# Source code for sympy.plotting.plot_implicit

"""Implicit plotting module for SymPy

The module implements a data series called ImplicitSeries which is used by
Plot class to plot implicit plots for different backends. The module,
by default, implements plotting using interval arithmetic. It switches to a
fall back algorithm if the expression cannot be plotted using interval arithmetic.
It is also possible to specify to use the fall back algorithm for all plots.

Boolean combinations of expressions cannot be plotted by the fall back
algorithm.

========
sympy.plotting.plot

References
==========
- Jeffrey Allen Tupper. Reliable Two-Dimensional Graphing Methods for
Mathematical Formulae with Two Free Variables.

- Jeffrey Allen Tupper. Graphing Equations with Generalized Interval
Arithmetic. Master's thesis. University of Toronto, 1996

"""

from __future__ import print_function, division

from .plot import BaseSeries, Plot
from .experimental_lambdify import experimental_lambdify, vectorized_lambdify
from .intervalmath import interval
from sympy.core.relational import (Equality, GreaterThan, LessThan,
Relational, StrictLessThan, StrictGreaterThan)
from sympy import Eq, Tuple, sympify, Symbol, Dummy
from sympy.external import import_module
from sympy.logic.boolalg import BooleanFunction
from sympy.polys.polyutils import _sort_gens
from sympy.utilities.decorator import doctest_depends_on
from sympy.utilities.iterables import flatten
import warnings

[docs]class ImplicitSeries(BaseSeries):
""" Representation for Implicit plot """
is_implicit = True

def __init__(self, expr, var_start_end_x, var_start_end_y,
has_equality, use_interval_math, depth, nb_of_points,
line_color):
super(ImplicitSeries, self).__init__()
self.expr = sympify(expr)
self.var_x = sympify(var_start_end_x[0])
self.start_x = float(var_start_end_x[1])
self.end_x = float(var_start_end_x[2])
self.var_y = sympify(var_start_end_y[0])
self.start_y = float(var_start_end_y[1])
self.end_y = float(var_start_end_y[2])
self.get_points = self.get_raster
self.has_equality = has_equality  # If the expression has equality, i.e.
#Eq, Greaterthan, LessThan.
self.nb_of_points = nb_of_points
self.use_interval_math = use_interval_math
self.depth = 4 + depth
self.line_color = line_color

def __str__(self):
return ('Implicit equation: %s for '
'%s over %s and %s over %s') % (
str(self.expr),
str(self.var_x),
str((self.start_x, self.end_x)),
str(self.var_y),
str((self.start_y, self.end_y)))

def get_raster(self):
func = experimental_lambdify((self.var_x, self.var_y), self.expr,
use_interval=True)
xinterval = interval(self.start_x, self.end_x)
yinterval = interval(self.start_y, self.end_y)
try:
temp = func(xinterval, yinterval)
except AttributeError:
if self.use_interval_math:
warnings.warn("Adaptive meshing could not be applied to the"
" expression. Using uniform meshing.")
self.use_interval_math = False

if self.use_interval_math:
return self._get_raster_interval(func)
else:
return self._get_meshes_grid()

def _get_raster_interval(self, func):
""" Uses interval math to adaptively mesh and obtain the plot"""
k = self.depth
interval_list = []
#Create initial 32 divisions
np = import_module('numpy')
xsample = np.linspace(self.start_x, self.end_x, 33)
ysample = np.linspace(self.start_y, self.end_y, 33)

#Add a small jitter so that there are no false positives for equality.
# Ex: y==x becomes True for x interval(1, 2) and y interval(1, 2)
#which will draw a rectangle.
jitterx = (np.random.rand(
len(xsample)) * 2 - 1) * (self.end_x - self.start_x) / 2**20
jittery = (np.random.rand(
len(ysample)) * 2 - 1) * (self.end_y - self.start_y) / 2**20
xsample += jitterx
ysample += jittery

xinter = [interval(x1, x2) for x1, x2 in zip(xsample[:-1],
xsample[1:])]
yinter = [interval(y1, y2) for y1, y2 in zip(ysample[:-1],
ysample[1:])]
interval_list = [[x, y] for x in xinter for y in yinter]
plot_list = []

#recursive call refinepixels which subdivides the intervals which are
#neither True nor False according to the expression.
def refine_pixels(interval_list):
""" Evaluates the intervals and subdivides the interval if the
expression is partially satisfied."""
temp_interval_list = []
plot_list = []
for intervals in interval_list:

#Convert the array indices to x and y values
intervalx = intervals[0]
intervaly = intervals[1]
func_eval = func(intervalx, intervaly)
#The expression is valid in the interval. Change the contour
#array values to 1.
if func_eval[1] is False or func_eval[0] is False:
pass
elif func_eval == (True, True):
plot_list.append([intervalx, intervaly])
elif func_eval[1] is None or func_eval[0] is None:
#Subdivide
avgx = intervalx.mid
avgy = intervaly.mid
a = interval(intervalx.start, avgx)
b = interval(avgx, intervalx.end)
c = interval(intervaly.start, avgy)
d = interval(avgy, intervaly.end)
temp_interval_list.append([a, c])
temp_interval_list.append([a, d])
temp_interval_list.append([b, c])
temp_interval_list.append([b, d])
return temp_interval_list, plot_list

while k >= 0 and len(interval_list):
interval_list, plot_list_temp = refine_pixels(interval_list)
plot_list.extend(plot_list_temp)
k = k - 1
#Check whether the expression represents an equality
#If it represents an equality, then none of the intervals
#would have satisfied the expression due to floating point
#differences. Add all the undecided values to the plot.
if self.has_equality:
for intervals in interval_list:
intervalx = intervals[0]
intervaly = intervals[1]
func_eval = func(intervalx, intervaly)
if func_eval[1] and func_eval[0] is not False:
plot_list.append([intervalx, intervaly])
return plot_list, 'fill'

def _get_meshes_grid(self):
"""Generates the mesh for generating a contour.

In the case of equality, contour function of matplotlib can
be used. In other cases, matplotlib's contourf is used.
"""
equal = False
if isinstance(self.expr, Equality):
expr = self.expr.lhs - self.expr.rhs
equal = True

elif isinstance(self.expr, (GreaterThan, StrictGreaterThan)):
expr = self.expr.lhs - self.expr.rhs

elif isinstance(self.expr, (LessThan, StrictLessThan)):
expr = self.expr.rhs - self.expr.lhs
else:
raise NotImplementedError("The expression is not supported for "
"plotting in uniform meshed plot.")
np = import_module('numpy')
xarray = np.linspace(self.start_x, self.end_x, self.nb_of_points)
yarray = np.linspace(self.start_y, self.end_y, self.nb_of_points)
x_grid, y_grid = np.meshgrid(xarray, yarray)

func = vectorized_lambdify((self.var_x, self.var_y), expr)
z_grid = func(x_grid, y_grid)
z_grid[np.ma.where(z_grid < 0)] = -1
z_grid[np.ma.where(z_grid > 0)] = 1
if equal:
return xarray, yarray, z_grid, 'contour'
else:
return xarray, yarray, z_grid, 'contourf'

[docs]@doctest_depends_on(modules=('matplotlib',))
def plot_implicit(expr, x_var=None, y_var=None, **kwargs):
"""A plot function to plot implicit equations / inequalities.

Arguments
=========

- expr : The equation / inequality that is to be plotted.
- x_var (optional) : symbol to plot on x-axis or tuple giving symbol
and range as (symbol, xmin, xmax)
- y_var (optional) : symbol to plot on y-axis or tuple giving symbol
and range as (symbol, ymin, ymax)

If neither x_var nor y_var are given then the free symbols in the
expression will be assigned in the order they are sorted.

The following keyword arguments can also be used:

- adaptive. Boolean. The default value is set to True. It has to be
set to False if you want to use a mesh grid.

- depth integer. The depth of recursion for adaptive mesh grid.
Default value is 0. Takes value in the range (0, 4).

- points integer. The number of points if adaptive mesh grid is not
used. Default value is 200.

- title string .The title for the plot.

- xlabel string. The label for the x-axis

- ylabel string. The label for the y-axis

Aesthetics options:

- line_color: float or string. Specifies the color for the plot.
See Plot to see how to set color for the plots.

plot_implicit, by default, uses interval arithmetic to plot functions. If
the expression cannot be plotted using interval arithmetic, it defaults to
a generating a contour using a mesh grid of fixed number of points. By
setting adaptive to False, you can force plot_implicit to use the mesh
grid. The mesh grid method can be effective when adaptive plotting using
interval arithmetic, fails to plot with small line width.

Examples
========

Plot expressions:

>>> from sympy import plot_implicit, cos, sin, symbols, Eq, And
>>> x, y = symbols('x y')

Without any ranges for the symbols in the expression

>>> p1 = plot_implicit(Eq(x**2 + y**2, 5))

With the range for the symbols

>>> p2 = plot_implicit(Eq(x**2 + y**2, 3),
...         (x, -3, 3), (y, -3, 3))

With depth of recursion as argument.

>>> p3 = plot_implicit(Eq(x**2 + y**2, 5),
...         (x, -4, 4), (y, -4, 4), depth = 2)

Using mesh grid and not using adaptive meshing.

>>> p4 = plot_implicit(Eq(x**2 + y**2, 5),
...         (x, -5, 5), (y, -2, 2), adaptive=False)

Using mesh grid with number of points as input.

>>> p5 = plot_implicit(Eq(x**2 + y**2, 5),
...         (x, -5, 5), (y, -2, 2),

Plotting regions.

>>> p6 = plot_implicit(y > x**2)

Plotting Using boolean conjunctions.

>>> p7 = plot_implicit(And(y > x, y > -x))

When plotting an expression with a single variable (y - 1, for example),
specify the x or the y variable explicitly:

>>> p8 = plot_implicit(y - 1, y_var=y)
>>> p9 = plot_implicit(x - 1, x_var=x)

"""
has_equality = False  # Represents whether the expression contains an Equality,
#GreaterThan or LessThan

def arg_expand(bool_expr):
"""
Recursively expands the arguments of an Boolean Function
"""
for arg in bool_expr.args:
if isinstance(arg, BooleanFunction):
arg_expand(arg)
elif isinstance(arg, Relational):
arg_list.append(arg)

arg_list = []
if isinstance(expr, BooleanFunction):
arg_expand(expr)

#Check whether there is an equality in the expression provided.
if any(isinstance(e, (Equality, GreaterThan, LessThan))
for e in arg_list):
has_equality = True

elif not isinstance(expr, Relational):
expr = Eq(expr, 0)
has_equality = True
elif isinstance(expr, (Equality, GreaterThan, LessThan)):
has_equality = True

xyvar = [i for i in (x_var, y_var) if i is not None]
free_symbols = expr.free_symbols
range_symbols = Tuple(*flatten(xyvar)).free_symbols
undeclared = free_symbols - range_symbols
if len(free_symbols & range_symbols) > 2:
raise NotImplementedError("Implicit plotting is not implemented for "
"more than 2 variables")

#Create default ranges if the range is not provided.
default_range = Tuple(-5, 5)
def _range_tuple(s):
if isinstance(s, Symbol):
return Tuple(s) + default_range
if len(s) == 3:
return Tuple(*s)
raise ValueError('symbol or (symbol, min, max) expected but got %s' % s)

if len(xyvar) == 0:
xyvar = list(_sort_gens(free_symbols))
var_start_end_x = _range_tuple(xyvar[0])
x = var_start_end_x[0]
if len(xyvar) != 2:
if x in undeclared or not undeclared:
xyvar.append(Dummy('f(%s)' % x.name))
else:
xyvar.append(undeclared.pop())
var_start_end_y = _range_tuple(xyvar[1])

nb_of_points = kwargs.pop('points', 300)
depth = kwargs.pop('depth', 0)
line_color = kwargs.pop('line_color', "blue")
#Check whether the depth is greater than 4 or less than 0.
if depth > 4:
depth = 4
elif depth < 0:
depth = 0

series_argument = ImplicitSeries(expr, var_start_end_x, var_start_end_y,
has_equality, use_interval, depth,
nb_of_points, line_color)
show = kwargs.pop('show', True)

#set the x and y limits
kwargs['xlim'] = tuple(float(x) for x in var_start_end_x[1:])
kwargs['ylim'] = tuple(float(y) for y in var_start_end_y[1:])
# set the x and y labels
kwargs.setdefault('xlabel', var_start_end_x[0].name)
kwargs.setdefault('ylabel', var_start_end_y[0].name)
p = Plot(series_argument, **kwargs)
if show:
p.show()
return p