 <= thirdSize * 3 ? shade : 255/2; const unsigned char color[] = {r,g,b};
visu.draw_line(result[i][1], result[i][2], 0, result[i + 1][1], result[i + 1][2], 0, color); } visu.display(draw_disp); drawTimer.stop();
draw_disp.set_title("Dragon Curve in SequenceL: %d Threads | Steps: %d | CompTime: %f Seconds | Draw Time: %f Seconds", threads, steps, drawTimer.getTime(), compTimer.getTime());
if(adding) steps++; else steps--;
if(steps <= 0) adding = true; else if(steps >= maxSteps) adding = false;
draw_disp.wait(waitTime); }
sl_done(); return 0; }</lang>
- Output:
Sidef
Uses the LSystem() class from Hilbert curve. <lang ruby>var rules = Hash(
x => 'x+yF+', y => '-Fx-y',
)
var lsys = LSystem(
width: 600, height: 600,
xoff: -430, yoff: -380,
len: 8, angle: 90, color: 'dark green',
)
lsys.execute('Fx', 11, "dragon_curve.png", rules)</lang> Output image: Dragon curve
Smalltalk
The classic book "Smalltalk-80 The Language and its Implementation" chapter 19 pages 372-3 includes a few lines for drawing the dragon curve (and the Hilbert curve too).
SPL
Animation of dragon curve. <lang spl>levels = 16 level = 0 step = 1 >
draw(level) level += step ? level>levels step = -1 level += step*2 . ? level=0, step = 1 #.delay(1)
<
draw(level)=
mx,my = #.scrsize()
fs = #.min(mx,my)/2
r = fs/2^((level-1)/2)
x = mx/2+fs*#.sqrt(2)/2
y = my/2+fs/4
a = #.pi/4*(level-2)
#.scroff()
#.scrclear()
#.drawline(x,y,x,y)
ss = 2^level-1
> i, 0..ss
? #.and(#.and(i,-i)*2,i)
a += #.pi/2
!
a -= #.pi/2
.
x += r*#.cos(a)
y += r*#.sin(a)
#.drawcolor(#.hsv2rgb(i/(ss+1)*360,1,1):3)
#.drawline(x,y)
<
#.scr()
.</lang>
SVG
SVG does not support recursion, but it does support transformations and multiple uses of the same graphic, so the fractal can be expressed linearly in the iteration count of the fractal.
This version also places circles at the endpoints of each subdivision, size varying with the scale of the fractal, so you can see the shape of each step somewhat.
Note: Some SVG implementations, particularly rsvg (as of v2.26.0), do not correctly interpret XML namespaces; in this case, replace the “l” namespace prefix with “xlink”.
<lang xml><?xml version="1.0" standalone="yes"?> <!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 20010904//EN"
"http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd">
<svg xmlns="http://www.w3.org/2000/svg"
xmlns:l="http://www.w3.org/1999/xlink" width="400" height="400"> <style type="text/css"><![CDATA[ line { stroke: black; stroke-width: .05; } circle { fill: black; } ]]></style>
<defs>
<g id="marks">
<circle cx="0" cy="0" r=".03"/>
<circle cx="1" cy="0" r=".03"/>
</g>
<g id="l0">
<line x1="0" y1="0" x2="1" y2="0"/>
</g>
<g id="l1"> <use l:href="#l0" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l0" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
<g id="l2"> <use l:href="#l1" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l1" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
<g id="l3"> <use l:href="#l2" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l2" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
<g id="l4"> <use l:href="#l3" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l3" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
<g id="l5"> <use l:href="#l4" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l4" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
<g id="l6"> <use l:href="#l5" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l5" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
<g id="l7"> <use l:href="#l6" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l6" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
<g id="l8"> <use l:href="#l7" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l7" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
<g id="l9"> <use l:href="#l8" transform="matrix( .5 .5 -.5 .5 0 0)"/>
<use l:href="#l8" transform="matrix(-.5 .5 -.5 -.5 1 0)"/>
<use l:href="#marks"/></g>
</defs>
<g transform="translate(100, 200) scale(200)">
<use l:href="#marks"/> <use l:href="#l9"/>
</g>
</svg></lang>
Tcl
<lang tcl>package require Tk
set pi [expr acos(-1)] set r2 [expr sqrt(2)]
proc turn {degrees} {
global a pi
set a [expr {$a + $degrees*$pi/180}]
} proc forward {len} {
global a coords
lassign [lrange $coords end-1 end] x y
lappend coords [expr {$x + cos($a)*$len}] [expr {$y + sin($a)*$len}]
} proc dragon {len split {d 1}} {
global r2 coords
if {$split == 0} {
forward $len return
}
# This next part is only necessary to allow the illustration of progress
if {$split == 10 && [llength $::coords]>2} {
.c coords dragon $::coords update
}
incr split -1
set sublen [expr {$len/$r2}]
turn [expr {$d*45}]
dragon $sublen $split 1
turn [expr {$d*-90}]
dragon $sublen $split -1
turn [expr {$d*45}]
}
set coords {150 180} set a 0.0 pack [canvas .c -width 700 -height 500] .c create line {0 0 0 0} -tag dragon dragon 400 17 .c coords dragon $coords</lang>
See also the Tcl/Tk wiki Dragon Curves and Recursive curves pages.
Plain TeX
PGF
The PGF package includes a "lindenmayersystems" library.
A dragon can be made with the "F-S" rule by defining S as a second drawing symbol.
So, for plainTeX,
<lang TeX>\input tikz.tex \usetikzlibrary{lindenmayersystems}
\pgfdeclarelindenmayersystem{Dragon curve}{
\symbol{S}{\pgflsystemdrawforward}
\rule{F -> F+S}
\rule{S -> F-S}
} \tikzpicture \draw
[lindenmayer system={Dragon curve, step=10pt, axiom=F, order=8}]
lindenmayer system;
\endtikzpicture \bye</lang>
LaTeX
Or fixed-direction variant to stay horizontal, this time for LaTeX,
<lang LaTeX>\documentclass{minimal} \usepackage{tikz} \usetikzlibrary{lindenmayersystems} \pgfdeclarelindenmayersystem{Dragon curve}{
\symbol{S}{\pgflsystemdrawforward}
\rule{F -> -F++S-}
\rule{S -> +F--S+}
} \foreach \i in {1,...,8} {
\hbox{
order=\i
\hspace{.5em}
\begin{tikzpicture}[baseline=0pt]
\draw
[lindenmayer system={Dragon curve, step=10pt,angle=45, axiom=F, order=\i}]
lindenmayer system;
\end{tikzpicture}
\hspace{1em}
}
\vspace{.5ex}
} \begin{document} \end{document}</lang>
TI-89 BASIC
<lang ti89b>Define dragon = (iter, xform) Prgm
Local a,b If iter > 0 Then dragon(iter-1, xform*[[.5,.5,0][–.5,.5,0][0,0,1]]) dragon(iter-1, xform*[[–.5,.5,0][–.5,–.5,1][0,0,1]]) Else xform*[0;0;1]→a xform*[0;1;1]→b PxlLine floor(a[1,1]), floor(a[2,1]), floor(b[1,1]), floor(b[2,1]) EndIf
EndPrgm
FnOff PlotsOff ClrDraw dragon(7, [[75,0,26] [0,75,47] [0,0,1]])</lang>
Valid coordinates on the TI-89's graph screen are x 0..76 and y 0..158. This and the outer size of the dragon curve were used to choose the position and scale determined by the transformation matrix initially passed to dragon such that the curve will fit onscreen no matter the number of recursions chosen. The height of the curve is 1 unit, so the vertical (and horizontal, to preserve proportions) scale is the height of the screen (rather, one less, to avoid rounding/FP error overrunning), or 75. The curve extends 1/3 unit above its origin, so the vertical translation is (one more than) 1/3 of the scale, or 26. The curve extends 1/3 to the left of its origin, or 25 pixels; the width of the curve is 1.5 units, or 1.5·76 = 114 pixels, and the screen is 159 pixels, so to center it we place the origin at 25 + (159-114)/2 = 47 pixels.
Vedit macro language
Vedit is a text editor, so obviously there is no graphics support in the macro language. However, since Vedit can edit any file, including graphics files, it is possible to do some graphics.
This implementation first creates a blank BMP file in an edit buffer, then plots the fractal in that file, and finally calls the application associated to BMP files to display the results.
The DRAGON routine combines two steps of the algorithm used in other implementations. As a result, each turn is 90 degrees and thus all lines are vertical or horizontal (or alternatively diagonal). In addition, the length is divided by 2 instead of square root of 2 on each step. This way we can avoid using any floating point calculations, trigonometric functions etc.
<lang vedit>File_Open("|(USER_MACRO)\dragon.bmp", OVERWRITE+NOEVENT) BOF Del_Char(ALL)
- 11 = 640 // width of the image
- 12 = 480 // height of the image
Call("CREATE_BMP")
- 1 = 384 // dx
- 2 = 0 // dy
- 3 = 6 // depth of recursion
- 4 = 1 // flip
- 5 = 150 // x
- 6 = 300 // y
Call("DRAGON") Buf_Close(NOMSG)
Sys(`start "" "|(USER_MACRO)\dragon.bmp"`, DOS+SUPPRESS+SIMPLE+NOWAIT) return
///////////////////////////////////////////////////////////////////// // // Dragon fractal, recursive //
- DRAGON:
if (#3 == 0) {
Call("DRAW_LINE")
} else {
#1 /= 2
#2 /= 2
#3--
if (#4) {
Num_Push(1,4) #4=1; #7=#1; #1=#2; #2=-#7; Call("DRAGON") Num_Pop(1,4) Num_Push(1,4) #4=0; Call("DRAGON") Num_Pop(1,4) Num_Push(1,4) #4=1; #7=#1; #1=-#2; #2=#7; Call("DRAGON") Num_Pop(1,4) Num_Push(1,4) #4=0; Call("DRAGON") Num_Pop(1,4)
} else {
Num_Push(1,4) #4=1; Call("DRAGON") Num_Pop(1,4) Num_Push(1,4) #4=0; #7=#1; #1=-#2; #2=#7; Call("DRAGON") Num_Pop(1,4) Num_Push(1,4) #4=1; Call("DRAGON") Num_Pop(1,4) Num_Push(1,4) #4=0; #7=#1; #1=#2; #2=-#7; Call("DRAGON") Num_Pop(1,4)
}
} return
///////////////////////////////////////////////////////////////////// // // Daw a horizontal, vertical or diagonal line. #1 = dx, #2 = dy //
- DRAW_LINE:
while (#1 || #2 ) {
#21 = (#1>0) - (#1<0) #22 = (#2>0) - (#2<0) #5 += #21; #1 -= #21 #6 += #22; #2 -= #22 Goto_Pos(1078 + #5 + #6*#11) IC(255, OVERWRITE) // plot a pixel
} return
///////////////////////////////////////////////////////////////////// // // Create a bitmap file //
- CREATE_BMP:
// BITMAPFILEHEADER: IT("BM") // bfType
- 10 = 1078+#11*#12 // file size
Call("INS_4BYTES") IC(0, COUNT, 4) // reserved
- 10 = 1078; Call("INS_4BYTES") // offset to bitmap data
// BITMAPINFOHEADER:
- 10 = 40; Call("INS_4BYTES") // size of BITMAPINFOHEADER
- 10 = #11; Call("INS_4BYTES") // width of image
- 10 = #12; Call("INS_4BYTES") // height of image
IC(1) IC(0) // number of bitplanes = 1 IC(8) IC(0) // bits/pixel = 8 IC(0, COUNT, 24) // compression, number of colors etc.
// Color table - create greyscale palette for (#1 = 0; #1 < 256; #1++) {
IC(#1) IC(#1) IC(#1) IC(0)
}
// Pixel data - init to black for (#1 = 0; #1 < #12; #1++) {
IC(0, COUNT, #11)
} return
// // Write 32 bit binary value from #10 in the file //
- INS_4BYTES:
for (#1 = 0; #1 < 4; #1++) {
Ins_Char(#10 & 0xff) #10 = #10 >> 8
} return</lang>
Visual Basic
<lang vb>Option Explicit Const Pi As Double = 3.14159265358979 Dim angle As Double Dim nDepth As Integer Dim nColor As Long
Private Sub Form_Load()
nColor = vbBlack nDepth = 12 DragonCurve
End Sub
Sub DragonProc(size As Double, ByVal split As Integer, d As Integer)
If split = 0 Then
xForm.Line -Step(-Cos(angle) * size, Sin(angle) * size), nColor
Else
angle = angle + d * Pi / 4
Call DragonProc(size / Sqr(2), split - 1, 1)
angle = angle - d * Pi / 2
Call DragonProc(size / Sqr(2), split - 1, -1)
angle = angle + d * Pi / 4
End If
End Sub
Sub DragonCurve()
Const xcoefi = 0.74 Const xcoefl = 0.59 xForm.PSet (xForm.Width * xcoefi, xForm.Height / 3), nColor Call DragonProc(xForm.Width * xcoefl, nDepth, 1)
End Sub</lang>
Visual Basic .NET
<lang vbnet>Option Explicit On Imports System.Math
Public Class DragonCurve
Dim nDepth As Integer = 12 Dim angle As Double Dim MouseX, MouseY As Integer Dim CurrentX, CurrentY As Integer Dim nColor As Color = Color.Black
Private Sub DragonCurve_Click(sender As Object, e As EventArgs) Handles Me.Click
SubDragonCurve()
End Sub
Sub DrawClear()
Me.CreateGraphics.Clear(Color.White)
End Sub
Sub DrawMove(ByVal X As Double, ByVal Y As Double)
CurrentX = X
CurrentY = Y
End Sub
Sub DrawLine(ByVal X As Double, ByVal Y As Double)
Dim MyGraph As Graphics = Me.CreateGraphics
Dim PenColor As Pen = New Pen(nColor)
Dim NextX, NextY As Long
NextX = CurrentX + X
NextY = CurrentY + Y
MyGraph.DrawLine(PenColor, CurrentX, CurrentY, NextX, NextY)
CurrentX = NextX
CurrentY = NextY
End Sub
Sub DragonProc(size As Double, ByVal split As Integer, d As Integer)
If split = 0 Then
DrawLine(-Cos(angle) * size, Sin(angle) * size)
Else
angle = angle + d * PI / 4
DragonProc(size / Sqrt(2), split - 1, 1)
angle = angle - d * PI / 2
DragonProc(size / Sqrt(2), split - 1, -1)
angle = angle + d * PI / 4
End If
End Sub
Sub SubDragonCurve()
Const xcoefi = 0.74, xcoefl = 0.59
DrawClear()
DrawMove(Me.Width * xcoefi, Me.Height / 3)
DragonProc(Me.Width * xcoefl, nDepth, 1)
End Sub
End Class</lang>
VBScript
VBScript does'nt have direct access to OS graphics, so I write SVG commands to an HTML file then I display it with the default browser. A turtle graphics class makes the definition of the curve very simple. <lang vb>
option explicit 'outputs turtle graphics to svg file and opens it
const pi180= 0.01745329251994329576923690768489 ' pi/180 const pi=3.1415926535897932384626433832795 'pi class turtle
dim fso dim fn dim svg dim iang 'radians dim ori 'radians dim incr dim pdown dim clr dim x dim y
public property let orient(n):ori = n*pi180 :end property
public property let iangle(n):iang= n*pi180 :end property
public sub pd() : pdown=true: end sub
public sub pu() :pdown=FALSE :end sub
public sub rt(i)
ori=ori - i*iang:
'if ori<0 then ori = ori+pi*2
end sub
public sub lt(i):
ori=(ori + i*iang)
'if ori>(pi*2) then ori=ori-pi*2
end sub
public sub bw(l)
x= x+ cos(ori+pi)*l*incr
y= y+ sin(ori+pi)*l*incr
' ori=ori+pi '?????
end sub
public sub fw(l)
dim x1,y1
x1=x + cos(ori)*l*incr
y1=y + sin(ori)*l*incr
if pdown then line x,y,x1,y1
x=x1:y=y1
end sub
Private Sub Class_Initialize()
initsvg
x=400:y=400:incr=100
ori=90*pi180
iang=90*pi180
clr=0
pdown=true
end sub
Private Sub Class_Terminate()
disply
end sub
private sub line (x,y,x1,y1)
svg.WriteLine "<line x1=""" & x & """ y1= """& y & """ x2=""" & x1& """ y2=""" & y1 & """/>"
end sub
private sub disply()
dim shell
svg.WriteLine "</svg></body></html>"
svg.close
Set shell = CreateObject("Shell.Application")
shell.ShellExecute fn,1,False
end sub
private sub initsvg()
dim scriptpath
Set fso = CreateObject ("Scripting.Filesystemobject")
ScriptPath= Left(WScript.ScriptFullName, InStrRev(WScript.ScriptFullName, "\"))
fn=Scriptpath & "SIERP.HTML"
Set svg = fso.CreateTextFile(fn,True)
if SVG IS nothing then wscript.echo "Can't create svg file" :vscript.quit
svg.WriteLine "<!DOCTYPE html>" &vbcrlf & "<html>" &vbcrlf & "<head>"
svg.writeline "<style>" & vbcrlf & "line {stroke:rgb(255,0,0);stroke-width:.5}" &vbcrlf &"</style>"
svg.writeline "</head>"&vbcrlf & "<body>"
svg.WriteLine "<svg xmlns=""http://www.w3.org/2000/svg"" width=""800"" height=""800"" viewBox=""0 0 800 800"">"
end sub
end class
sub dragon(st,le,dir) if st=0 then x.fw le: exit sub
x.rt dir dragon st-1, le/1.41421 ,1 x.rt dir*2 dragon st-1, le/1.41421 ,-1 x.rt dir
end sub
dim x
set x=new turtle
x.iangle=45
x.orient=45
x.incr=1
x.x=200:x.y=200
dragon 12,300,1
set x=nothing 'this displays the image
</lang>
Wren
<lang ecmascript>import "graphics" for Canvas, Color import "dome" for Window
class Game {
static init() {
Window.title = "Dragon curve"
Window.resize(800, 600)
Canvas.resize(800, 600)
var iter = 14
var turns = getSequence(iter)
var startingAngle = -iter * Num.pi / 4
var side = 400 / 2.pow(iter/2)
dragon(turns, startingAngle, side)
}
static getSequence(iterations) {
var turnSequence = []
for (i in 0...iterations) {
var copy = []
copy.addAll(turnSequence)
if (copy.count > 1) copy = copy[-1..0]
turnSequence.add(1)
copy.each { |i| turnSequence.add(-i) }
}
return turnSequence
}
static dragon(turns, startingAngle, side) {
var col = Color.blue
var angle = startingAngle
var x1 = 230
var y1 = 350
var x2 = x1 + (angle.cos * side).truncate
var y2 = y1 + (angle.sin * side).truncate
Canvas.line(x1, y1, x2, y2, col)
x1 = x2
y1 = y2
for (turn in turns) {
angle = angle + turn*Num.pi/2
x2 = x1 + (angle.cos * side).truncate
y2 = y1 + (angle.sin * side).truncate
Canvas.line(x1, y1, x2, y2, col)
x1 = x2
y1 = y2
}
}
static update() {}
static draw(alpha) {}
}</lang>
X86 Assembly
Translation of XPL0. Assemble with tasm, tlink /t <lang asm> .model tiny
.code
.486
org 100h ;assume ax=0, bx=0, sp=-2
start: mov al, 13h ;(ah=0) set 320x200 video graphics mode
int 10h
push 0A000h
pop es
mov si, 8000h ;color
mov cx, 75*256+100 ;coordinates of initial horizontal line segment
mov dx, 75*256+164 ;use power of 2 for length
call dragon
mov ah, 0 ;wait for keystroke
int 16h
mov ax, 0003h ;restore normal text mode
int 10h
ret
dragon: cmp sp, -100 ;at maximum recursion depth?
jne drag30 ;skip if not
mov bl, dh ;draw at max depth to get solid image
imul di, bx, 320 ;(bh=0) plot point at X=dl, Y=dh
mov bl, dl
add di, bx
mov ax, si ;color
shr ax, 13
or al, 8 ;use bright colors 8..F
stosb ;es:[di++]:= al
inc si
ret
drag30:
push cx ;preserve points P and Q
push dx
xchg ax, dx ;DX:= Q(0)-P(0);
sub al, cl
sub ah, ch ;DY:= Q(1)-P(1);
mov dx, ax ;new point
sub dl, ah ;R(0):= P(0) + (DX-DY)/2
jns drag40
inc dl
drag40: sar dl, 1 ;dl:= (al-ah)/2 + cl
add dl, cl
add dh, al ;R(1):= P(1) + (DX+DY)/2;
jns drag45
inc dh
drag45: sar dh, 1 ;dh:= (al+ah)/2 + ch
add dh, ch
call dragon ;Dragon(P, R);
pop cx ;get Q
push cx
call dragon ;Dragon(Q, R);
pop dx ;restore points
pop cx
ret
end start</lang>
Xfractint
The xfractint program includes two dragon curves in its lsystem/fractint.l. Here is another version. Xfractint has only a single "F" drawing symbol, so empty symbols X and Y are used for even and odd positions to control the expansion. Each X and each Y is always followed by a single F each.
<lang Xfractint>Dragon3 {
Angle 4 Axiom XF X=XF+Y Y=XF-Y }</lang>
Put this in a file dragon3.l and run as follows. params=8 means an 8-order curve.
<lang sh>xfractint type=lsystem lfile=dragon3.l lname=Dragon3 params=8</lang>
XPL0
<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations
proc Dragon(D, P, Q); \Draw a colorful dragon curve int D, P, Q; \recursive depth, coordinates of line segment int R(2), \coordinates of generated new point
DX, DY, C; \deltas, color
[C:= [0]; \color is a local, static-like variable D:= D+1; \depth of recursion increases if D >= 13 then \draw lines at maximum depth to get solid image
[Move(P(0), P(1)); Line(Q(0), Q(1), C(0)>>9+4!8); C(0):= C(0)+1; return];
DX:= Q(0)-P(0); DY:= Q(1)-P(1); R(0):= P(0) + (DX-DY)/2; \new point R(1):= P(1) + (DX+DY)/2; Dragon(D, P, R); \draw two segments that include the new point Dragon(D, Q, R); ];
int X, Y, P(2), Q(2); [SetVid($101); \set 640x480 video graphics mode X:= 32; Y:= 32; \coordinates of initial horizontal line segment P(0):= X; P(1):= Y; Q(0):= X+64; Q(1):= Y; \(power of two length works best for integers) Dragon(0, P, Q); \draw its dragon curve X:= ChIn(1); \wait for keystroke SetVid(3); \restore normal text mode ]</lang>
Yabasic
<lang Yabasic>w = 390 : h = int(w * 11 / 16) open window w, h level = 18 : insize = 247 x = 92 : y = 94
iters = 2^level qiter = 510/iters SQ = sqrt(2) : QPI = pi/4
rotation = 0 : iter = 0 : rq = 1.0 dim rqs(level)
color 0,0,0 clear window dragon()
sub dragon() if level<=0 then yn = sin(rotation)*insize + y xn = cos(rotation)*insize + x if iter*2<iters then color 0,iter*qiter,255-iter*qiter else color qiter*iter-255,(iters-iter)*qiter,0 end if line x,y,xn,yn iter = iter + 1 x = xn : y = yn return end if insize = insize/SQ rotation = rotation + rq*QPI level = level - 1 rqs(level) = rq : rq = 1 dragon() rotation = rotation - rqs(level)*QPI*2 rq = -1 dragon() rq = rqs(level) rotation = rotation + rq*QPI level = level + 1 insize = insize*SQ return end sub</lang>
zkl
Draw the curve in SVG to stdout.
<lang zkl>println(0'|<?xml version='1.0' encoding='utf-8' standalone='no'?>|"\n"
0'|<!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.1//EN'|"\n" 0'|'http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd'>|"\n" 0'|<svg width='100%' height='100%' version='1.1'|"\n" 0'|xmlns='http://www.w3.org/2000/svg'>|);
order:=13.0; # akin to number of recursion steps d_size:=1000.0; # size in pixels pi:=(1.0).pi; turn_angle:=pi/2; # turn angle of each segment, 90 degrees for the canonical dragon
angle:=pi - (order * (pi/4)); # starting angle len:=(d_size/1.5) / (2.0).sqrt().pow(order); # size of each segment x:=d_size*5/6; y:=d_size*1/3; # starting point
foreach i in ([0 .. (2.0).pow(order-1)]){
# find which side to turn based on the iteration angle += i.bitAnd(-i).shiftLeft(1).bitAnd(i) and -turn_angle or turn_angle;
dx:=x + len * angle.sin(); dy:=y - len * angle.cos();
println("<line x1='",x,"' y1='",y,"' x2='",dx,"' y2='",dy,
"' style='stroke:rgb(0,0,0);stroke-width:1'/>");
x=dx; y=dy;
} println("</svg>");</lang>
- Output:
$zkl bbb > dragon.svg $ls -l dragon.svg ... 408780 May 18 00:29 dragon.svg $less dragon.svg <?xml version='1.0' encoding='utf-8' standalone='no'?> <!DOCTYPE svg PUBLIC '-//W3C//DTD SVG 1.1//EN' 'http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd'> <svg width='100%' height='100%' version='1.1' xmlns='http://www.w3.org/2000/svg'> <line x1='833.333' y1='333.333' x2='838.542' y2='328.125' style='stroke:rgb(0,0,0);stroke-width:1'/> .... Visiting file:///home/craigd/Projects/ZKL/Tmp/dragon.svg shows a nice dragon curve
http://home.comcast.net/~zenkinetic/Images/dragon.svg
ZX Spectrum Basic
<lang zxbasic>10 LET level=15: LET insize=120 20 LET x=80: LET y=70 30 LET iters=2^level 40 LET qiter=256/iters 50 LET sq=SQR (2): LET qpi=PI/4 60 LET rotation=0: LET iter=0: LET rq=1 70 DIM r(level) 75 GO SUB 80: STOP 80 REM Dragon 90 IF level>1 THEN GO TO 200 100 LET yn=SIN (rotation)*insize+y 110 LET xn=COS (rotation)*insize+x 120 PLOT x,y: DRAW xn-x,yn-y 130 LET iter=iter+1 140 LET x=xn: LET y=yn 150 RETURN 200 LET insize=insize/sq 210 LET rotation=rotation+rq*qpi 220 LET level=level-1 230 LET r(level)=rq: LET rq=1 240 GO SUB 80 250 LET rotation=rotation-r(level)*qpi*2 260 LET rq=-1 270 GO SUB 80 280 LET rq=r(level) 290 LET rotation=rotation+rq*qpi 300 LET level=level+1 310 LET insize=insize*sq 320 RETURN </lang>