def poly_plot(poly, size=[5, 5], show_plot=True):
"""Automatically make a plot that shows the roots and critical points of the polynomial"""
if type(poly) == ZPoly:
if poly.M:
pts = zpoly_plot(poly)
return pts
else:
poly = QPoly(poly.coef)
if type(poly) == QPoly:
r = qpoly_roots(poly)
s = stationary_points(poly)
c = inflection_points(poly)
interesting_points = r + s + c
elif type(poly) == RFunc:
r = rfunc_roots(poly)
a = rfunc_asymptotes(poly)
interesting_points = r + a
else:
raise TypeError("Input must be QPoly, RFunc, or ZPoly")
interesting_values = poly.evaluate(interesting_points)
# Pick the minimum possible axes for the graph
xlimit = [min(interesting_points), max(interesting_points)]
ylimit = [min(interesting_values), max(interesting_values)]
# Choose how much to pad
xmargin = max(abs(xlimit[0] - xlimit[1]) / 10, 1)
ymargin = max(abs(ylimit[0] - ylimit[1]) / 10, 1)
# create the actual axes
xwidth = [float(xlimit[0] - xmargin), float(xlimit[1] + xmargin)]
ywidth = [float(ylimit[0] - ymargin), float(ylimit[1] + ymargin)]
step_size = abs(xwidth[0] - xwidth[1]) / 100
x = rational_seq(xwidth[0], xwidth[1], step_size)
y = [float(i) for i in poly.evaluate(x)]
x = [float(i) for i in x]
pts = [i for i in zip(x, y)]
if show_plot:
make_canvas(xwidth,
ywidth,
size=size,
show_axes=True,
title=poly.pretty_name)
plot_points(pts)
connect([xwidth[0], 0], [xwidth[1], 0], color="gray", zorder=-1)
return [(a, b) for a, b in zip(x, y)]
def zpoly_plot(poly,size=[5,5]):
"""Automatically make a plot that shows the roots and critical points of the polynomial"""
M = poly.M
xwidth = [-1,M+1]
ywidth = [-1,M+1]
pts = zpoly_points(poly)
make_canvas(xwidth,ywidth,size=size,show_axes=True,title=poly.pretty_name)
scatter_points(pts)
return pts
b = b0
P = [[0, 0]]
old = [0, 0]
for x, y in [(a * 2, 0), (b, b), (0, a * 2), (-b, b), (-a * 2, 0),
(-b, -b), (0, -2 * a)]:
new = [old[0] + x, old[1] + y]
P.append(new)
old = new
return P
if __name__ == '__main__':
from Utility import make_canvas, plot_points, connect
make_canvas([-10, 10], size=6, show_axes=False)
for xy in range(-10, 11):
connect([-10, xy], [10, xy], color='gray', linewidth=.5)
connect([xy, -10], [xy, 10], color='gray', linewidth=.5)
for i in range(3):
pts = integer_octagon(i)
# Shift to center
x_mn = sum([i[0] for i in pts]) / 8
y_mn = sum([i[1] for i in pts]) / 8
pts = [[x - x_mn, y - y_mn] for x, y in pts]
plot_points(pts, color='k')
connect(pts[0], pts[-1], color='k')
def prim_pythag_triples(N):
"""N is the limit of m and n in Euclid's formula"""
for m in range(1, N):
for n in range(m + 1, N):
if gcd(n, m) == 1: # must be coprime
if not (n % 2 == 1 and m % 2 == 1): # can't both be odd
a = n * n - m * m
b = 2 * m * n
c = m * m + n * n
yield sorted([a, b, c])
if __name__ == '__main__':
from Utility import make_canvas, scatter_points
for i in prim_pythag_triples(10):
print(i)
make_canvas([-1.2, 1.2], size=6, title="Rational Points on a Circle")
pts = []
for i in prim_pythag_triples(11):
pts += [[i[0] / i[2], i[1] / i[2]]]
pts += [[-i[0] / i[2], i[1] / i[2]]]
pts += [[i[0] / i[2], -i[1] / i[2]]]
pts += [[-i[0] / i[2], -i[1] / i[2]]]
scatter_points(pts, s=2)
ctr += 1
k *= ctr
yield out
if __name__ == '__main__':
from Utility import make_canvas, scatter_points, show_plot
print("Farey Sequence F_12")
F12 = [i for i in farey_sequence(12)]
print(F12)
print()
xy = question_mark_func(25)
make_canvas(
[-.02, 1.02],
size=6,
title="Minkowski's Question-mark Function\nAt Rational Arguments")
scatter_points(xy, s=1)
show_plot()
print("\n\nHarmonic Progressions")
S1 = harmonic_progression(a=1, d=1)
for pos, val in enumerate(S1):
if pos > 10:
break
print(val)
print("\n\nA Harmonic Progression")
S2 = harmonic_progression(a="1/3", d="2/9")
for pos, val in enumerate(S2):
if pos > 10: