Peano curve
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Produce a graphical or ASCII-art representation of a Peano curve of at least order 3.
Action!
Action! language does not support recursion. Therefore an iterative approach with a stack has been proposed.
DEFINE MAXSIZE="12"
INT ARRAY
angleStack(MAXSIZE)
BYTE ARRAY
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 angle BYTE depth,stage)
IF IsFull() THEN Break() FI
angleStack(stacksize)=angle
depthStack(stacksize)=depth
stageStack(stackSize)=stage
stacksize==+1
RETURN
PROC Pop(INT POINTER angle BYTE POINTER depth,stage)
IF IsEmpty() THEN Break() FI
stacksize==-1
angle^=angleStack(stacksize)
depth^=depthStack(stacksize)
stage^=stageStack(stacksize)
RETURN
INT FUNC Sin(INT a)
WHILE a<0 DO a==+360 OD
WHILE a>360 DO a==-360 OD
IF a=90 THEN
RETURN (1)
ELSEIF a=270 THEN
RETURN (-1)
FI
RETURN (0)
INT FUNC Cos(INT a)
RETURN (Sin(a-90))
PROC DrawPeano(INT x BYTE y,len BYTE depth)
BYTE stage
INT angle=[90],a
Plot(x,y)
Push(90,depth,0)
WHILE IsEmpty()=0
DO
Pop(@a,@depth,@stage)
IF stage<3 THEN
Push(a,depth,stage+1)
FI
IF stage=0 THEN
angle==+a
IF depth>1 THEN
Push(-a,depth-1,0)
FI
ELSEIF stage=1 THEN
x==+len*Cos(angle)
y==-len*Sin(angle)
DrawTo(x,y)
IF depth>1 THEN
Push(a,depth-1,0)
FI
ELSEIF stage=2 THEN
x==+len*Cos(angle)
y==-len*Sin(angle)
DrawTo(x,y)
IF depth>1 THEN
Push(-a,depth-1,0)
FI
ELSEIF stage=3 THEN
angle==-a
FI
OD
RETURN
PROC Main()
BYTE CH=$02FC,COLOR1=$02C5,COLOR2=$02C6
Graphics(8+16)
Color=1
COLOR1=$0C
COLOR2=$02
DrawPeano(69,186,7,6)
DO UNTIL CH#$FF OD
CH=$FF
RETURN- Output:
Screenshot from Atari 8-bit computer
Ada
with PDF_Out; use PDF_Out;
procedure Peano_Curve is
Filename : constant String := "peano-curve.pdf";
Order : constant Positive := 4;
Scale : constant Real := 2.1;
Line_Width : constant Real := 2.5;
Corner : constant Point := (150.0, 50.0);
Background : constant Color_Type := (0.827, 0.816, 0.016);
Frame : constant Rectangle := (10.0, 10.0, 820.0, 575.0);
PDF : PDF_Out_File;
type Coord is record
X, Y : Natural;
end record;
function "+" (Left : Coord; Right : Coord) return Coord
is ((Left.X + Right.X, Left.Y + Right.Y));
function "*" (Left : Natural; Right : Coord) return Coord
is ((Left * Right.X, Left * Right.Y));
procedure Peano (Pos : Coord; Length : Positive; I1, I2 : Integer) is
Len : constant Integer := Length / 3;
begin
if Length = 1 then
PDF.Line (Corner + Scale * (Real (3 * Pos.X), Real (3 * Pos.Y)));
else
Peano (Pos + Len * (2 * I1, 2 * I1), Len, I1, I2);
Peano (Pos + Len * (I1 - I2 + 1, I1 + I2), Len, I1, 1 - I2);
Peano (Pos + Len * (1, 1), Len, I1, 1 - I2);
Peano (Pos + Len * (I1 + I2, I1 - I2 + 1), Len, 1 - I1, 1 - I2);
Peano (Pos + Len * (2 * I2, 2 - 2 * I2), Len, I1, I2);
Peano (Pos + Len * (1 + I2 - I1, 2 - I1 - I2), Len, I1, I2);
Peano (Pos + Len * (2 - 2 * I1, 2 - 2 * I1), Len, I1, I2);
Peano (Pos + Len * (2 - I1 - I2, 1 + I2 - I1), Len, 1 - I1, I2);
Peano (Pos + Len * (2 - 2 * I2, 2 * I2), Len, 1 - I1, I2);
end if;
end Peano;
procedure Draw_Peano is
begin
PDF.Stroking_Color (Black);
PDF.Line_Width (Line_Width);
PDF.Move (Corner);
Peano ((0, 0), 3**Order, 0, 0);
PDF.Finish_Path (Close_Path => False,
Rendering => Stroke,
Rule => Nonzero_Winding_Number);
end Draw_Peano;
begin
PDF.Create (Filename);
PDF.Page_Setup (A4_Landscape);
PDF.Color (Background);
PDF.Draw (Frame, Fill);
Draw_Peano;
PDF.Close;
end Peano_Curve;
ALGOL 68
Generates an SVG file containing the curve using the L-System. Very similar to the Algol 68 Sierpinski square curve sample. Note the Algol 68 L-System library source code is on a separate page on Rosetta Code - follow the above link and then to the Talk page.
BEGIN # Peano Curve in SVG #
# uses the RC Algol 68 L-System library for the L-System evaluation & #
# interpretation #
PR read "lsystem.incl.a68" PR # include L-System utilities #
PROC peano curve = ( STRING fname, INT size, length, order, init x, init y )VOID:
IF FILE svg file;
BOOL open error := IF open( svg file, fname, stand out channel ) = 0
THEN
# opened OK - file already exists and #
# will be overwritten #
FALSE
ELSE
# failed to open the file #
# - try creating a new file #
establish( svg file, fname, stand out channel ) /= 0
FI;
open error
THEN # failed to open the file #
print( ( "Unable to open ", fname, newline ) );
stop
ELSE # file opened OK #
REAL x := init x;
REAL y := init y;
INT angle := 90;
put( svg file, ( "<svg xmlns='http://www.w3.org/2000/svg' width='"
, whole( size, 0 ), "' height='", whole( size, 0 ), "'>"
, newline, "<rect width='100%' height='100%' fill='white'/>"
, newline, "<path stroke-width='1' stroke='black' fill='none' d='"
, newline, "M", whole( x, 0 ), ",", whole( y, 0 ), newline
)
);
LSYSTEM ssc = ( "L"
, ( "L" -> "LFRFL-F-RFLFR+F+LFRFL"
, "R" -> "RFLFR+F+LFRFL-F-RFLFR"
)
);
STRING curve = ssc EVAL order;
curve INTERPRET ( ( CHAR c )VOID:
IF c = "F" THEN
x +:= length * cos( angle * pi / 180 );
y +:= length * sin( angle * pi / 180 );
put( svg file, ( " L", whole( x, 0 ), ",", whole( y, 0 ), newline ) )
ELIF c = "+" THEN
angle +:= 90 MODAB 360
ELIF c = "-" THEN
angle -:= 90 MODAB 360
FI
);
put( svg file, ( "'/>", newline, "</svg>", newline ) );
close( svg file )
FI # sierpinski square # ;
peano curve( "peano.svg", 1200, 12, 3, 50, 50 )
ENDAutoHotkey
Requires Gdip Library
gdip1()
PeanoX := A_ScreenWidth/2 - 100, PeanoY := A_ScreenHeight/2 - 100
Peano(PeanoX, PeanoY, 3**3, 5, 5, Arr:=[])
xmin := xmax := ymin := ymax := 0
for i, point in Arr
{
xmin := A_Index = 1 ? point.x : xmin < point.x ? xmin : point.x
xmax := point.x > xmax ? point.x : xmax
ymin := A_Index = 1 ? point.y : ymin < point.y ? ymin : point.y
ymax := point.y > ymax ? point.y : ymax
}
for i, point in Arr
points .= point.x - xmin + PeanoX "," point.y - ymin + PeanoY "|"
points := Trim(points, "|")
Gdip_DrawLines(G, pPen, Points)
UpdateLayeredWindow(hwnd1, hdc, 0, 0, Width, Height)
return
; ---------------------------------------------------------------
Peano(x, y, lg, i1, i2, Arr) {
if (lg =1 )
{
Arr[Arr.count()+1, "x"] := x
Arr[Arr.count(), "y"] := y
return
}
lg := lg/3
Peano(x+(2*i1*lg) , y+(2*i1*lg) , lg , i1 , i2 , Arr)
Peano(x+((i1-i2+1)*lg) , y+((i1+i2)*lg) , lg , i1 , 1-i2 , Arr)
Peano(x+lg , y+lg , lg , i1 , 1-i2 , Arr)
Peano(x+((i1+i2)*lg) , y+((i1-i2+1)*lg) , lg , 1-i1 , 1-i2 , Arr)
Peano(x+(2*i2*lg) , y+(2*(1-i2)*lg) , lg , i1 , i2 , Arr)
Peano(x+((1+i2-i1)*lg) , y+((2-i1-i2)*lg) , lg , i1 , i2 , Arr)
Peano(x+(2*(1-i1)*lg) , y+(2*(1-i1)*lg) , lg , i1 , i2 , Arr)
Peano(x+((2-i1-i2)*lg) , y+((1+i2-i1)*lg) , lg , 1-i1 , i2 , Arr)
Peano(x+(2*(1-i2)*lg) , y+(2*i2*lg) , lg , 1-i1 , i2 , Arr)
}
; ---------------------------------------------------------------
gdip1(){
global
If !pToken := Gdip_Startup()
{
MsgBox, 48, gdiplus error!, Gdiplus failed to start. Please ensure you have gdiplus on your system
ExitApp
}
OnExit, Exit
Width := A_ScreenWidth, Height := A_ScreenHeight
Gui, 1: -Caption +E0x80000 +LastFound +OwnDialogs +Owner +AlwaysOnTop
Gui, 1: Show, NA
hwnd1 := WinExist()
hbm := CreateDIBSection(Width, Height)
hdc := CreateCompatibleDC()
obm := SelectObject(hdc, hbm)
G := Gdip_GraphicsFromHDC(hdc)
Gdip_SetSmoothingMode(G, 4)
pPen := Gdip_CreatePen(0xFFFF0000, 2)
}
; ---------------------------------------------------------------
gdip2(){
global
Gdip_DeleteBrush(pBrush)
Gdip_DeletePen(pPen)
SelectObject(hdc, obm)
DeleteObject(hbm)
DeleteDC(hdc)
Gdip_DeleteGraphics(G)
}
; ---------------------------------------------------------------
Exit:
gdip2()
Gdip_Shutdown(pToken)
ExitApp
Return
C
Adaptation of the C program in the Breinholt-Schierz paper , requires the WinBGIm library.
/*Abhishek Ghosh, 14th September 2018*/
#include <graphics.h>
#include <math.h>
void Peano(int x, int y, int lg, int i1, int i2) {
if (lg == 1) {
lineto(3*x,3*y);
return;
}
lg = lg/3;
Peano(x+(2*i1*lg), y+(2*i1*lg), lg, i1, i2);
Peano(x+((i1-i2+1)*lg), y+((i1+i2)*lg), lg, i1, 1-i2);
Peano(x+lg, y+lg, lg, i1, 1-i2);
Peano(x+((i1+i2)*lg), y+((i1-i2+1)*lg), lg, 1-i1, 1-i2);
Peano(x+(2*i2*lg), y+(2*(1-i2)*lg), lg, i1, i2);
Peano(x+((1+i2-i1)*lg), y+((2-i1-i2)*lg), lg, i1, i2);
Peano(x+(2*(1-i1)*lg), y+(2*(1-i1)*lg), lg, i1, i2);
Peano(x+((2-i1-i2)*lg), y+((1+i2-i1)*lg), lg, 1-i1, i2);
Peano(x+(2*(1-i2)*lg), y+(2*i2*lg), lg, 1-i1, i2);
}
int main(void) {
initwindow(1000,1000,"Peano, Peano");
Peano(0, 0, 1000, 0, 0); /* Start Peano recursion. */
getch();
cleardevice();
return 0;
}
C++
#include <cmath>
#include <fstream>
#include <iostream>
#include <string>
class peano_curve {
public:
void write(std::ostream& out, int size, int length, int order);
private:
static std::string rewrite(const std::string& s);
void line(std::ostream& out);
void execute(std::ostream& out, const std::string& s);
double x_;
double y_;
int angle_;
int length_;
};
void peano_curve::write(std::ostream& out, int size, int length, int order) {
length_ = length;
x_ = length;
y_ = length;
angle_ = 90;
out << "<svg xmlns='http://www.w3.org/2000/svg' width='"
<< size << "' height='" << size << "'>\n";
out << "<rect width='100%' height='100%' fill='white'/>\n";
out << "<path stroke-width='1' stroke='black' fill='none' d='";
std::string s = "L";
for (int i = 0; i < order; ++i)
s = rewrite(s);
execute(out, s);
out << "'/>\n</svg>\n";
}
std::string peano_curve::rewrite(const std::string& s) {
std::string t;
for (char c : s) {
switch (c) {
case 'L':
t += "LFRFL-F-RFLFR+F+LFRFL";
break;
case 'R':
t += "RFLFR+F+LFRFL-F-RFLFR";
break;
default:
t += c;
break;
}
}
return t;
}
void peano_curve::line(std::ostream& out) {
double theta = (3.14159265359 * angle_)/180.0;
x_ += length_ * std::cos(theta);
y_ += length_ * std::sin(theta);
out << " L" << x_ << ',' << y_;
}
void peano_curve::execute(std::ostream& out, const std::string& s) {
out << 'M' << x_ << ',' << y_;
for (char c : s) {
switch (c) {
case 'F':
line(out);
break;
case '+':
angle_ = (angle_ + 90) % 360;
break;
case '-':
angle_ = (angle_ - 90) % 360;
break;
}
}
}
int main() {
std::ofstream out("peano_curve.svg");
if (!out) {
std::cerr << "Cannot open output file\n";
return 1;
}
peano_curve pc;
pc.write(out, 656, 8, 4);
return 0;
}
- Output:
EasyLang
proc lsysexp level . axiom$ rules$[] .
for l to level
an$ = ""
for c$ in strchars axiom$
for i = 1 step 2 to len rules$[]
if rules$[i] = c$
c$ = rules$[i + 1]
break 1
.
.
an$ &= c$
.
swap axiom$ an$
.
.
proc lsysdraw axiom$ x y ang . .
linewidth 0.3
move x y
for c$ in strchars axiom$
if c$ = "F"
x += cos dir
y += sin dir
line x y
elif c$ = "-"
dir -= ang
elif c$ = "+"
dir += ang
.
.
.
axiom$ = "L"
rules$[] = [ "L" "LFRFL-F-RFLFR+F+LFRFL" "R" "RFLFR+F+LFRFL-F-RFLFR" ]
lsysexp 4 axiom$ rules$[]
lsysdraw axiom$ 5 90 90Factor
USING: accessors L-system ui ;
: peano ( L-system -- L-system )
L-parser-dialect >>commands
[ 90 >>angle ] >>turtle-values
"L" >>axiom
{
{ "L" "LFRFL-F-RFLFR+F+LFRFL" }
{ "R" "RFLFR+F+LFRFL-F-RFLFR" }
} >>rules ;
[ <L-system> peano "Peano curve" open-window ] with-ui
When using the L-system visualizer, the following controls apply:
| Button | Command |
|---|---|
| a | zoom in |
| z | zoom out |
| left arrow | turn left |
| right arrow | turn right |
| up arrow | pitch down |
| down arrow | pitch up |
| q | roll left |
| w | roll right |
| Button | Command |
|---|---|
| x | iterate L-system |
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.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
Variant 1, by recursion
Test cases
Varian 2, by L-system
There are generic functions written in Fōrmulæ to compute an L-system in the page L-system.
The program that creates a Peano curve is:
FreeBASIC
Const anchura = 243 'una potencia de 3 para una curva uniformemente espaciada
Screenres 700,700
Sub Peano(x As Integer, y As Integer, lg As Integer, i1 As Integer, i2 As Integer)
If lg = 1 Then
Line - (x * 3, y * 3)
Return
End If
lg /= 3
Peano(x + (2 * i1 * lg), y + (2 * i1 * lg), lg, i1, i2)
Peano(x + ((i1 - i2 + 1) * lg), y + ((i1 + i2) * lg), lg, i1, 1 - i2)
Peano(x + lg, y + lg, lg, i1, 1 - i2)
Peano(x + ((i1 + i2) * lg), y + ((i1 - i2 + 1) * lg), lg, 1 - i1, 1 - i2)
Peano(x + (2 * i2 * lg), y + (2 * (1 - i2) * lg), lg, i1, i2)
Peano(x + ((1 + i2 - i1) * lg), y + ((2 - i1 - i2) * lg), lg, i1, i2)
Peano(x + (2 * (1 - i1) * lg), y + (2 * (1 - i1) * lg), lg, i1, i2)
Peano(x + ((2 - i1 - i2) * lg), y + ((1 + i2 - i1) * lg), lg, 1 - i1, i2)
Peano(x + (2 * (1 - i2) * lg), y + (2 * i2 * lg), lg, 1 - i1, i2)
End Sub
Peano(0, 0, anchura, 0, 0)
SleepFutureBasic
Couldn't help adding a little bling.
_window = 1
_curveSize = 7
_curveOrder = 3 * 3 * 3 * 3 // 3^4
void local fn BuildWindow
CGRect r = fn CGRectMake( 0, 0, 565, 570 )
window _window, @"Peano Curve In FutureBasic", r, NSWindowStyleMaskTitled
WindowSetBackgroundColor( _window, fn ColorBlack )
end fn
local fn Peano( x as long, y as long, lg as long, i1 as long, i2 as long )
ColorRef color = fn ColorRed
if ( lg == 1 ) then line to x * _curveSize, y * _curveSize : exit fn
lg /= 3
select ( rnd(8) )
case 1 : color = fn ColorBrown
case 2 : color = fn ColorRed
case 3 : color = fn ColorOrange
case 4 : color = fn ColorYellow
case 5 : color = fn ColorGreen
case 6 : color = fn ColorBlue
case 7 : color = fn ColorPurple
case 8 : color = fn ColorWhite
end select
pen 2, color
fn Peano( x + 2*i1* lg, y + 2*i1* lg, lg, i1, i2 )
fn Peano( x + (i1-i2+1)*lg, y + (i1+i2)* lg, lg, i1, 1-i2 )
fn Peano( x + lg, y + lg, lg, i1, 1-i2 )
fn Peano( x + (i1+i2)* lg, y + (i1-i2+1)*lg, lg, 1-i1, 1-i2 )
fn Peano( x + 2*i2* lg, y + 2*(1-i2)* lg, lg, i1, i2 )
fn Peano( x + (1+i2-i1)*lg, y + (2-i1-i2)*lg, lg, i1, i2 )
fn Peano( x + 2*(1-i1)* lg, y + 2*(1-i1)* lg, lg, i1, i2 )
fn Peano( x + (2-i1-i2)*lg, y + (1+i2-i1)*lg, lg, 1-i1, i2 )
fn Peano( x + 2*(1-i2)* lg, y + 2*i2* lg, lg, 1-i1, i2 )
end fn
randomize
fn BuildWindow
fn Peano( 0, 0, _curveOrder, 0, 0 )
HandleEvents- Output:
Go
The following is based on the recursive algorithm and C code in this paper scaled up to 81 x 81 points. The image produced is a variant known as a Peano-Meander curve (see Figure 1(b) here).
package main
import "github.com/fogleman/gg"
var points []gg.Point
const width = 81
func peano(x, y, lg, i1, i2 int) {
if lg == 1 {
px := float64(width-x) * 10
py := float64(width-y) * 10
points = append(points, gg.Point{px, py})
return
}
lg /= 3
peano(x+2*i1*lg, y+2*i1*lg, lg, i1, i2)
peano(x+(i1-i2+1)*lg, y+(i1+i2)*lg, lg, i1, 1-i2)
peano(x+lg, y+lg, lg, i1, 1-i2)
peano(x+(i1+i2)*lg, y+(i1-i2+1)*lg, lg, 1-i1, 1-i2)
peano(x+2*i2*lg, y+2*(1-i2)*lg, lg, i1, i2)
peano(x+(1+i2-i1)*lg, y+(2-i1-i2)*lg, lg, i1, i2)
peano(x+2*(1-i1)*lg, y+2*(1-i1)*lg, lg, i1, i2)
peano(x+(2-i1-i2)*lg, y+(1+i2-i1)*lg, lg, 1-i1, i2)
peano(x+2*(1-i2)*lg, y+2*i2*lg, lg, 1-i1, i2)
}
func main() {
peano(0, 0, width, 0, 0)
dc := gg.NewContext(820, 820)
dc.SetRGB(1, 1, 1) // White background
dc.Clear()
for _, p := range points {
dc.LineTo(p.X, p.Y)
}
dc.SetRGB(1, 0, 1) // Magenta curve
dc.SetLineWidth(1)
dc.Stroke()
dc.SavePNG("peano.png")
}
IS-BASIC
100 PROGRAM "PeanoC.bas"
110 OPTION ANGLE DEGREES
120 SET VIDEO MODE 5:SET VIDEO COLOR 0:SET VIDEO X 40:SET VIDEO Y 27
130 OPEN #101:"video:"
140 DISPLAY #101:AT 1 FROM 1 TO 27
150 PLOT 280,240,ANGLE 90;
160 CALL PEANO(28,90,6)
170 DEF PEANO(D,A,LEV)
180 IF LEV=0 THEN EXIT DEF
190 PLOT RIGHT A;
200 CALL PEANO(D,-A,LEV-1)
210 PLOT FORWARD D;
220 CALL PEANO(D,A,LEV-1)
230 PLOT FORWARD D;
240 CALL PEANO(D,-A,LEV-1)
250 PLOT LEFT A;
260 END DEFJ
This Hilbert variant is taken directly from the j lab "viewmat".
load'viewmat'
hp=: 3 : '(|.,]) 1 (0 _2 _2 ,&.> _2 _1 0 + #y) } (,.|:) y'
HP=: 3 3 $ 0 0 0 0 1
WR=: 16777215 16711680
viewrgb WR {~ hp ^:7 HP
Or, a smaller example (6 iterations instead of 7) and a different color selection:
require 'viewmat'
hp=: 3 : '(|.,]) 1 (0 _2 _2 ,&.> _2 _1 0 + #y) } (,.|:) y'
MG=: 256 #. 128 0 128,:0 192 0
viewrgb 2 ([ # #"1) MG {~ hp ^:6 [ 0, 0 1 0 ,: 0,
Java
Output is a file in SVG format.
import java.io.*;
public class PeanoCurve {
public static void main(final String[] args) {
try (Writer writer = new BufferedWriter(new FileWriter("peano_curve.svg"))) {
PeanoCurve s = new PeanoCurve(writer);
final int length = 8;
s.currentAngle = 90;
s.currentX = length;
s.currentY = length;
s.lineLength = length;
s.begin(656);
s.execute(rewrite(4));
s.end();
} catch (final Exception ex) {
ex.printStackTrace();
}
}
private PeanoCurve(final Writer writer) {
this.writer = writer;
}
private void begin(final int size) throws IOException {
write("<svg xmlns='http://www.w3.org/2000/svg' width='%d' height='%d'>\n", size, size);
write("<rect width='100%%' height='100%%' fill='white'/>\n");
write("<path stroke-width='1' stroke='black' fill='none' d='");
}
private void end() throws IOException {
write("'/>\n</svg>\n");
}
private void execute(final String s) throws IOException {
write("M%g,%g\n", currentX, currentY);
for (int i = 0, n = s.length(); i < n; ++i) {
switch (s.charAt(i)) {
case 'F':
line(lineLength);
break;
case '+':
turn(ANGLE);
break;
case '-':
turn(-ANGLE);
break;
}
}
}
private void line(final double length) throws IOException {
final double theta = (Math.PI * currentAngle) / 180.0;
currentX += length * Math.cos(theta);
currentY += length * Math.sin(theta);
write("L%g,%g\n", currentX, currentY);
}
private void turn(final int angle) {
currentAngle = (currentAngle + angle) % 360;
}
private void write(final String format, final Object... args) throws IOException {
writer.write(String.format(format, args));
}
private static String rewrite(final int order) {
String s = "L";
for (int i = 0; i < order; ++i) {
final StringBuilder sb = new StringBuilder();
for (int j = 0, n = s.length(); j < n; ++j) {
final char ch = s.charAt(j);
if (ch == 'L')
sb.append("LFRFL-F-RFLFR+F+LFRFL");
else if (ch == 'R')
sb.append("RFLFR+F+LFRFL-F-RFLFR");
else
sb.append(ch);
}
s = sb.toString();
}
return s;
}
private final Writer writer;
private double lineLength;
private double currentX;
private double currentY;
private int currentAngle;
private static final int ANGLE = 90;
}
- Output:
jq
Works with gojq, the Go implementation of jq
This entry includes two distinct solutions. In both cases, the jq program generates a SVG document, which can be viewed directly in the browser, at least if the file suffix is ".svg".
Using a Lindenmayer System
In this section, a Lindenmayer system of rules is used with turtle graphics.
The output converted to a .png file can be viewed at https://imgur.com/gallery/4QbUN7I
Simple Turtle Graphics
# => = 0 degrees
# ^ = 90 degrees
# <= = 180 degrees
# v = 270 degrees
# $start : [$x, $y]
def turtle($start):
$start
| if type == "array" then "M \($start|join(","))" else "M 0,0" end
| {svg: ., up:true, angle:0};
def turtleUp: .up=true;
def turtleDown: .up=false;
def turtleRotate($angle): .angle = (360 + (.angle + $angle)) % 360;
def turtleForward($d):
if .up
then if .angle== 0 then .svg += " m \($d),0"
elif .angle== 90 then .svg += " m 0,-\($d)"
elif .angle==180 then .svg += " m -\($d),0"
elif .angle==270 then .svg += " m 0,\($d)"
else "unsupported angle \(.angle)" | error
end
else if .angle== 0 then .svg += " h \($d)"
elif .angle== 90 then .svg += " v -\($d)"
elif .angle==180 then .svg += " h -\($d)"
elif .angle==270 then .svg += " v \($d)"
else "unsupported angle \(.angle)" | error
end
end;
def svg($size):
"<svg viewBox=\"0 0 \($size) \($size)\" xmlns=\"http://www.w3.org/2000/svg\">",
.,
"</svg>";
def path($fill; $stroke; $width):
"<path fill=\"\($fill)\" stroke=\"\($stroke)\" stroke-width=\"\($width)\" d=\"\(.svg)\" />";
def draw:
path("none"; "red"; "0.1") | svg(100) ;Peano Curve
# Compute the curve using a Lindenmayer system of rules
def rules:
{ L: "LFRFL-F-RFLFR+F+LFRFL",
R: "RFLFR+F+LFRFL-F-RFLFR" } ;
def peano($count):
rules as $rules
| def p($count):
if $count <= 0 then .
else gsub("L"; "l") | gsub("R"; $rules["R"]) | gsub("l"; $rules["L"]) | p($count-1)
end;
"L" | p($count) ;
def interpret($x):
if $x == "+" then turtleRotate(90)
elif $x == "-" then turtleRotate(-90)
elif $x == "F" then turtleForward(1)
else .
end;
def peano_curve($n):
peano($n)
| split("")
| reduce .[] as $action (turtle([1,1]) | turtleDown;
interpret($action) ) ;
peano_curve(4)
| draw- Output:
See https://imgur.com/gallery/4QbUN7I
Peano-Meander curve
Adapted from #Go
A .png version of the SVG image generated using an invocation such as the following can be viewed at https://imgur.com/a/RGQr17J
jq -nr -f peano-curve.jq > peano-curve.svg
# Input: an array
# Output: the array augmented with another x,y pair
def peano($x; $y; $lg; $i1; $i2):
$lg as $width
| def p($x; $y; $lg; $i1; $i2):
if $lg == 1
then (($width - $x) * 10) as $px
| (($width - $y) * 10) as $py
| . + [[$px,$py]]
else (($lg/3) | floor) as $lg
| p($x+2*$i1*$lg; $y+2*$i1*$lg; $lg; $i1; $i2)
| p($x+($i1-$i2+1)*$lg; $y+($i1+$i2)*$lg; $lg; $i1; 1-$i2)
| p($x+$lg; $y+$lg; $lg; $i1; 1-$i2)
| p($x+($i1+$i2)*$lg; $y+($i1-$i2+1)*$lg; $lg; 1-$i1; 1-$i2)
| p($x+2*$i2*$lg; $y+2*(1-$i2)*$lg; $lg; $i1; $i2)
| p($x+(1+$i2-$i1)*$lg; $y+(2-$i1-$i2)*$lg; $lg; $i1; $i2)
| p($x+2*(1-$i1)*$lg; $y+2*(1-$i1)*$lg; $lg; $i1; $i2)
| p($x+(2-$i1-$i2)*$lg; $y+(1+$i2-$i1)*$lg; $lg; 1-$i1; $i2)
| p($x+2*(1-$i2)*$lg; $y+2*$i2*$lg; $lg; 1-$i1; $i2)
end;
p($x; $y; $lg; $i1; $i2);
def svg:
"<svg viewBox=\"0 0 820 820\" xmlns=\"http://www.w3.org/2000/svg\">" ;
def path($fill; $stroke):
"<path fill=\"\($fill)\" stroke=\"\($stroke)\" d=\"M ";
def endpath:
" \" /> </svg>";
def peanoCurve:
null | peano(0; 0; 81; 0; 0) | map(join(",")) | join(" ");
svg,
( path("none"; "red") + peanoCurve + endpath)- Output:
Julia
The peano function is from the C version.
using Gtk, Graphics, Colors
function peano(ctx, x, y, lg, i1, i2)
if lg < 3
line_to(ctx, x - 250, y - 250)
stroke(ctx)
move_to(ctx, x - 250 , y - 250)
else
lg = div(lg, 3)
peano(ctx, x + (2 * i1 * lg), y + (2 * i1 * lg), lg, i1, i2)
peano(ctx, x + ((i1 - i2 + 1) * lg), y + ((i1 + i2) * lg), lg, i1, 1 - i2)
peano(ctx, x + lg, y + lg, lg, i1, 1 - i2)
peano(ctx, x + ((i1 + i2) * lg), y + ((i1 - i2 + 1) * lg), lg, 1 - i1, 1 - i2)
peano(ctx, x + (2 * i2 * lg), y + ( 2 * (1-i2) * lg), lg, i1, i2)
peano(ctx, x + ((1 + i2 - i1) * lg), y + ((2 - i1 - i2) * lg), lg, i1, i2)
peano(ctx, x + (2 * (1 - i1) * lg), y + (2 * (1 - i1) * lg), lg, i1, i2)
peano(ctx, x + ((2 - i1 - i2) * lg), y + ((1 + i2 - i1) * lg), lg, 1 - i1, i2)
peano(ctx, x + (2 * (1 - i2) * lg), y + (2 * i2 * lg), lg, 1 - i1, i2)
end
end
const can = @GtkCanvas()
const win = GtkWindow(can, "Peano Curve", 500, 500)
@guarded draw(can) do widget
ctx = getgc(can)
h = height(can)
w = width(can)
set_source(ctx, colorant"blue")
set_line_width(ctx, 1)
peano(ctx, w/2, h/2, 500, 0, 0)
end
show(can)
const cond = Condition()
endit(w) = notify(cond)
signal_connect(endit, win, :destroy)
wait(cond)
Lua
Using the Bitmap class here, with an ASCII pixel representation, then extending..
local PeanoLSystem = {
axiom = "L",
rules = {
L = "LFRFL-F-RFLFR+F+LFRFL",
R = "RFLFR+F+LFRFL-F-RFLFR"
},
eval = function(self, n)
local source, result = self.axiom
for i = 1, n do
result = ""
for j = 1, #source do
local ch = source:sub(j,j)
result = result .. (self.rules[ch] and self.rules[ch] or ch)
end
source = result
end
return result
end
}
function Bitmap:drawPath(path, x, y, dx, dy)
self:set(x, y, "@")
for i = 1, #path do
local ch = path:sub(i,i)
if (ch == "F") then
local reps = dx==0 and 1 or 3 -- aspect correction
for r = 1, reps do
x, y = x+dx, y+dy
self:set(x, y, dx==0 and "|" or "-")
end
x, y = x+dx, y+dy
self:set(x, y, "+")
elseif (ch =="-") then
dx, dy = dy, -dx
elseif (ch == "+") then
dx, dy = -dy, dx
end
end
self:set(x, y, "X")
end
function Bitmap:render()
for y = 1, self.height do
print(table.concat(self.pixels[y]))
end
end
bitmap = Bitmap(53*2,53)
bitmap:clear(" ")
bitmap:drawPath(PeanoLSystem:eval(3), 0, 0, 0, 1)
bitmap:render()
- Output:
@ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ +
| | | | | | | | |
+---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+
| | | | | | | | |
+ +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+---+ +---+ +---+ +---+ + + +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +
| | |
+---+ +---+ +---+ +---+ + + +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+
| | | | | | | | |
+ +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ +
| | | | | | | | |
+---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ + + +---+ +---+ +---+ +---+
| | |
+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ + + +---+ +---+ +---+ +---+
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ +
| | | | | | | | |
+---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+
| | | | | | | | |
+ +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+ +---+ + + +---+
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
| | | | | | | | | | | | | | | | | | | | | | | | | | |
+---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ X
M2000 Interpreter
Draw to M2000 console (which has graphic capabilities). Sub is a copy from freebasic, changed *Line* statement to *Draw to*
Module Peano_curve {
Cls 1, 0
Const Center=2
Report Center, "Peano curve"
let factor=.9, k=3, wi=k^4
let wi2=min.data(scale.x*factor, scale.y*factor), n=wi2/wi
let (dx, dy)=((scale.x-wi2)/2, (scale.y-wi2)/2)
dx+=k*1.2*twipsX
dy+=k*1.2*twipsY
move dx, dy
pen 11 {
Peano(0,0,wi,0, 0)
}
sub Peano(x, y, lg, i1, i2)
if lg ==1 then
draw to x*n+dx , y*n+dy
return
end if
lg/=k
Peano(x+2*i1*lg, y+2*i1*lg, lg, i1, i2)
Peano(x+(i1-i2+1)*lg, y+(i1+i2)*lg, lg, i1, 1-i2)
Peano(x+lg, y+lg, lg, i1, 1-i2)
Peano(x+(i1+i2)*lg, y+(i1-i2+1)*lg, lg, 1-i1, 1-i2)
Peano(x+2*i2*lg, y+2*(1-i2)*lg, lg, i1, i2)
Peano(x+(1+i2-i1)*lg, y+(2-i1-i2)*lg, lg, i1, i2)
Peano(x+2*(1-i1)*lg, y+2*(1-i1)*lg, lg, i1, i2)
Peano(x+(2-i1-i2)*lg, y+(1+i2-i1)*lg, lg, 1-i1, i2)
Peano(x+2*(1-i2)*lg, y+2*i2*lg, lg, 1-i1, i2)
end sub
}
Peano_curveMathematica/Wolfram Language
Graphics[PeanoCurve[4]]
- Output:
Outputs a graphical representation of a 4th order PeanoCurve.
Nim
import imageman
const Width = 81
proc peano(points: var seq[Point]; x, y, lg, i1, i2: int) =
if lg == 1:
points.add ((Width - x) * 10, (Width - y) * 10)
return
let lg = lg div 3
points.peano(x + 2 * i1 * lg, y + 2 * i1 * lg, lg, i1, i2)
points.peano(x + (i1 - i2 + 1) * lg, y + (i1 + i2) * lg, lg, i1, 1 - i2)
points.peano(x + lg, y + lg, lg, i1, 1 - i2)
points.peano(x + (i1 + i2) * lg, y + (i1 - i2 + 1) * lg, lg, 1 - i1, 1 - i2)
points.peano(x + 2 * i2 * lg, y + 2 * (1-i2) * lg, lg, i1, i2)
points.peano(x + (1 + i2 - i1) * lg, y + (2 - i1 - i2) * lg, lg, i1, i2)
points.peano(x + 2 * (1 - i1) * lg, y + 2 * (1 - i1) * lg, lg, i1, i2)
points.peano(x + (2 - i1 - i2) * lg, y + (1 + i2 - i1) * lg, lg, 1 - i1, i2)
points.peano(x + 2 * (1 - i2) * lg, y + 2 * i2 * lg, lg, 1 - i1, i2)
var points: seq[Point]
points.peano(0, 0, Width, 0, 0)
var image = initImage[ColorRGBU](820, 820)
let color = ColorRGBU([byte 255, 255, 0])
for i in 1..points.high:
image.drawLine(points[i - 1], points[i], color)
image.savePNG("peano.png", compression = 9)
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 = (
L => 'LFRFL-F-RFLFR+F+LFRFL',
R => 'RFLFR+F+LFRFL-F-RFLFR'
);
my $peano = 'L';
$peano =~ s/([LR])/$rules{$1}/eg for 1..4;
# Draw the curve in SVG
($x, $y) = (0, 0);
$theta = pi/2;
$r = 4;
for (split //, $peano) {
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; }
}
$max = max(@X,@Y);
$xt = -min(@X)+10;
$yt = -min(@Y)+10;
$svg = SVG->new(width=>$max+20, height=>$max+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, '>', 'peano_curve.svg';
print $fh $svg->xmlify(-namespace=>'svg');
close $fh;
Peano curve (offsite image)
Phix
You can run this online here. Space key toggles between switchback and meander curves.
-- -- demo\rosetta\peano_curve.exw -- ============================ -- -- Draws a peano curve. Space key toggles between switchback and meander curves. -- with javascript_semantics include pGUI.e Ihandle dlg, canvas cdCanvas cddbuffer, cdcanvas bool meander = false -- space toggles (false==draw switchback curve) constant width = 81 sequence points = {} -- switchback peano: -- -- There are (as per wp) four shapes to draw: -- -- 1: +-v ^ 2: ^ v-+ 3: v ^-+ 2: +-^ v -- | | | | | | | | | | | | -- ^ v-+ +-v ^ +-^ v v ^-+ -- -- 1 starts bottom left, ends top right -- 2 starts bottom right, ends top left -- 3 starts top left, ends bottom right -- 4 starts top right, ends bottom left -- -- given the centre point (think {1,1}), and using {0,0} as the bottom left: -- constant shapes = {{{-1,-1},{-1,0},{-1,+1},{0,+1},{0,0},{0,-1},{+1,-1},{+1,0},{+1,+1}}, {{+1,-1},{+1,0},{+1,+1},{0,+1},{0,0},{0,-1},{-1,-1},{-1,0},{-1,+1}}, -- (== sq_mul(shapes[1],{-1,0})) {{-1,+1},{-1,0},{-1,-1},{0,-1},{0,0},{0,+1},{+1,+1},{+1,0},{+1,-1}}, -- (== reverse(shapes[2])) {{+1,+1},{+1,0},{+1,-1},{0,-1},{0,0},{0,+1},{-1,+1},{-1,0},{-1,-1}}} -- (== reverse(shapes[1])) constant subshapes = {{1,2,1,3,4,3,1,2,1}, {2,1,2,4,3,4,2,1,2}, -- == sq_sub({3,3,3,7,7,7,3,3,3},subshapes[1]) {3,4,3,1,2,1,3,4,3}, -- == sq_sub(5,subshapes[2]) {4,3,4,2,1,2,4,3,4}} -- == sq_sub(5,subshapes[1]) -- As noted, it should theoretically be possible to simplify/shorten/remove/inline those tables procedure switchback_peano(integer x, y, level, shape) -- (written from scratch, with a nod to the meander algorithm [below]) if level<=1 then points = append(points, {x*10, y*10}) return end if level /= 3 for i=1 to 9 do integer {dx,dy} = shapes[shape][i] switchback_peano(x+dx*level,y+dy*level,level,subshapes[shape][i]) end for end procedure procedure meander_peano(integer x, y, lg, i1, i2) -- (translated from Go) if lg=1 then integer px := (width-x) * 10, py := (width-y) * 10 points = append(points, {px, py}) return end if lg /= 3 meander_peano(x+2*i1*lg, y+2*i1*lg, lg, i1, i2) meander_peano(x+(i1-i2+1)*lg, y+(i1+i2)*lg, lg, i1, 1-i2) meander_peano(x+lg, y+lg, lg, i1, 1-i2) meander_peano(x+(i1+i2)*lg, y+(i1-i2+1)*lg, lg, 1-i1, 1-i2) meander_peano(x+2*i2*lg, y+2*(1-i2)*lg, lg, i1, i2) meander_peano(x+(1+i2-i1)*lg, y+(2-i1-i2)*lg, lg, i1, i2) meander_peano(x+2*(1-i1)*lg, y+2*(1-i1)*lg, lg, i1, i2) meander_peano(x+(2-i1-i2)*lg, y+(1+i2-i1)*lg, lg, 1-i1, i2) meander_peano(x+2*(1-i2)*lg, y+2*i2*lg, lg, 1-i1, i2) end procedure function redraw_cb(Ihandle /*ih*/, integer /*posx*/, /*posy*/) if length(points)=0 then if meander then meander_peano(0, 0, width, 0, 0) else switchback_peano(41, 41, width, 1) end if end if cdCanvasActivate(cddbuffer) cdCanvasBegin(cddbuffer, CD_OPEN_LINES) for i=1 to length(points) do integer {x,y} = points[i] cdCanvasVertex(cddbuffer, x, y) end for cdCanvasEnd(cddbuffer) 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_WHITE) cdCanvasSetForeground(cddbuffer, CD_MAGENTA) return IUP_DEFAULT end function function key_cb(Ihandle /*ih*/, atom c) if c=K_ESC then return IUP_CLOSE end if if c=' ' then meander = not meander points = {} cdCanvasClear(cddbuffer) IupUpdate(canvas) end if return IUP_CONTINUE end function procedure main() IupOpen() canvas = IupCanvas("RASTERSIZE=822x822") IupSetCallbacks(canvas, {"MAP_CB", Icallback("map_cb"), "ACTION", Icallback("redraw_cb")}) dlg = IupDialog(canvas,`TITLE="Peano Curve"`) IupSetAttribute(dlg, "DIALOGFRAME", "YES") -- no resize here IupSetCallback(dlg, "KEY_CB", Icallback("key_cb")) IupShow(dlg) if platform()!=JS then IupMainLoop() IupClose() end if end procedure main()
Processing
//Abhishek Ghosh, 28th June 2022
void Peano(int x, int y, int lg, int i1, int i2) {
if (lg == 1) {
ellipse(x,y,1,1);
return;
}
lg = lg/3;
Peano(x+(2*i1*lg), y+(2*i1*lg), lg, i1, i2);
Peano(x+((i1-i2+1)*lg), y+((i1+i2)*lg), lg, i1, 1-i2);
Peano(x+lg, y+lg, lg, i1, 1-i2);
Peano(x+((i1+i2)*lg), y+((i1-i2+1)*lg), lg, 1-i1, 1-i2);
Peano(x+(2*i2*lg), y+(2*(1-i2)*lg), lg, i1, i2);
Peano(x+((1+i2-i1)*lg), y+((2-i1-i2)*lg), lg, i1, i2);
Peano(x+(2*(1-i1)*lg), y+(2*(1-i1)*lg), lg, i1, i2);
Peano(x+((2-i1-i2)*lg), y+((1+i2-i1)*lg), lg, 1-i1, i2);
Peano(x+(2*(1-i2)*lg), y+(2*i2*lg), lg, 1-i1, i2);
}
void setup(){
size(1000,1000);
Peano(0, 0, 1000, 0, 0);
}
Prolog
main:-
write_peano_curve('peano_curve.svg', 656, 4).
write_peano_curve(File, Size, Order):-
open(File, write, Stream),
format(Stream,
"<svg xmlns='http://www.w3.org/2000/svg' width='~d' height='~d'>\n",
[Size, Size]),
write(Stream, "<rect width='100%' height='100%' fill='white'/>\n"),
peano_curve(Stream, "L", 8, 8, 8, 90, Order),
write(Stream, "</svg>\n"),
close(Stream).
peano_curve(Stream, Axiom, X, Y, Length, Angle, Order):-
write(Stream, "<path stroke-width='1' stroke='black' fill='none' d='"),
format(Stream, 'M~g,~g\n', [X, Y]),
rewrite(Axiom, Order, S),
string_chars(S, Chars),
execute(Stream, X, Y, Length, Angle, Chars),
write(Stream, "'/>\n").
rewrite(S, 0, S):-!.
rewrite(S0, N, S):-
string_chars(S0, Chars0),
rewrite1(Chars0, '', S1),
N1 is N - 1,
rewrite(S1, N1, S).
rewrite1([], S, S):-!.
rewrite1([C|Chars], T, S):-
rewrite2(C, X),
string_concat(T, X, T1),
rewrite1(Chars, T1, S).
rewrite2('L', "LFRFL-F-RFLFR+F+LFRFL"):-!.
rewrite2('R', "RFLFR+F+LFRFL-F-RFLFR"):-!.
rewrite2(X, X).
execute(_, _, _, _, _, []):-!.
execute(Stream, X, Y, Length, Angle, ['F'|Chars]):-
!,
Theta is (pi * Angle) / 180.0,
X1 is X + Length * cos(Theta),
Y1 is Y + Length * sin(Theta),
format(Stream, 'L~g,~g\n', [X1, Y1]),
execute(Stream, X1, Y1, Length, Angle, Chars).
execute(Stream, X, Y, Length, Angle, ['+'|Chars]):-
!,
Angle1 is (Angle + 90) mod 360,
execute(Stream, X, Y, Length, Angle1, Chars).
execute(Stream, X, Y, Length, Angle, ['-'|Chars]):-
!,
Angle1 is (Angle - 90) mod 360,
execute(Stream, X, Y, Length, Angle1, Chars).
execute(Stream, X, Y, Length, Angle, [_|Chars]):-
execute(Stream, X, Y, Length, Angle, Chars).
- Output:
Python
This implementation is, as I think, a variant of the Peano curve (just because it's different in the images). And it's supposed to be like a fullscreen app. And because of scale, the "steps" of the curve will be different in horizontal and vertical (it'll be quite nice if your screen is square :D). If peano(4) (in the last line of code) is runned with the input higher than 8, the void between the steps will be filled with steps, and the hole screen will fill (of course, this depends of your screen size). It's also produces a graphic with the current stack pilled, timed by the current function runned.
import turtle as tt
import inspect
stack = [] # Mark the current stacks in run.
def peano(iterations=1):
global stack
# The turtle Ivan:
ivan = tt.Turtle(shape = "classic", visible = True)
# The app window:
screen = tt.Screen()
screen.title("Desenhin do Peano")
screen.bgcolor("#232323")
screen.delay(0) # Speed on drawing (if higher, more slow)
screen.setup(width=0.95, height=0.9)
# The size of each step walked (here, named simply "walk"). It's not a pixel scale. This may stay still:
walk = 1
def screenlength(k):
# A function to make the image good to see (without it would result in a partial image).
# This will guarantee that we can see the the voids and it's steps.
if k != 0:
length = screenlength(k-1)
return 2*length + 1
else: return 0
kkkj = screenlength(iterations)
screen.setworldcoordinates(-1, -1, kkkj + 1, kkkj + 1)
ivan.color("#EEFFFF", "#FFFFFF")
# The magic \(^-^)/:
def step1(k):
global stack
stack.append(len(inspect.stack()))
if k != 0:
ivan.left(90)
step2(k - 1)
ivan.forward(walk)
ivan.right(90)
step1(k - 1)
ivan.forward(walk)
step1(k - 1)
ivan.right(90)
ivan.forward(walk)
step2(k - 1)
ivan.left(90)
def step2(k):
global stack
stack.append(len(inspect.stack()))
if k != 0:
ivan.right(90)
step1(k - 1)
ivan.forward(walk)
ivan.left(90)
step2(k - 1)
ivan.forward(walk)
step2(k - 1)
ivan.left(90)
ivan.forward(walk)
step1(k - 1)
ivan.right(90)
# Making the program work:
ivan.left(90)
step2(iterations)
tt.done()
if __name__ == "__main__":
peano(4)
import pylab as P # This plot, after closing the drawing window, the "stack" graphic.
P.plot(stack)
P.show()
You can see the output image here and the output stack graphic here.
Quackery
[ $ "turtleduck.qky" loadfile ] now!
[ stack ] is switch.arg ( --> [ )
[ switch.arg put ] is switch ( x --> )
[ switch.arg release ] is otherwise ( --> )
[ switch.arg share
!= iff ]else[ done
otherwise ]'[ do ]done[ ] is case ( x --> )
[ $ "" swap witheach
[ nested quackery join ] ] is expand ( $ --> $ )
[ $ "F" ] is F ( $ --> $ )
[ $ "L" ] is L ( $ --> $ )
[ $ "R" ] is R ( $ --> $ )
[ $ "AFBFARFRBFAFBLFLAFBFA" ] is A ( $ --> $ )
[ $ "BFAFBLFLAFBFARFRBFAFB" ] is B ( $ --> $ )
$ "A"
4 times expand
turtle
witheach
[ switch
[ char F case [ 4 1 walk ]
char L case [ -1 4 turn ]
char R case [ 1 4 turn ]
otherwise ( ignore ) ] ]- Output:
R
HilberCurve Library from bioconductor is used to produce 4th order peano curve, for more details please refer to Bioconductor[1]
#to install hilbercurve library, biocmanager needs to be installed first
install.packages("BiocManager")
BiocManager::install("HilbertCurve")
#loading library and setting seed for random numbers
library(HilbertCurve)
library(circlize)
set.seed(123)
#4th order peano curve is generated
for(i in 1:512) {
peano = HilbertCurve(1, 512, level = 4, reference = TRUE, arrow = FALSE)
hc_points(peano, x1 = i, np = NULL, pch = 16, size = unit(3, "mm"))
}Racket
Draw the Peano curve using the classical turtle style known from Logo.
The MetaPict library is used to implement a turtle.
See also https://pdfs.semanticscholar.org/fee6/187cc2dd1679d4976db9522b06a49f63be46.pdf
/* Jens Axel Søgaard, 27th December 2018*/
#lang racket
(require metapict metapict/mat)
;;; Turtle State
(define p (pt 0 0)) ; current position
(define d (vec 0 1)) ; current direction
(define c '()) ; line segments drawn so far
;;; Turtle Operations
(define (jump q) (set! p q))
(define (move q) (set! c (cons (curve p -- q) c)) (set! p q))
(define (forward x) (move (pt+ p (vec* x d))))
(define (left a) (set! d (rot a d)))
(define (right a) (left (- a)))
;;; Peano
(define (peano n a h)
(unless (= n 0)
(right a)
(peano (- n 1) (- a) h)
(forward h)
(peano (- n 1) a h)
(forward h)
(peano (- n 1) (- a) h)
(left a)))
;;; Produce image
(set-curve-pict-size 400 400)
(with-window (window -1 81 -1 82)
(peano 6 90 3)
(draw* c))
Raku
(formerly Perl 6)
use SVG;
role Lindenmayer {
has %.rules;
method succ {
self.comb.map( { %!rules{$^c} // $c } ).join but Lindenmayer(%!rules)
}
}
my $peano = 'L' but Lindenmayer( { 'L' => 'LFRFL-F-RFLFR+F+LFRFL', 'R' => 'RFLFR+F+LFRFL-F-RFLFR' } );
$peano++ xx 4;
my @points = (10, 10);
for $peano.comb {
state ($x, $y) = @points[0,1];
state $d = 0 + 8i;
when 'F' { @points.append: ($x += $d.re).round(1), ($y += $d.im).round(1) }
when /< + - >/ { $d *= "{$_}1i" }
default { }
}
say SVG.serialize(
svg => [
:660width, :660height, :style<stroke:lime>,
:rect[:width<100%>, :height<100%>, :fill<black>],
:polyline[ :points(@points.join: ','), :fill<black> ],
],
);
See: Peano curve (SVG image)
Ruby
Implemented as a Lindenmayer System, depends on JRuby or JRubyComplete see Hilbert for grammar
load_library :grammar
# Peano class
class Peano
include Processing::Proxy
attr_reader :draw_length, :vec, :theta, :axiom, :grammar
DELTA = 60 # degrees
def initialize(vec)
@axiom = 'XF' # Axiom
rules = {
'X' => 'X+YF++YF-FX--FXFX-YF+', # LSystem Rules
'Y' => '-FX+YFYF++YF+FX--FX-Y'
}
@grammar = Grammar.new(axiom, rules)
@theta = 0
@draw_length = 100
@vec = vec
end
def generate(gen)
@draw_length = draw_length * 0.6**gen
grammar.generate gen
end
def translate_rules(prod)
coss = ->(orig, alpha, len) { orig + len * DegLut.cos(alpha) }
sinn = ->(orig, alpha, len) { orig - len * DegLut.sin(alpha) }
[].tap do |pts| # An array to store line vertices as Vec2D
prod.scan(/./) do |ch|
case ch
when 'F'
pts << vec.copy
@vec = Vec2D.new(
coss.call(vec.x, theta, draw_length),
sinn.call(vec.y, theta, draw_length)
)
pts << vec
when '+'
@theta += DELTA
when '-'
@theta -= DELTA
when 'X', 'Y'
else
puts("character #{ch} not in grammar")
end
end
end
end
end
attr_reader :points
def setup
sketch_title 'Peano'
peano = Peano.new(Vec2D.new(width * 0.65, height * 0.9))
production = peano.generate 4 # 4 generations looks OK
@points = peano.translate_rules(production)
no_loop
end
def draw
background(0)
render points
end
def render(points)
no_fill
stroke 200.0
stroke_weight 3
begin_shape
points.each_slice(2) do |v0, v1|
v0.to_vertex(renderer)
v1.to_vertex(renderer)
end
end_shape
end
def renderer
@renderer ||= GfxRender.new(g)
end
def settings
size(800, 800)
end
Rust
// [dependencies]
// svg = "0.8.0"
use svg::node::element::path::Data;
use svg::node::element::Path;
struct PeanoCurve {
current_x: f64,
current_y: f64,
current_angle: i32,
line_length: f64,
}
impl PeanoCurve {
fn new(x: f64, y: f64, length: f64, angle: i32) -> PeanoCurve {
PeanoCurve {
current_x: x,
current_y: y,
current_angle: angle,
line_length: length,
}
}
fn rewrite(order: usize) -> String {
let mut str = String::from("L");
for _ in 0..order {
let mut tmp = String::new();
for ch in str.chars() {
match ch {
'L' => tmp.push_str("LFRFL-F-RFLFR+F+LFRFL"),
'R' => tmp.push_str("RFLFR+F+LFRFL-F-RFLFR"),
_ => tmp.push(ch),
}
}
str = tmp;
}
str
}
fn execute(&mut self, order: usize) -> Path {
let mut data = Data::new().move_to((self.current_x, self.current_y));
for ch in PeanoCurve::rewrite(order).chars() {
match ch {
'F' => data = self.draw_line(data),
'+' => self.turn(90),
'-' => self.turn(-90),
_ => {}
}
}
Path::new()
.set("fill", "none")
.set("stroke", "black")
.set("stroke-width", "1")
.set("d", data)
}
fn draw_line(&mut self, data: Data) -> Data {
let theta = (self.current_angle as f64).to_radians();
self.current_x += self.line_length * theta.cos();
self.current_y += self.line_length * theta.sin();
data.line_to((self.current_x, self.current_y))
}
fn turn(&mut self, angle: i32) {
self.current_angle = (self.current_angle + angle) % 360;
}
fn save(file: &str, size: usize, order: usize) -> std::io::Result<()> {
use svg::node::element::Rectangle;
let rect = Rectangle::new()
.set("width", "100%")
.set("height", "100%")
.set("fill", "white");
let mut p = PeanoCurve::new(8.0, 8.0, 8.0, 90);
let document = svg::Document::new()
.set("width", size)
.set("height", size)
.add(rect)
.add(p.execute(order));
svg::save(file, &document)
}
}
fn main() {
PeanoCurve::save("peano_curve.svg", 656, 4).unwrap();
}
- Output:
Sidef
Uses the LSystem class defined at Hilbert curve.
var rules = Hash(
l => 'lFrFl-F-rFlFr+F+lFrFl',
r => 'rFlFr+F+lFrFl-F-rFlFr',
)
var lsys = LSystem(
width: 500,
height: 500,
xoff: -50,
yoff: -50,
len: 5,
angle: 90,
color: 'dark green',
)
lsys.execute('l', 4, "peano_curve.png", rules)
Output image: Peano curve
VBA
Const WIDTH = 243 'a power of 3 for a evenly spaced curve
Dim n As Long
Dim points() As Single
Dim flag As Boolean
'Store the coordinate pairs (x, y) generated by Peano into
'a SafeArrayOfPoints with lineto. The number of points
'generated depend on WIDTH. Peano is called twice. Once
'to count the number of points, and twice to generate
'the points after the dynamic array has been
'redimensionalised.
'VBA doesn't have a lineto method. Instead of AddLine, which
'requires four parameters, including the begin pair of
'coordinates, the method AddPolyline is used, which is
'called from main after all the points are generated.
'This creates a single object, whereas AddLine would
'create thousands of small unconnected line objects.
Private Sub lineto(x As Integer, y As Integer)
If flag Then
points(n, 1) = x
points(n, 2) = y
End If
n = n + 1
End Sub
Private Sub Peano(ByVal x As Integer, ByVal y As Integer, ByVal lg As Integer, _
ByVal i1 As Integer, ByVal i2 As Integer)
If (lg = 1) Then
Call lineto(x * 3, y * 3)
Exit Sub
End If
lg = lg / 3
Call Peano(x + (2 * i1 * lg), y + (2 * i1 * lg), lg, i1, i2)
Call Peano(x + ((i1 - i2 + 1) * lg), y + ((i1 + i2) * lg), lg, i1, 1 - i2)
Call Peano(x + lg, y + lg, lg, i1, 1 - i2)
Call Peano(x + ((i1 + i2) * lg), y + ((i1 - i2 + 1) * lg), lg, 1 - i1, 1 - i2)
Call Peano(x + (2 * i2 * lg), y + (2 * (1 - i2) * lg), lg, i1, i2)
Call Peano(x + ((1 + i2 - i1) * lg), y + ((2 - i1 - i2) * lg), lg, i1, i2)
Call Peano(x + (2 * (1 - i1) * lg), y + (2 * (1 - i1) * lg), lg, i1, i2)
Call Peano(x + ((2 - i1 - i2) * lg), y + ((1 + i2 - i1) * lg), lg, 1 - i1, i2)
Call Peano(x + (2 * (1 - i2) * lg), y + (2 * i2 * lg), lg, 1 - i1, i2)
End Sub
Sub main()
n = 1: flag = False
Call Peano(0, 0, WIDTH, 0, 0) 'Start Peano recursion to count number of points
ReDim points(1 To n - 1, 1 To 2)
n = 1: flag = True
Call Peano(0, 0, WIDTH, 0, 0) 'Start Peano recursion to generate and store points
ActiveSheet.Shapes.AddPolyline points 'Excel assumed
End SubVBScript
VBSCript does'nt have access to Windows graphics so I write SVG commands into an HML file and display it using the default browser. A turtle graphics class makes the recursive definition of the curve easy.
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
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()
setlocale "us"
initsvg
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 peano (n,a)
if n=0 then exit sub
x.rt a
peano n-1, -a
x.fw 1
peano n-1, a
x.fw 1
peano n-1, -a
x.lt a
end sub
dim x,i
set x=new turtle
x.iangle=90
x.orient=0
x.incr=7
x.x=100:x.y=500
peano 7,1
set x=nothing 'show image in browserWren
import "graphics" for Canvas, Color, Point
import "dome" for Window
class Game {
static init() {
Window.title = "Peano curve"
Canvas.resize(820, 820)
Window.resize(820, 820)
Canvas.cls(Color.white) // white background
__points = []
__width = 81
peano(0, 0, __width, 0, 0)
var col = Color.rgb(255, 0, 255) // magenta
var prev = __points[0]
for (p in __points.skip(1)) {
var curr = p
Canvas.line(prev.x, prev.y, curr.x, curr.y, col)
prev = curr
}
}
static peano(x, y, lg, i1, i2) {
if (lg == 1) {
var px = (__width - x) * 10
var py = (__width - y) * 10
__points.add(Point.new(px, py))
return
}
lg = (lg/3).floor
peano(x+2*i1*lg, y+2*i1*lg, lg, i1, i2)
peano(x+(i1-i2+1)*lg, y+(i1+i2)*lg, lg, i1, 1-i2)
peano(x+lg, y+lg, lg, i1, 1-i2)
peano(x+(i1+i2)*lg, y+(i1-i2+1)*lg, lg, 1-i1, 1-i2)
peano(x+2*i2*lg, y+2*(1-i2)*lg, lg, i1, i2)
peano(x+(1+i2-i1)*lg, y+(2-i1-i2)*lg, lg, i1, i2)
peano(x+2*(1-i1)*lg, y+2*(1-i1)*lg, lg, i1, i2)
peano(x+(2-i1-i2)*lg, y+(1+i2-i1)*lg, lg, 1-i1, i2)
peano(x+2*(1-i2)*lg, y+2*i2*lg, lg, 1-i1, i2)
}
static update() {}
static draw(dt) {}
}
XPL0
proc Peano(X, Y, Lg, I1, I2);
int X, Y, Lg, I1, I2;
[if Lg = 1 then
[Line(3*X, 3*Y, 11 \cyan\);
return;
];
Lg:= Lg/3;
Peano(X + 2*I1* Lg, Y + 2*I1* Lg, Lg, I1, I2);
Peano(X + (I1-I2+1)*Lg, Y + (I1+I2)* Lg, Lg, I1, 1-I2);
Peano(X + Lg, Y + Lg, Lg, I1, 1-I2);
Peano(X + (I1+I2)* Lg, Y + (I1-I2+1)*Lg, Lg, 1-I1, 1-I2);
Peano(X + 2*I2* Lg, Y + 2*(1-I2)* Lg, Lg, I1, I2);
Peano(X + (1+I2-I1)*Lg, Y + (2-I1-I2)*Lg, Lg, I1, I2);
Peano(X + 2*(1-I1)* Lg, Y + 2*(1-I1)* Lg, Lg, I1, I2);
Peano(X + (2-I1-I2)*Lg, Y + (1+I2-I1)*Lg, Lg, 1-I1, I2);
Peano(X + 2*(1-I2)* Lg, Y + 2*I2* Lg, Lg, 1-I1, I2);
];
[SetVid($13);
Peano(0, 0, 3*3*3*3, 0, 0); \start Peano recursion
]- Output:
Yabasic
WIDTH = 243 //a power of 3 for a evenly spaced curve
open window 700, 700
Peano(0, 0, WIDTH, 0, 0)
Sub Peano(x, y, lg, i1, i2)
If (lg = 1) Then
line x * 3, y * 3
return
End If
lg = lg / 3
Peano(x + (2 * i1 * lg), y + (2 * i1 * lg), lg, i1, i2)
Peano(x + ((i1 - i2 + 1) * lg), y + ((i1 + i2) * lg), lg, i1, 1 - i2)
Peano(x + lg, y + lg, lg, i1, 1 - i2)
Peano(x + ((i1 + i2) * lg), y + ((i1 - i2 + 1) * lg), lg, 1 - i1, 1 - i2)
Peano(x + (2 * i2 * lg), y + (2 * (1 - i2) * lg), lg, i1, i2)
Peano(x + ((1 + i2 - i1) * lg), y + ((2 - i1 - i2) * lg), lg, i1, i2)
Peano(x + (2 * (1 - i1) * lg), y + (2 * (1 - i1) * lg), lg, i1, i2)
Peano(x + ((2 - i1 - i2) * lg), y + ((1 + i2 - i1) * lg), lg, 1 - i1, i2)
Peano(x + (2 * (1 - i2) * lg), y + (2 * i2 * lg), lg, 1 - i1, i2)
End Subzkl
Using a Lindenmayer system and turtle graphics & turned 90°:
lsystem("L", // axiom
Dictionary("L","LFRFL-F-RFLFR+F+LFRFL", "R","RFLFR+F+LFRFL-F-RFLFR"), # rules
"+-F", 4) // constants, order
: turtle(_);
fcn lsystem(axiom,rules,consts,n){ // Lindenmayer system --> string
foreach k in (consts){ rules.add(k,k) }
buf1,buf2 := Data(Void,axiom).howza(3), Data().howza(3); // characters
do(n){
buf1.pump(buf2.clear(), rules.get);
t:=buf1; buf1=buf2; buf2=t; // swap buffers
}
buf1.text // n=4 --> 16,401 characters
}Using Image Magick and the PPM class from http://rosettacode.org/wiki/Bitmap/Bresenham%27s_line_algorithm#zkl
fcn turtle(koch){
const D=10.0;
dir,angle, x,y := 0.0, (90.0).toRad(), 20.0, 830.0; // turtle; x,y are float
img,color := PPM(850,850), 0x00ff00;
foreach c in (koch){
switch(c){
case("F"){ // draw forward
dx,dy := D.toRectangular(dir);
tx,ty := x,y; x,y = (x+dx),(y+dy);
img.line(tx.toInt(),ty.toInt(), x.toInt(),y.toInt(), color);
}
case("-"){ dir-=angle } // turn right
case("+"){ dir+=angle } // turn left
}
}
img.writeJPGFile("peanoCurve.zkl.jpg");
}- Output:
Image at Peano curve
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