The most known method to draw a Dragon curve is by using turtle graphics. Here we implement a method visually illustrated in a video posted by Numberphile: https://www.youtube.com/watch?v=NajQEiKFom4. We are starting with a vertical segment and the successive rotations are counterclockwise.
import numpy as np
from numpy import pi
import plotly.graph_objects as go
def rot_matrix(alpha):
#Define the matrix of rotation about origin with an angle of alpha radians:
return np.array([[np.cos(alpha), -np.sin(alpha)],
[np.sin(alpha), np.cos(alpha)]])
def rotate_dragon(x, y, alpha=pi/2):
#x,y lists or 1D-array containng the (x, y)-coordinates of the turn points on the dragon curve constructed
# in a single step
X, Y = rot_matrix(alpha).dot(np.stack((x, y))) # the lists of coordinates of turn points on the rotated curve
return X, Y
#the initial step dragon cuvre is represented by a vertical line of length L
L = 0.12
X = np.array([0, 0])
Y = np.array([-L, 0])
fig = go.Figure(data=[go.Scatter(x=X,y=Y,
mode='lines',
line_color='#0000ee',
line_width=1.5,
showlegend=False)
])
title = "Animated construction of the Dragon curve,<br>through successive rotations"
fig.update_layout(title_text=title, title_x=0.5,
font=dict(family='Balto', size=16),
width=700, height=700,
xaxis_visible=False,
yaxis_visible=False,
xaxis_range=[-11, 6],
yaxis_range=[-11, 3],
#margin_l=40,
);
The frame 0 displays the initial vertical segment, as the dragon cuve defined in step 0 of the iterative process of construction.
alpha = pi/10 # The rotation of 90 degrees is defined as 5 successive rotations of 18 degrees=pi10 radians
n_rot90 = 13 # we have 13 steps
frames = []
for k in range(n_rot90):
#Record the last point on the dragon, defined in the previous step
x0, y0 = X[-1], Y[-1]
x = X-x0 #Translation with origin at (x0, y0) to be the center of rotation
y = Y-y0
for j in range(5):
X, Y = rotate_dragon(x, y, alpha=(j+1)*alpha)
X = np.concatenate((x[:-1], X[::-1]), axis=None) #concatenate to the (k-1)^th step dragon its rotated version
Y = np.concatenate((y[:-1], Y[::-1]), axis=None)
X = X+x0
Y = Y+y0
frames.append(go.Frame(data=[go.Scatter(x=X,y=Y)],
traces=[0]))
Define a button that triggers the animation:
buttonPlay = {'args': [None,
{'frame': {'duration': 100,
'redraw': False},
'transition': {'duration': 0},
'fromcurrent': True,
'mode': 'immediate'}],
'label': 'Play',
'method': 'animate'}
fig.update_layout(updatemenus=[{'buttons': [buttonPlay],
'showactive': False,
'type': 'buttons',
'x': 1,
'xanchor': 'left',
'y': 1,
'yanchor': 'top'
}])
fig.frames=frames
import chart_studio.plotly as py
py.iplot(fig, filename='rot-dragon1')
A gif file derived from this animation is posted on Wikimedia.