380 lines
12 KiB
Python
380 lines
12 KiB
Python
#!/usr/bin/env sage -python
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from collections import Counter
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import matplotlib.pyplot as plt
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import inspect
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from PIL import Image
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from pathlib import Path
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import warnings
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from .utility import mod_one
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# 9.11 (9.8)
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# 9.15 (9.9)
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import sys
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import os
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JUPYTER = 'ipykernel'
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IPy_TERMINAL = 'IPython'
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def get_ipython_info():
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if JUPYTER in sys.modules:
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return JUPYTER
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elif IPy_TERMINAL in sys.modules:
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return IPy_TERMINAL
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return False
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global ipython_info
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ipython_info = get_ipython_info()
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class SignatureFunction:
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def __init__(self, values=None, counter=None, plot_title=''):
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# counter of signature jumps
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if counter is None:
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counter = Counter()
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values = values or []
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for k, v in values:
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counter[k] += v
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counter = Counter({k : v for k, v in counter.items() if v != 0})
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if any(k >= 1 for k in counter.keys()):
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msg = "Signature function is defined on the interval [0, 1)."
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raise ValueError(msg)
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counter[0] += 0
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counter[1] += 0
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self.jumps_counter = counter
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self.plot_title = plot_title
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def __rshift__(self, shift):
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# A shift of the signature functions corresponds to the rotation.
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counter = Counter({mod_one(k + shift) : v \
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for k, v in self.jumps_counter.items()})
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return SignatureFunction(counter=counter)
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def __lshift__(self, shift):
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return self.__rshift__(-shift)
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def __neg__(self):
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counter = Counter()
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counter.subtract(self.jumps_counter)
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return SignatureFunction(counter=counter)
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def __add__(self, other):
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counter = copy(self.jumps_counter)
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counter.update(other.jumps_counter)
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if self.plot_title and other.plot_title:
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title = self.plot_title + " + " + other.plot_title
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else:
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title = self.plot_title or other.plot_title
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return SignatureFunction(counter=counter, plot_title=title)
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def __sub__(self, other):
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counter = copy(self.jumps_counter)
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counter.subtract(other.jumps_counter)
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return SignatureFunction(counter=counter)
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def __mul__(self, number):
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# scalar multiplication
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counter = Counter({k : number * v \
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for k, v in self.jumps_counter.items()})
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return SignatureFunction(counter=counter)
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def __rmul__(self, number):
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return(self.__mul__(number))
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def __eq__(self, other):
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return self.jumps_counter == other.jumps_counter
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def __str__(self):
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result = ''.join([str(jump_arg) + ": " + str(jump) + "\n"
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for jump_arg, jump in sorted(self.jumps_counter.items())])
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return result
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def __repr__(self):
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result = ''.join([str(jump_arg) + ": " + str(jump) + ", "
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for jump_arg, jump in sorted(self.jumps_counter.items())])
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return result[:-2] + "."
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def __call__(self, arg):
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# return the value of the signature function at the point arg, i.e.
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# sum of all signature jumps that occur before arg
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items = self.jumps_counter.items()
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result = [jump for jump_arg, jump in items if jump_arg < mod_one(arg)]
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return 2 * sum(result) + self.jumps_counter[arg]
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def double_cover(self):
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# to read values for t^2
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items = self.jumps_counter.items()
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counter = Counter({(1 + k) / 2 : v for k, v in items})
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counter.update(Counter({k / 2 : v for k, v in items}))
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return SignatureFunction(counter=counter)
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def square_root(self):
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# to read values for t^(1/2)
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counter = Counter()
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for jump_arg, jump in self.jumps_counter.items():
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if jump_arg < 1/2:
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counter[2 * jump_arg] = jump
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return SignatureFunction(counter=counter)
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def minus_square_root(self):
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# to read values for t^(1/2)
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items = self.jumps_counter.items()
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counter = Counter({mod_one(2 * k) : v for k, v in items if k >= 1/2})
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return SignatureFunction(counter=counter)
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def is_zero_everywhere(self):
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return not any(self.jumps_counter.values())
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def extremum(self, limit=math.inf):
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max_point = (0, 0)
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current = 0
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items = sorted(self.jumps_counter.items())
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for arg, jump in items:
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current += 2 * jump
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assert current == self(arg) + jump
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if abs(current) > abs(max_point[1]):
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max_point = (arg, current)
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if abs(current) > limit:
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break
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return max_point
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def total_sign_jump(self):
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# Total signature jump is the sum of all jumps.
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return sum([j[1] for j in sorted(self.jumps_counter.items())])
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def plot(self, *args, **kargs):
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SignaturePloter.plot(self, *args, **kargs)
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class SignaturePloter:
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@classmethod
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def plot_many(cls, *sf_list, save_path=None, title='', cols=None):
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axes_num = len(sf_list)
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if axes_num > 36:
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sf_list = sf_list[36]
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axes_num = 36
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msg = "To many functions for the plot were given. "
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msg += "Only 36 can be plotted "
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warnings.warn(msg)
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# print war, set val in conf
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cols = cols or ceil(sqrt(axes_num))
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rows = ceil(axes_num/cols)
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fig, axes_matrix = plt.subplots(rows, cols,
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sharex='col', sharey='row',
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gridspec_kw={'hspace': 0, 'wspace': 0},
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# sharey=True,
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# sharex=True,
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)
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for i, sf in enumerate(sf_list):
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col = i % cols
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row = (i - col)/cols
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sf.plot(subplot=True,
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ax=axes_matrix[row][col],
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title=sf.plot_title)
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fig.suptitle(title)
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plt.tight_layout()
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cls.show_and_save(save_path)
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@classmethod
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def plot_sum_of_two(cls, sf1, sf2, save_path=None, title=''):
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sf = sf1 + sf2
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fig, axes_matrix = plt.subplots(2, 2, sharey=True, figsize=(10,5))
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sf1.plot(subplot=True,
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ax=axes_matrix[0][1])
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sf2.plot(subplot=True,
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ax=axes_matrix[1][0],
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color='red',
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linestyle='dotted')
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sf.plot(subplot=True,
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ax=axes_matrix[0][0],
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color='black')
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sf1.plot(subplot=True,
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ax=axes_matrix[1][1],
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alpha=0.3)
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sf2.plot(subplot=True,
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ax=axes_matrix[1][1],
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color='red', alpha=0.3,
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linestyle='dotted')
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sf.plot(subplot=True,
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ax=axes_matrix[1][1],
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color='black',
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alpha=0.7,)
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fig.suptitle(title)
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plt.tight_layout()
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cls.show_and_save(save_path)
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@classmethod
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def plot(cls, sf, subplot=False, ax=None,
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save_path=None,
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title='',
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alpha=1,
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color='blue',
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linestyle='solid',
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special_point=None,
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special_label='',
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extraticks=None,
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ylabel=''):
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if ax is None:
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fig, ax = plt.subplots(1, 1)
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keys = sorted(sf.jumps_counter.keys())
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y = [(sf(k) + sf.jumps_counter[k]) for k in keys[:-1]]
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xmax = keys[1:]
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xmin = keys[:-1]
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ax.set(ylabel=ylabel)
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ax.set(title=title)
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ax.hlines(y, xmin, xmax, color=color, linestyle=linestyle, alpha=alpha)
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if special_point is not None:
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arg, val = special_point
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extraticks = extraticks or []
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plt.xticks(list(plt.xticks()[0]) + extraticks)
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ext = sf.extremum()[1]
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ytext = ext/2 + 1/2
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xtext = arg + 1/5
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ax.annotate(special_label, xy=(arg, val), xytext=(xtext, ytext),
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arrowprops=dict(facecolor='black', shrink=0.05,
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alpha=0.7, width=2),)
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if subplot:
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return ax
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cls.show_and_save(save_path)
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@staticmethod
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def show_and_save(save_path):
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if save_path is not None:
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save_path = Path(save_path)
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save_path = save_path.with_suffix('.png')
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plt.savefig(save_path)
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if ipython_info == JUPYTER:
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plt.show()
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elif True: # save_path is None:
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plt.savefig('tmp.png')
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plt.close()
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image = Image.open('tmp.png')
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image.show()
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# msg = "For interactive shell set save_path."
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# warnings.warn(msg)
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@staticmethod
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def step_function_data(sf):
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# Transform the signature jump data to a format understandable
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# by the plot function.
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result = [(k, sf.sf(k) + sf.jumps_counter[k])
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for k in sorted(sf.jumps_counter.keys())]
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return result
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@staticmethod
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def tikz_plot(sf, save_as):
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plt_sin = plot(sin(x), (x, 0, 2*pi))
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# plt_sin.show()
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plt_sin.save("MyPic.pdf")
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return
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# Draw the graph of the signature and transform it into TiKz.
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# header of the LaTeX file
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head = inspect.cleandoc(
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r"""
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\documentclass{standalone}
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\usepackage{tikz}
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\usetikzlibrary{calc}
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\begin{document}
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\begin{tikzpicture}
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""")
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body = \
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r"""
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%A piecewise linear function is drawn over the interval.
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\draw (5,0) -- (6,-4);
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%The axes are drawn.
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\draw[latex-latex] ($(0,{-4*(2/5)}) +(0pt,-12.5pt)$) --
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($(0,{4*(2/5)}) +(0pt,12.5pt)$) node[above right]{$y$};
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\draw[latex-latex] ($({-4*(2/5)},0) +(-12.5pt,0pt)$) --
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($({12*(2/5)},0) +(12.5pt,0pt)$) node[below right]{$x$};
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"""
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tail = \
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r"""
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\end{tikzpicture}
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\end{document}
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"""
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tikzpicture = re.sub(r' +', ' ', ''.join([head, body, tail]))
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tikzpicture = re.sub(r'\n ', '\n', tikzpicture)
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with open("tmp.tex", "w") as f:
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f.write(tikzpicture)
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data = self.step_function_data()
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with open(save_as, "w") as f:
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head = \
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r"""
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\documentclass[tikz]{{standalone}}
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%\usepackage{{tikz}}
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\usetikzlibrary{{datavisualization}}
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\usetikzlibrary{{datavisualization.formats.functions}}
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%\usetikzlibrary{{calc}}
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\begin{{document}}
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\begin{{tikzpicture}}
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\datavisualization[scientific axes, visualize as smooth line,
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x axis={{ticks={{none,major={{at={{, {arg0} " as \\( {val0} \\
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%]
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""".format(arg0=str(N(data[0][0] ,digits=4)), val0=str(data[0][0]))
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f.write(head)
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# f.write(", " + str(N(data[0][0],digits=4)) + " as \\(" + \
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# str(data[0][0]) + "\\)")
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for jump_arg, jump in data[1:3]:
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f.write(", " + str(N(jump_arg,digits=4)) +
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" as \\(" + str(jump_arg) + "\\)")
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f.write("}}}}\n")
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f.write(" ]\n")
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f.write("data [format=function]{\n")
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f.write("var x : interval [0:1];\n")
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f.write("func y = \\value x;\n")
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f.write("};\n")
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# close LaTeX enviroments
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tail = \
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r"""
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%};
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\end{tikzpicture}
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\end{document}
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"""
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f.write(tail)
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SignatureFunction.__doc__ = \
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"""
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This simple class encodes twisted and untwisted signature functions
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of knots. Since the signature function is entirely encoded by its signature
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jump, the class stores only information about signature jumps
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in a dictionary self.jumps_counter.
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The dictionary stores data of the signature jump as a key/values pair,
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where the key is the argument at which the functions jumps
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and value encodes the value of the jump. Remember that we treat
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signature functions as defined on the interval [0,1).
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"""
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