Source code for diofant.plotting.plot_implicit

"""Implicit plotting module for Diofant

The module implements a data series called ImplicitSeries which is used by
``Plot`` class to plot implicit plots for different backends.

See Also


from ..core import Dummy, Eq, Symbol, Tuple, sympify
from ..core.relational import Equality, GreaterThan, LessThan, Relational
from ..external import import_module
from ..logic.boolalg import BooleanFunction
from ..polys.polyutils import _sort_gens
from ..utilities import flatten, lambdify
from ..utilities.decorator import doctest_depends_on
from .plot import BaseSeries, Plot

[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().__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 # If the expression has equality, i.e. Eq, Greaterthan, LessThan. self.has_equality = has_equality self.nb_of_points = nb_of_points self.use_interval_math = use_interval_math self.depth = 4 + depth self.line_color = line_color def get_raster(self): return self._get_meshes_grid() 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 self.expr.has(Equality): # pragma: no cover raise NotImplementedError("The expression is not supported for " "plotting in uniform meshed plot.") else: expr = self.expr 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 = lambdify((self.var_x, self.var_y), expr, "numpy") z_grid = func(x_grid, y_grid) z_grid[ < 0)] = -1 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. Parameters ========== ``expr`` : Expr The equation / inequality that is to be plotted. ``x_var`` : symbol or tuple, optional symbol to plot on x-axis or tuple giving symbol and range as ``(symbol, xmin, xmax)`` ``y_var`` : symbol or tuple, 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, optional 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`` : str The title for the plot. ``xlabel`` : str The label for the x-axis ``ylabel`` : string The label for the y-axis Aesthetics options: ``line_color`` : float or str 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: 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), ... adaptive=False, points=400) 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) """ # Represents whether the expression contains an Equality, # GreaterThan or LessThan has_equality = False 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: # pragma: no cover 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, (Dummy, Symbol)): return Tuple(s) + default_range if len(s) == 3: return Tuple(*s) raise ValueError('symbol or `(symbol, min, max)` expected but got %s' % str(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)' % else: xyvar.append(undeclared.pop()) var_start_end_y = _range_tuple(xyvar[1]) use_interval = kwargs.pop('adaptive', False) 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: return p