Interactive fun
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Inspired by:
Lattice + LLL + Fundamental mesh plot
@interact
def draw_lattice(v1x = input_box(label = "v1 x =", default = 1),
v1y = input_box(label = "v1 y =", default = 0),
v2x = input_box(label = "v2 x =", default = 0),
v2y = input_box(label = "v2 y =", default = 1),
box = 5, search = 10,
plot_LLL = True,
plot_C = True,
plot_F = True):
v1 = vector((v1x, v1y))
v2 = vector((v2x, v2y))
vecs = []
# Generate vectors
for i in range(-search,search):
for j in range(-search,search):
vecs.append(i*v1 + j*v2)
# Plot stuff
G = Graphics()
for p1 in vecs:
x1, y1 = p1
if x1 > -box and x1 < box and y1 > -box and y1 < box:
G += point(p1, color = 'green', size = 30)
G += line([p1, p1 + v2], linestyle = '--', alpha = .20)
G += line([p1, p1 + v1], linestyle = '--', alpha = .20)
G+= arrow((0, 0), v1, color = 'red', arrowsize = 2)
G+= arrow((0, 0), v2, color = 'red', arrowsize = 2)
G+= text('v1', v1 + .2 * v1, color = 'red')
G+= text('v2', v2 + .2 * v2, color = 'red')
# LLL
if plot_LLL:
v1_, v2_ = matrix([v1, v2]).LLL()
G+= arrow((0, 0), v1_, color = 'purple', arrowsize = 2)
G+= arrow((0, 0), v2_, color = 'purple', arrowsize = 2)
if plot_C:
G += circle(center = (0, 0), radius = norm(v1_) if norm(v1_) > norm(v2_) else norm(v2_), alpha = .5, color = 'purple')
# Fundamental mesh
if plot_F:
F = polygon([[0, 0], v1, v1 + v2, v2], color='red', alpha = .1)
G += F
G.show(axes = False, figsize = (7, 7))
@interact
def draw_cvp(v1x = input_box(label = "v1 x =", default = 1),
v1y = input_box(label = "v1 y =", default = 0),
v2x = input_box(label = "v2 x =", default = 0),
v2y = input_box(label = "v2 y =", default = 1),
wx = input_box(label = "w x =", default = 1.8),
wy = input_box(label = "w y =", default = 1.7),
box = 5, search = 10):
v1 = vector((v1x, v1y))
v2 = vector((v2x, v2y))
print(v1, v2)
vecs = []
# Generate vectors
for i in range(-search,search):
for j in range(-search,search):
vecs.append(i*v1 + j*v2)
# Plot stuff
G = Graphics()
for p1 in vecs:
x1, y1 = p1
if x1 > -box and x1 < box and y1 > -box and y1 < box:
G += point(p1, color = 'green', size = 30)
G += line([p1, p1 + v2], linestyle = '--', alpha = .20)
G += line([p1, p1 + v1], linestyle = '--', alpha = .20)
G+= arrow((0, 0), v1, color = 'red', arrowsize = 2)
G+= arrow((0, 0), v2, color = 'red', arrowsize = 2)
G+= text('v1', v1 + .2 * v1, color = 'red')
G+= text('v2', v2 + .2 * v2, color = 'red')
# Cvp
L = IntegerLattice(matrix([v1, v2]))
w = vector((wx, wy))
t = L.closest_vector(w)
G += point(w, color = 'purple', size = 30)
G+= text('w', w + .2 * v1, color = 'purple')
G += point(t, color = 'red', size = 30)
G+= text('t', t + .2 * v1, color = 'red')
G+= circle(center = w, radius=norm(t - w), color = 'purple', alpha = .5)
G.show(axes = False, figsize = (7, 7))@interact
def draw_dual(v1x = input_box(label = "v1 x =", default = 1),
v1y = input_box(label = "v1 y =", default = 0),
v2x = input_box(label = "v2 x =", default = 0),
v2y = input_box(label = "v2 y =", default = 1),
box = 3, search = 6,
plot_lattice_points = True,
plot_lattice_lines = True,
plot_dual_points = True,
plot_dual_lines = True):
v1 = vector((v1x, v1y))
v2 = vector((v2x, v2y))
vecs = []
v1_, v2_ = matrix([v1,v2]).inverse().T
vecs_ = []
# Generate vectors
for i in range(-search,search):
for j in range(-search,search):
vecs.append(i*v1 + j*v2)
for i in range(-search,search):
for j in range(-search,search):
vecs_.append(i*v1_ + j*v2_)
# Plot stuff
G = Graphics()
for p1 in vecs:
x1, y1 = p1
if x1 > -box and x1 < box and y1 > -box and y1 < box:
if plot_lattice_points:
G += point(p1, color = 'green', size = 70)
if plot_lattice_lines:
G += line([p1, p1 + v2], linestyle = '--', alpha = .20, color = 'green')
G += line([p1, p1 + v1], linestyle = '--', alpha = .20, color = 'green')
if plot_lattice_lines:
G+= arrow((0, 0), v1, color = 'green', arrowsize = 2)
G+= arrow((0, 0), v2, color = 'green', arrowsize = 2)
G+= text('v1', v1 + .2 * v1, color = 'green')
G+= text('v2', v2 + .2 * v2, color = 'green')
# Dual
for p1 in vecs_:
x1, y1 = p1
if x1 > -box and x1 < box and y1 > -box and y1 < box:
if plot_dual_points:
G += point(p1, color = 'red', size = 25)
if plot_dual_lines:
G += line([p1, p1 + v2_], linestyle = '--', alpha = .20, color = 'red')
G += line([p1, p1 + v1_], linestyle = '--', alpha = .20, color = 'red')
if plot_dual_lines:
G+= arrow((0, 0), v1_, color = 'red', arrowsize = 2)
G+= arrow((0, 0), v2_, color = 'red', arrowsize = 2)
G+= text('v1\'', v1_ + .2 * v1, color = 'red')
G+= text('v2\'', v2_ + .2 * v2, color = 'red')
G.show(axes = False, figsize = (7, 7))Q-ary plots
# I'm not sure this does what is supposed to do
# but the plots are nice
from sage.modules.free_module_integer import IntegerLattice
@interact
def draw_qary(v1x = input_box(label = "v1 x =", default = 1),
v1y = input_box(label = "v1 y =", default = 0),
v2x = input_box(label = "v2 x =", default = 0),
v2y = input_box(label = "v2 y =", default = 1),
q = input_box(label = "q =", default = 5),
box = 7, search = 6,
plot_lattice_points = True,
plot_lattice_lines = True,
plot_qary_points = True,
plot_qary_lines = True):
v1 = vector((v1x, v1y))
v2 = vector((v2x, v2y))
L = IntegerLattice(matrix([v1, v2]))
vecs = []
v1_ = v1.change_ring(Zmod(q))
v2_ = v2.change_ring(Zmod(q))
vecs_ = []
# Generate vectors
for i in range(-search,search):
for j in range(-search,search):
v = i*v1 + j*v2
vecs.append(v)
v = v.change_ring(Zmod(q)).change_ring(ZZ)
if v not in vecs_:
vecs_.append(v)
# Lattice
G = Graphics()
for p1 in vecs:
x1, y1 = p1
if x1 > -box and x1 < box and y1 > -box and y1 < box:
if plot_lattice_points:
G += point(p1, color = 'green', size = 70)
if plot_lattice_lines:
G += line([p1, p1 + v2], linestyle = '--', alpha = .20, color = 'green')
G += line([p1, p1 + v1], linestyle = '--', alpha = .20, color = 'green')
if plot_lattice_lines:
G+= arrow((0, 0), v1, color = 'green', arrowsize = 2)
G+= arrow((0, 0), v2, color = 'green', arrowsize = 2)
G+= text('v1', v1 + .2 * v1, color = 'green')
G+= text('v2', v2 + .2 * v2, color = 'green')
# qary
for p1 in vecs_:
p1 = p1
x1, y1 = p1
if x1 > -box and x1 < box and y1 > -box and y1 < box:
if plot_qary_points:
G += point(p1, color = 'red', size = 25)
if plot_qary_lines:
G += line([p1, p1 + v2_], linestyle = '--', alpha = .20, color = 'red')
G += line([p1, p1 + v1_], linestyle = '--', alpha = .20, color = 'red')
if plot_qary_lines:
G+= arrow((0, 0), v1_, color = 'purple', arrowsize = 2)
G+= arrow((0, 0), v2_, color = 'purple', arrowsize = 2)
G+= text('v1\'', v1_.change_ring(ZZ) + .2 * v1, color = 'purple')
G+= text('v2\'', v2_.change_ring(ZZ) + .2 * v2, color = 'purple')
G.show(axes = False, figsize = (10, 10))
