Introduction
Spirograph toy that is used to produce complex patterns using plastic cogs and colored pens. A fractal is a curve, that is developed using a recurring pattern that repeats itself infinitely on a low scale. Fractals are used for modeling structures (such as snowflakes) or for describing partly chaotic phenomena.
Spirograph can be used to draw various fractals. Some of them are given below
You can visit benice-equation-blogspot.in for more fractals design with their parametric equation. Some of them are given below
Mathematics behind the curtain
These are the two parametric equation to form a spirograph fractals, to understand these equations you have to consider a generalized figure of spirograph.
For the mathematics part you can refer to Wiki although i’ll try to explain a little of that mathematics in a short here. If we are interested behind the maths then you can check out the referred links. So as of now, these various curve can be drawn by using a parametric equation and varying some values of that equation we can get different fractals. So here’s the parametric equation:
![x(t) = R[(1-k)cost + lkcos(((1-k)/k)t)], y(t) = R[(1-k)sint - lksin(((1-k)/k)t).](https://geeksforgeeks.org/wp-content/uploads/2023/10/quicklatex.com-70d4c5d7e0bd78e6f26123e5b5e412f1_l3.png "Rendered by QuickLaTeX.com")
where,
R is a scaling parameter and does not affect the structure of the Spirograph.
and,
So, now let’s try to implement this in code.
#importing the required libraries import random, argparse import math import turtle from PIL import Image from datetime import datetime from fractions import gcd # A class that draws a spirograph class Spiro: # constructor def __init__(self, xc, yc, col, R, r, l): # create own turtle self.t = turtle.Turtle() # set cursor shape self.t.shape('turtle') # set step in degrees self.step = 5 # set drawing complete flag self.drawingComplete = False # set parameters self.setparams(xc, yc, col, R, r, l) # initiatize drawing self.restart() # set parameters def setparams(self, xc, yc, col, R, r, l): # spirograph parameters self.xc = xc self.yc = yc self.R = int(R) self.r = int(r) self.l = l self.col = col # reduce r/R to smallest form by dividing with GCD gcdVal = gcd(self.r, self.R) self.nRot = self.r//gcdVal # get ratio of radii self.k = r/float(R) # set color self.t.color(*col) # current angle self.a = 0 # restart drawing def restart(self): # set flag self.drawingComplete = False # show turtle self.t.showturtle() # go to first point self.t.up() R, k, l = self.R, self.k, self.l a = 0.0 x = R*((1-k)*math.cos(a) + l*k*math.cos((1-k)*a/k)) y = R*((1-k)*math.sin(a) - l*k*math.sin((1-k)*a/k)) self.t.setpos(self.xc + x, self.yc + y) self.t.down() # draw the whole thing def draw(self): # draw rest of points R, k, l = self.R, self.k, self.l for i in range(0, 360*self.nRot + 1, self.step): a = math.radians(i) x = R*((1-k)*math.cos(a) + l*k*math.cos((1-k)*a/k)) y = R*((1-k)*math.sin(a) - l*k*math.sin((1-k)*a/k)) self.t.setpos(self.xc + x, self.yc + y) # done - hide turtle self.t.hideturtle() # update by one step def update(self): # skip if done if self.drawingComplete: return # increment angle self.a += self.step # draw step R, k, l = self.R, self.k, self.l # set angle a = math.radians(self.a) x = self.R*((1-k)*math.cos(a) + l*k*math.cos((1-k)*a/k)) y = self.R*((1-k)*math.sin(a) - l*k*math.sin((1-k)*a/k)) self.t.setpos(self.xc + x, self.yc + y) # check if drawing is complete and set flag if self.a >= 360*self.nRot: self.drawingComplete = True # done - hide turtle self.t.hideturtle() # clear everything def clear(self): self.t.clear() # A class for animating spirographs class SpiroAnimator: # constructor def __init__(self, N): # timer value in milliseconds self.deltaT = 10 # get window dimensions self.width = turtle.window_width() self.height = turtle.window_height() # create spiro objects self.spiros = [] for i in range(N): # generate random parameters rparams = self.genRandomParams() # set spiro params spiro = Spiro(*rparams) self.spiros.append(spiro) # call timer turtle.ontimer(self.update, self.deltaT) # restart sprio drawing def restart(self): for spiro in self.spiros: # clear spiro.clear() # generate random parameters rparams = self.genRandomParams() # set spiro params spiro.setparams(*rparams) # restart drawing spiro.restart() # generate random parameters def genRandomParams(self): width, height = self.width, self.height R = random.randint(50, min(width, height)//2) r = random.randint(10, 9*R//10) l = random.uniform(0.1, 0.9) xc = random.randint(-width//2, width//2) yc = random.randint(-height//2, height//2) col = (random.random(), random.random(), random.random()) return (xc, yc, col, R, r, l) def update(self): # update all spiros nComplete = 0 for spiro in self.spiros: # update spiro.update() # count completed ones if spiro.drawingComplete: nComplete+= 1 # if all spiros are complete, restart if nComplete == len(self.spiros): self.restart() # call timer turtle.ontimer(self.update, self.deltaT) # toggle turtle on/off def toggleTurtles(self): for spiro in self.spiros: if spiro.t.isvisible(): spiro.t.hideturtle() else: spiro.t.showturtle() # save spiros to image def saveDrawing(): # hide turtle turtle.hideturtle() # generate unique file name dateStr = (datetime.now()).strftime("%d%b%Y-%H%M%S") fileName = 'spiro-' + dateStr print('saving drawing to %s.eps/png' % fileName) # get tkinter canvas canvas = turtle.getcanvas() # save postscipt image canvas.postscript(file = fileName + '.eps') # use PIL to convert to PNG img = Image.open(fileName + '.eps') img.save(fileName + '.png', 'png') # show turtle turtle.showturtle() # main() function def main(): # use sys.argv if needed print('generating spirograph...') # create parser descStr = """This program draws spirographs using the Turtle module. When run with no arguments, this program draws random spirographs. Terminology: R: radius of outer circle. r: radius of inner circle. l: ratio of hole distance to r. """ parser = argparse.ArgumentParser(description=descStr) # add expected arguments parser.add_argument('--sparams', nargs=3, dest='sparams', required=False, help="The three arguments in sparams: R, r, l.") # parse args args = parser.parse_args() # set to 80% screen width turtle.setup(width=0.8) # set cursor shape turtle.shape('turtle') # set title turtle.title("Spirographs!") # add key handler for saving images turtle.onkey(saveDrawing, "s") # start listening turtle.listen() # hide main turtle cursor turtle.hideturtle() # checks args and draw if args.sparams: params = [float(x) for x in args.sparams] # draw spirograph with given parameters # black by default col = (0.0, 0.0, 0.0) spiro = Spiro(0, 0, col, *params) spiro.draw() else: # create animator object spiroAnim = SpiroAnimator(4) # add key handler to toggle turtle cursor turtle.onkey(spiroAnim.toggleTurtles, "t") # add key handler to restart animation turtle.onkey(spiroAnim.restart, "space") # start turtle main loop turtle.mainloop() # call main if __name__ == '__main__': main() |
Output:
The above program draws 4 different kinds of spirograph fractals, try to generate other fractals and then upload your github links in the comment. I’ll be happy to help you out if any error comes up.
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