def plot_cell_mayavi(lines,
boundary=None,
clf=True,
smooth=True,
min_seg_length=5,
**kwargs):
import mayavi.mlab as may
may.clf()
hues = n.linspace(0, 1, len(lines) + 1)[:-1]
colours = [hsv_to_rgb(hue, 1, 1) for hue in hues]
random.shuffle(colours)
i = 0
for (line, colour) in zip(lines, colours):
vprint('Plotting line {} / {}\r'.format(i, len(lines) - 1), False)
i += 1
for segment in line:
if len(segment) < min_seg_length:
continue
plot_line(segment,
mode='mayavi',
clf=False,
color=colour,
**kwargs)
if boundary is not None:
draw_bounding_box_mayavi(boundary, **kwargs)
def octree_simplify(self,
runs=1,
plot=False,
rotate=True,
obey_knotting=False,
**kwargs):
'''
Simplifies the curves via the octree reduction of
:module:`pyknotid.simplify.octree`.
Parameters
----------
runs : int
The number of times to run the octree simplification.
Defaults to 1.
plot : bool
Whether to plot the curve after each run. Defaults to False.
rotate : bool
Whether to rotate the space curve before each run. Defaults
to True as this can make things much faster.
obey_knotting : bool
Whether to not let the line pass through itself. Defaults to
False - knotting of individual components will be ignored!
This is *much* faster than the alternative.
kwargs are passed to the :class:`pyknotid.simplify.octree.OctreeCell`
constructor.
'''
from ..simplify.octree import OctreeCell, remove_nearby_points
for line in self.lines:
line.points = remove_nearby_points(line.points)
for i in range(runs):
if n.sum([len(knot.points) for knot in self.lines]) > 30:
vprint(
'\rRun {} of {}, {} points remain'.format(
i, runs, len(self)), False, self.verbose)
if rotate:
rot_mat = get_rotation_matrix(n.random.random(3))
for line in self.lines:
line._apply_matrix(rot_mat)
oc = OctreeCell.from_lines([line.points for line in self.lines],
**kwargs)
oc.simplify(obey_knotting)
self._recent_octree = oc
self.lines = [Knot(line) for line in oc.get_lines()]
for line in self.lines:
line.points = remove_nearby_points(line.points)
if rotate:
for line in self.lines:
line._apply_matrix(rot_mat.T)
if plot:
self.plot()
vprint('\nReduced to {} points'.format(len(self)))
def plot_cell_vispy(lines,
boundary=None,
clf=True,
colours=None,
randomise_colours=True,
tube_radius=1.,
**kwargs):
if clf:
clear_vispy_canvas()
if colours is None:
hues = n.linspace(0, 1, len(lines) + 1)[:-1]
colours = [hsv_to_rgb(hue, 1, 1) for hue in hues]
if randomise_colours:
random.shuffle(colours)
elif not isinstance(colours, (list, tuple)):
colours = [colours for _ in lines]
i = 0
segments = []
segment_colours = []
segment_radii = []
if not isinstance(tube_radius, list):
tube_radius = [tube_radius for _ in range(len(lines))]
for (line, colour, radius) in zip(lines, colours, tube_radius):
vprint('Plotting line {} / {}\r'.format(i, len(lines) - 1), False)
i += 1
if len(colour) == 4 and colour[-1] == 0:
continue
for segment in line:
if len(segment) < 4:
continue
segments.append(segment)
segment_colours.append(colour)
segment_radii.append(radius)
# plot_line_vispy(segment,
# clf=False, colour=colour, **kwargs)
plot_lines_vispy(segments,
colours=segment_colours,
tube_radius=segment_radii,
clf=clf,
**kwargs)
if boundary is not None:
draw_bounding_box_vispy(boundary, tube_radius=tube_radius[0])
def writhing_numbers(gc, diagrams, based=False):
'''Returns the signed sum of representations of the given Arrow
diagrams in the given representation.
Parameters
----------
gc : A :class:`~pyknotid.representations.gausscode.GaussCode` or
equivalent representation.
The knot for which to find the writhes.
diagrams : str or list or tuple
A list of strings, or single string, representing Arrow
diagrams, e.g. '1-2+1+2-' for Vassiliev 2.
based : bool
Whether the diagrams have basepoints (if True, assumed to be
just before the first entry).
'''
if not isinstance(diagrams, (list, tuple)):
diagrams = [diagrams]
multipliers = defaultdict(lambda: {})
use_multipliers = False
if any(['d' in diagram for diagram in diagrams]):
use_multipliers = True
if use_multipliers:
for di, diagram in enumerate(diagrams):
terms = diagram.split(',')
for term in terms:
if term[-1] == 'd':
multiplier = 2
else:
multiplier = 1
term = int(term[:-2]) if 'd' in term else int(term[:-1])
multipliers[diagram.replace('d', '')][term] = multiplier
diagrams[di] = diagram.replace('d', '')
for d in diagrams:
validate_diagram(d)
level = 0
code = gc._gauss_code
code = code[0]
gc_len = len(gc)
code_len = len(code)
from pyknotid.invariants import _crossing_arrows_and_signs
arrows, signs = _crossing_arrows_and_signs(code, gc.crossing_numbers)
crossing_numbers = list(gc.crossing_numbers)
# degrees = [len(diagram.split(',')) for diagram in diagrams]
degrees = defaultdict(lambda: [])
for diagram in diagrams:
degrees[len(diagram.split(',')) // 2].append(diagram)
relations = {diagram: [] for diagram in diagrams}
for diagram in diagrams:
degree = len(diagram.split(',')) // 2
num_relations = factorial(degree - 1) * 4
terms = diagram.split(',')
numbers = [term[:-1] for term in terms]
number_strs = list(sorted(set(numbers), key=lambda j: int(j)))
for i, number in enumerate(number_strs):
for oi, other_number in enumerate(number_strs[i + 1:]):
oi += i + 1
if i != 0:
if terms.index(number + '-') < terms.index(other_number +
'-'):
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][0] < l[oi][0])
else:
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][0] > l[oi][0])
if terms.index(number + '-') < terms.index(other_number + '+'):
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][0] < l[oi][1])
else:
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][0] > l[oi][1])
if terms.index(number + '+') < terms.index(other_number + '-'):
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][1] < l[oi][0])
else:
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][1] > l[oi][0])
# This one seems to not be necessary for diagrams where all arrows cross?
if terms.index(number + '+') < terms.index(other_number + '+'):
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][1] < l[oi][1])
else:
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][1] > l[oi][1])
if i == 0:
continue
if terms.index(number + '+') < terms.index(number + '-'):
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][1] < l[i][0])
else:
relations[diagram].append(
lambda l, i=i, oi=oi: l[i][1] > l[i][0])
max_degree = max(degrees.keys())
representations_sums = {d: 0 for d in diagrams}
used_sets = {d: set() for d in diagrams}
combs = combinations(crossing_numbers, max_degree)
try:
num_combs = (factorial(len(crossing_numbers)) //
factorial(max_degree) //
factorial(len(crossing_numbers) - max_degree))
except ValueError:
num_combs = 0
strs = [None for _ in range(max_degree)] * 2
order = [None for _ in range(max_degree)] * 2
for ci, comb in enumerate(combs):
if ci % 10000 == 0:
vprint('\rCombination {} of {} '.format(ci + 1, num_combs),
newline=False,
condition=(ci % 10000) == 0)
if based:
perms = [comb]
else:
perms = permutations(comb)
ordered_indices = tuple(sorted(comb))
for diagram in diagrams:
if ordered_indices not in used_sets[diagram]:
break
else:
continue
for perm in perms:
cur_arrows = [list(arrows[i]) for i in perm]
a1s = cur_arrows[0][0]
if based:
a1s = 0
for i, arrow in enumerate(cur_arrows):
arrow[0] = (arrow[0] - a1s) % code_len
arrow[1] = (arrow[1] - a1s) % code_len
for diagram in diagrams:
if ordered_indices in used_sets[diagram]:
continue
for relation in relations[diagram]:
if not relation(cur_arrows):
break
else:
if use_multipliers:
representations_sums[diagram] += (reduce(
lambda x, y: x * y, [
signs[arrow_i]**multipliers[diagram][num + 1]
for num, arrow_i in enumerate(perm)
]))
else:
representations_sums[diagram] += (reduce(
lambda x, y: x * y,
[signs[arrow_i] for arrow_i in perm]))
used_sets[diagram].add(ordered_indices)
vprint()
return representations_sums
def _vprint(self, s, newline=True):
'''Prints s, with optional newline. Intended for internal use
in displaying progress.'''
if self.verbose:
vprint(s, newline)
def plot_sphere_mollweide_vispy(func,
circle_points=50,
depth=2,
edge_color=None,
cmap='hsv',
smooth=0,
mesh='circles',
**kwargs):
'''func must be a function of sphere angles theta, phi'''
# from vispy.scene import Sphere, ArcballCamera
from vispy.scene import TurntableCamera, Mesh
if mesh == 'circles':
vertices, indices = circles_ellipse_mesh(circle_points, depth)
else:
vertices, indices = recursive_ellipse_mesh(circle_points, depth)
vertices[:, 0] *= 2 * n.sqrt(2)
vertices[:, 1] *= n.sqrt(2)
mesh = Mesh(vertices, indices)
md = mesh._meshdata
vertices = md.get_vertices()
values = n.zeros(len(vertices))
print('pre')
thetas = []
phis = []
for i, vertex in enumerate(vertices):
if i % 10 == 0:
vprint('\ri = {} / {}'.format(i, len(vertices)), newline=False)
intermediate = n.arcsin(vertex[1] / n.sqrt(2))
theta = n.arcsin((2 * intermediate + n.sin(2 * intermediate)) / n.pi)
phi = n.pi * vertex[0] / (2 * n.sqrt(2) * n.cos(intermediate))
# theta = n.arccos(vertex[2])
# phi = n.arctan2(vertex[1], vertex[0])
if n.isnan(theta):
theta = 0.0
print('theta', vertex)
if n.isnan(phi):
phi = 0.0
print('phi', vertex)
thetas.append(theta)
phis.append(phi)
values[i] = func(theta + n.pi / 2, phi + n.pi)
vprint()
print('thetas', n.min(thetas), n.max(thetas))
print('phis', n.min(phis), n.max(phis))
colours = n.zeros((len(values), 4))
max_val = n.max(values)
min_val = n.min(values)
unique_values = n.unique(colours)
max_val += (1. + 1. / len(unique_values)) * (max_val - min_val)
diff = (max_val - min_val)
import matplotlib.pyplot as plt
cm = plt.get_cmap(cmap)
for i in range(len(colours)):
colours[i] = cm(((values[i] - min_val) / diff))
colours[:, -1] = 1.
faces = md.get_faces()
for si in range(smooth):
new_colours = [[n.array(row) for _ in range(3)] for row in colours]
for i, face in enumerate(faces):
new_colours[face[0]].append(colours[face[1]])
new_colours[face[0]].append(colours[face[2]])
new_colours[face[1]].append(colours[face[0]])
new_colours[face[1]].append(colours[face[2]])
new_colours[face[2]].append(colours[face[0]])
new_colours[face[2]].append(colours[face[1]])
new_colours = n.array([n.average(cs, axis=0) for cs in new_colours])
colours = new_colours
md.set_vertex_colors(colours)
ensure_vispy_canvas()
clear_vispy_canvas()
vispy_canvas.view.camera = TurntableCamera(fov=0, elevation=90.)
vispy_canvas.view.add(mesh)
vispy_canvas.show()
def plot_sphere_shell_vispy(func,
rows=100,
cols=100,
radius=1.,
opacity=1.,
translation=(0., 0., 0.),
method='latitude',
edge_color=None,
cmap='hsv',
smooth=0,
cutoff=0.4,
cutoff_max=0.8,
transparent_side=True,
**kwargs):
'''func must be a function of sphere angles theta, phi'''
from vispy.scene import Sphere, ArcballCamera
s = Sphere(rows=rows,
cols=cols,
method=method,
edge_color=edge_color,
radius=radius)
mesh = s.mesh
md = mesh._meshdata
vertices = md.get_vertices()
md.set_vertices(vertices + n.array(translation))
values = n.zeros(len(vertices))
opacities = n.ones(len(vertices))
print('pre')
for i, vertex in enumerate(vertices):
if i % 10 == 0:
vprint('\ri = {} / {}'.format(i, len(vertices)), newline=False)
vertex = vertex / n.sqrt(n.sum(vertex * vertex))
theta = n.arccos(vertex[2])
phi = n.arctan2(vertex[1], vertex[0])
if n.isnan(theta):
theta = 0.0
values[i] = func(theta, phi)
distance = vertex[1] - cutoff
distance_frac = distance / (cutoff_max - cutoff)
opacity = max(0, 1. - distance_frac)
opacities[i] = 1. if vertex[1] < cutoff else opacity
vprint()
colours = n.zeros((len(values), 4))
max_val = n.max(values)
min_val = n.min(values)
unique_values = n.unique(colours)
# max_val += (1. + 1./len(unique_values))*(max_val - min_val)
diff = (max_val - min_val)
import matplotlib.pyplot as plt
cm = plt.get_cmap(cmap)
for i in range(len(colours)):
colours[i] = cm(((values[i] - min_val) / diff))
colours[:, -1] = opacity
faces = md.get_faces()
for si in range(smooth):
new_colours = [[n.array(row) for _ in range(3)] for row in colours]
for i, face in enumerate(faces):
new_colours[face[0]].append(colours[face[1]])
new_colours[face[0]].append(colours[face[2]])
new_colours[face[1]].append(colours[face[0]])
new_colours[face[1]].append(colours[face[2]])
new_colours[face[2]].append(colours[face[0]])
new_colours[face[2]].append(colours[face[1]])
new_colours = n.array([n.average(cs, axis=0) for cs in new_colours])
colours = new_colours
colours = [(c[0], c[1], c[2], o) for c, o in zip(colours, opacities)]
md.set_vertex_colors(colours)
ensure_vispy_canvas()
vispy_canvas.view.camera = ArcballCamera(fov=30)
vispy_canvas.view.add(s)
vispy_canvas.show()
return colours
def plot_sphere_lambert_sharp_vispy(func,
circle_points=50,
depth=2,
output_size=500,
edge_color=None,
cmap='brg',
smooth=0,
mesh='circles',
**kwargs):
'''func must be a function of sphere angles theta, phi'''
from vispy.scene import TurntableCamera, Mesh
if mesh == 'circles':
vertices, indices = circles_ellipse_mesh(circle_points, depth)
else:
vertices, indices = recursive_ellipse_mesh(circle_points, depth)
vertices[:, 0] *= 2
vertices[:, 1] *= 2
mesh = Mesh(vertices, indices)
md = mesh._meshdata
vertices = md.get_vertices()
values = n.zeros(len(vertices))
print('pre')
thetas = []
phis = []
for i, vertex in enumerate(vertices):
if i % 10 == 0:
vprint('\ri = {} / {}'.format(i, len(vertices)), newline=False)
theta = 2 * np.arccos(np.sqrt(vertex[0]**2 + vertex[1]**2) / 2.)
phi = np.arctan2(vertex[1], vertex[0])
if n.isnan(theta):
theta = 0.0
print('theta', vertex)
if n.isnan(phi):
phi = 0.0
print('phi', vertex)
thetas.append(theta)
phis.append(phi)
values[i] = func(theta + n.pi / 2, phi + n.pi)
vprint()
return thetas, phis, values
import svgwrite as svg
d = svg.Drawing()
import matplotlib.pyplot as plt
cmap = plt.get_cmap(cmap)
max_value = n.max(values)
min_value = n.min(values)
def normalise(v):
return (v - min_value) / (max_value - min_value)
for tri_i, triangle in enumerate(indices):
v1 = vertices[triangle[0]]
v2 = vertices[triangle[1]]
v3 = vertices[triangle[2]]
c1 = values[triangle[0]]
c2 = values[triangle[1]]
c3 = values[triangle[2]]
offset_x = 2.001412
offset_y = 2.0214
v1_x = (v1[0] + offset_x) / 4. * output_size
v1_y = (v1[1] + offset_y) / 4. * output_size
v2_x = (v2[0] + offset_x) / 4. * output_size
v2_y = (v2[1] + offset_y) / 4. * output_size
v3_x = (v3[0] + offset_x) / 4. * output_size
v3_y = (v3[1] + offset_y) / 4. * output_size
c_x = n.average([v1_x, v2_x, v3_x])
c_y = n.average([v1_y, v2_y, v3_y])
v12_x = (v1_x + v2_x) / 2.
v12_y = (v1_y + v2_y) / 2.
v13_x = (v1_x + v3_x) / 2.
v13_y = (v1_y + v3_y) / 2.
v23_x = (v2_x + v3_x) / 2.
v23_y = (v2_y + v3_y) / 2.
cmap_1 = cmap(normalise(c1))
rgb_cmap1 = [int(c * 255) for c in cmap_1[:3]]
d.add(
d.polygon(
[[v1_x, v1_y], [v12_x, v12_y], [c_x, c_y], [v13_x, v13_y]],
fill='rgb({},{},{})'.format(*rgb_cmap1),
stroke='rgb({},{},{})'.format(*rgb_cmap1),
stroke_width='0.5'))
cmap_2 = cmap(normalise(c2))
rgb_cmap2 = [int(c * 255) for c in cmap_2[:3]]
d.add(
d.polygon(
[[v2_x, v2_y], [v12_x, v12_y], [c_x, c_y], [v23_x, v23_y]],
fill='rgb({},{},{})'.format(*rgb_cmap2),
stroke='rgb({},{},{})'.format(*rgb_cmap2),
stroke_width='0.5'))
cmap_3 = cmap(normalise(c3))
rgb_cmap3 = [int(c * 255) for c in cmap_3[:3]]
d.add(
d.polygon(
[[v3_x, v3_y], [v13_x, v13_y], [c_x, c_y], [v23_x, v23_y]],
fill='rgb({},{},{})'.format(*rgb_cmap3),
stroke='rgb({},{},{})'.format(*rgb_cmap3),
stroke_width='0.5'))
# print('cols', c1, cmap_1, c2, cmap_2, c3, cmap_3)
return d