Inspired by David Richeson's applet on GeoGebra the Python script below plots the lines joining pairs of points, evenly spaced on the perimeter of a circle according to the following algorithm: label the \(n\) points \(k=0, 1, 2, \ldots n-1\); then join point \(k\) to point \(mk\,\mathrm{mod}\,n\).
The line segments inside and outside the circle can be plotted in different colours to make pleasing images. Pass \(m\) and \(n\) as arguments on the command line, e.g.
python circlines.py 82 110
python circlines.py 20 40
import sys
import numpy as np
import matplotlib.pyplot as plt
def add_lines(ax, m, n, line_segments=True, c='w', lw=0.5, *args, **kwargs):
ax.add_patch(plt.Circle((0, 0), 1, fc='none', ec='w', lw=2))
# Pop off oc ("outer line colour") if presentm otherwise set to the
# same as the inside colour.
oc = kwargs.pop('oc', c)
dtheta = 2 * np.pi / n
theta = np.arange(0, 2 * np.pi, dtheta)
pts = np.array([np.cos(theta), np.sin(theta)]).T
def get_bounded_coords(X, x1, y1, g):
"""
Get the coordinates of the intersection of the line through
(x1, y1) with gradient g at with x-coordinate X, unless this
has a y-coordinate outside the range (YMIN, YMAX), in which case
set y appropriately and determine the correspinding x-coordinate.
"""
def get_x_at_y(Y):
"""Return the x-coordinate at y=Y for the line."""
return (Y - y1) / g + x1
yA = y1 + g * (X - x1)
if yA > YMAX:
return get_x_at_y(YMAX), YMAX
elif yA < YMIN:
return get_x_at_y(YMIN), YMIN
return X, yA
for k, pt1 in enumerate(pts):
pt2 = pts[m * k % n]
x1, y1 = pt1
x2, y2 = pt2
if x1 > x2:
x1, y1, x2, y2 = x2, y2, x1, y1
if line_segments:
# Just join the line segments inside the circle.
plt.plot([x1, x2], [y1, y2], c=c, lw=lw, **kwargs)
continue
if x1 == x2:
# Vertical lines are a special case.
xA, xB, yA, yB = x1, x2, YMIN, YMAX
else:
# The gradient of the line.
g = (y2 - y1) / (x2 - x1)
# Get the start and end points of the line on the boundary
# rectangle, (XMIN, YMIN), (XMAX, YMAX).
xA, yA = get_bounded_coords(XMIN, x1, y1, g)
xB, yB = get_bounded_coords(XMAX, x1, y1, g)
if oc == c:
# Same colour inside and outside the circle.
plt.plot([xA, xB], [yA, yB], c=c, lw=lw, **kwargs)
else:
plt.plot([xA, x1], [yA, y1], c=oc, lw=lw, **kwargs)
plt.plot([x1, x2], [y1, y2], c=c, lw=lw, **kwargs)
plt.plot([x2, xB], [y2, yB], c=oc, lw=lw, **kwargs)
if __name__ == '__main__':
m, n = map(int, sys.argv[1:3])
XMIN, XMAX, YMIN, YMAX = -4, 4, -3, 3
fig, ax = plt.subplots(facecolor='k')
add_lines(ax, m, n, c='y', line_segments=False, oc='r')
ax.set_xlim(XMIN, XMAX)
ax.set_ylim(YMIN, YMAX)
ax.axis('off')
plt.show()