def plot_implicit(
expr,
x_var=None,
y_var=None,
adaptive=True,
depth=0,
points=300,
line_color="blue",
show=True,
**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 300.
- ``show`` Boolean. Default value is True. If set to False, the plot will
not be shown. See ``Plot`` for further information.
- ``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.
Default value is "Blue"
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),
... 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)
"""
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])
# 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,
adaptive,
depth,
points,
line_color,
)
# 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
def test__sort_gens():
assert _sort_gens([]) == ()
assert _sort_gens([x]) == (x,)
assert _sort_gens([p]) == (p,)
assert _sort_gens([q]) == (q,)
assert _sort_gens([x, p]) == (x, p)
assert _sort_gens([p, x]) == (x, p)
assert _sort_gens([q, p]) == (p, q)
assert _sort_gens([q, p, x]) == (x, p, q)
assert _sort_gens([x, p, q], wrt=x) == (x, p, q)
assert _sort_gens([x, p, q], wrt=p) == (p, x, q)
assert _sort_gens([x, p, q], wrt=q) == (q, x, p)
assert _sort_gens([x, p, q], wrt='x') == (x, p, q)
assert _sort_gens([x, p, q], wrt='p') == (p, x, q)
assert _sort_gens([x, p, q], wrt='q') == (q, x, p)
assert _sort_gens([x, p, q], wrt='x,q') == (x, q, p)
assert _sort_gens([x, p, q], wrt='q,x') == (q, x, p)
assert _sort_gens([x, p, q], wrt='p,q') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q,p') == (q, p, x)
assert _sort_gens([x, p, q], wrt='x, q') == (x, q, p)
assert _sort_gens([x, p, q], wrt='q, x') == (q, x, p)
assert _sort_gens([x, p, q], wrt='p, q') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q, p') == (q, p, x)
assert _sort_gens([x, p, q], wrt=[x, 'q']) == (x, q, p)
assert _sort_gens([x, p, q], wrt=[q, 'x']) == (q, x, p)
assert _sort_gens([x, p, q], wrt=[p, 'q']) == (p, q, x)
assert _sort_gens([x, p, q], wrt=[q, 'p']) == (q, p, x)
assert _sort_gens([x, p, q], wrt=['x', 'q']) == (x, q, p)
assert _sort_gens([x, p, q], wrt=['q', 'x']) == (q, x, p)
assert _sort_gens([x, p, q], wrt=['p', 'q']) == (p, q, x)
assert _sort_gens([x, p, q], wrt=['q', 'p']) == (q, p, x)
assert _sort_gens([x, p, q], sort='x > p > q') == (x, p, q)
assert _sort_gens([x, p, q], sort='p > x > q') == (p, x, q)
assert _sort_gens([x, p, q], sort='p > q > x') == (p, q, x)
assert _sort_gens([x, p, q], wrt='x', sort='q > p') == (x, q, p)
assert _sort_gens([x, p, q], wrt='p', sort='q > x') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q', sort='p > x') == (q, p, x)
# https://github.com/sympy/sympy/issues/19353
n1 = Symbol('\n1')
assert _sort_gens([n1]) == (n1,)
assert _sort_gens([x, n1]) == (x, n1)
X = symbols('x0,x1,x2,x10,x11,x12,x20,x21,x22')
assert _sort_gens(X) == X
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
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),
... adaptive=False, points=400)
Plotting regions.
>>> p6 = plot_implicit(y > x**2)
Plotting Using boolean conjunctions.
>>> p7 = plot_implicit(And(y > x, y > -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])
use_interval = kwargs.pop("adaptive", True)
nb_of_points = kwargs.pop("points", 300)
depth = kwargs.pop("depth", 0)
# 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
)
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
def test__sort_gens():
assert _sort_gens([]) == ()
assert _sort_gens([x]) == (x, )
assert _sort_gens([p]) == (p, )
assert _sort_gens([q]) == (q, )
assert _sort_gens([x, p]) == (x, p)
assert _sort_gens([p, x]) == (x, p)
assert _sort_gens([q, p]) == (p, q)
assert _sort_gens([q, p, x]) == (x, p, q)
assert _sort_gens([x, p, q], wrt=x) == (x, p, q)
assert _sort_gens([x, p, q], wrt=p) == (p, x, q)
assert _sort_gens([x, p, q], wrt=q) == (q, x, p)
assert _sort_gens([x, p, q], wrt='x') == (x, p, q)
assert _sort_gens([x, p, q], wrt='p') == (p, x, q)
assert _sort_gens([x, p, q], wrt='q') == (q, x, p)
assert _sort_gens([x, p, q], wrt='x,q') == (x, q, p)
assert _sort_gens([x, p, q], wrt='q,x') == (q, x, p)
assert _sort_gens([x, p, q], wrt='p,q') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q,p') == (q, p, x)
assert _sort_gens([x, p, q], wrt='x, q') == (x, q, p)
assert _sort_gens([x, p, q], wrt='q, x') == (q, x, p)
assert _sort_gens([x, p, q], wrt='p, q') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q, p') == (q, p, x)
assert _sort_gens([x, p, q], wrt=[x, 'q']) == (x, q, p)
assert _sort_gens([x, p, q], wrt=[q, 'x']) == (q, x, p)
assert _sort_gens([x, p, q], wrt=[p, 'q']) == (p, q, x)
assert _sort_gens([x, p, q], wrt=[q, 'p']) == (q, p, x)
assert _sort_gens([x, p, q], wrt=['x', 'q']) == (x, q, p)
assert _sort_gens([x, p, q], wrt=['q', 'x']) == (q, x, p)
assert _sort_gens([x, p, q], wrt=['p', 'q']) == (p, q, x)
assert _sort_gens([x, p, q], wrt=['q', 'p']) == (q, p, x)
assert _sort_gens([x, p, q], sort='x > p > q') == (x, p, q)
assert _sort_gens([x, p, q], sort='p > x > q') == (p, x, q)
assert _sort_gens([x, p, q], sort='p > q > x') == (p, q, x)
assert _sort_gens([x, p, q], wrt='x', sort='q > p') == (x, q, p)
assert _sort_gens([x, p, q], wrt='p', sort='q > x') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q', sort='p > x') == (q, p, x)
X = symbols('x0,x1,x2,x10,x11,x12,x20,x21,x22')
assert _sort_gens(X) == X
def test__sort_gens():
assert _sort_gens([]) == ()
assert _sort_gens([x]) == (x,)
assert _sort_gens([p]) == (p,)
assert _sort_gens([q]) == (q,)
assert _sort_gens([x, p]) == (x, p)
assert _sort_gens([p, x]) == (x, p)
assert _sort_gens([q, p]) == (p, q)
assert _sort_gens([q, p, x]) == (x, p, q)
assert _sort_gens([x, p, q], wrt=x) == (x, p, q)
assert _sort_gens([x, p, q], wrt=p) == (p, x, q)
assert _sort_gens([x, p, q], wrt=q) == (q, x, p)
assert _sort_gens([x, p, q], wrt='x') == (x, p, q)
assert _sort_gens([x, p, q], wrt='p') == (p, x, q)
assert _sort_gens([x, p, q], wrt='q') == (q, x, p)
assert _sort_gens([x, p, q], wrt='x,q') == (x, q, p)
assert _sort_gens([x, p, q], wrt='q,x') == (q, x, p)
assert _sort_gens([x, p, q], wrt='p,q') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q,p') == (q, p, x)
assert _sort_gens([x, p, q], wrt='x, q') == (x, q, p)
assert _sort_gens([x, p, q], wrt='q, x') == (q, x, p)
assert _sort_gens([x, p, q], wrt='p, q') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q, p') == (q, p, x)
assert _sort_gens([x, p, q], wrt=[x, 'q']) == (x, q, p)
assert _sort_gens([x, p, q], wrt=[q, 'x']) == (q, x, p)
assert _sort_gens([x, p, q], wrt=[p, 'q']) == (p, q, x)
assert _sort_gens([x, p, q], wrt=[q, 'p']) == (q, p, x)
assert _sort_gens([x, p, q], wrt=['x', 'q']) == (x, q, p)
assert _sort_gens([x, p, q], wrt=['q', 'x']) == (q, x, p)
assert _sort_gens([x, p, q], wrt=['p', 'q']) == (p, q, x)
assert _sort_gens([x, p, q], wrt=['q', 'p']) == (q, p, x)
assert _sort_gens([x, p, q], sort='x > p > q') == (x, p, q)
assert _sort_gens([x, p, q], sort='p > x > q') == (p, x, q)
assert _sort_gens([x, p, q], sort='p > q > x') == (p, q, x)
assert _sort_gens([x, p, q], wrt='x', sort='q > p') == (x, q, p)
assert _sort_gens([x, p, q], wrt='p', sort='q > x') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q', sort='p > x') == (q, p, x)
X = symbols('x0,x1,x2,x10,x11,x12,x20,x21,x22')
assert _sort_gens(X) == X
def test__sort_gens():
assert _sort_gens([]) == ()
assert _sort_gens([x]) == (x, )
assert _sort_gens([p]) == (p, )
assert _sort_gens([q]) == (q, )
assert _sort_gens([x, p]) == (x, p)
assert _sort_gens([p, x]) == (x, p)
assert _sort_gens([q, p]) == (p, q)
assert _sort_gens([q, p, x]) == (x, p, q)
assert _sort_gens([x, p, q], wrt='x') == (x, p, q)
assert _sort_gens([x, p, q], wrt='p') == (p, x, q)
assert _sort_gens([x, p, q], wrt='q') == (q, x, p)
assert _sort_gens([x, p, q], sort='x < p < q') == (x, p, q)
assert _sort_gens([x, p, q], sort='p < x < q') == (p, x, q)
assert _sort_gens([x, p, q], sort='p < q < x') == (p, q, x)
assert _sort_gens([x, p, q], wrt='x', sort='q < p') == (x, q, p)
assert _sort_gens([x, p, q], wrt='p', sort='q < x') == (p, q, x)
assert _sort_gens([x, p, q], wrt='q', sort='p < x') == (q, p, x)
def test__sort_gens():
assert _sort_gens([]) == ()
assert _sort_gens([x]) == (x,)
assert _sort_gens([p]) == (p,)
assert _sort_gens([q]) == (q,)
assert _sort_gens([x,p]) == (x,p)
assert _sort_gens([p,x]) == (x,p)
assert _sort_gens([q,p]) == (p,q)
assert _sort_gens([q,p,x]) == (x,p,q)
assert _sort_gens([x,p,q], wrt='x') == (x,p,q)
assert _sort_gens([x,p,q], wrt='p') == (p,x,q)
assert _sort_gens([x,p,q], wrt='q') == (q,x,p)
assert _sort_gens([x,p,q], sort='x < p < q') == (x, p, q)
assert _sort_gens([x,p,q], sort='p < x < q') == (p, x, q)
assert _sort_gens([x,p,q], sort='p < q < x') == (p, q, x)
assert _sort_gens([x,p,q], wrt='x', sort='q < p') == (x, q, p)
assert _sort_gens([x,p,q], wrt='p', sort='q < x') == (p, q, x)
assert _sort_gens([x,p,q], wrt='q', sort='p < x') == (q, p, x)
def test__sort_gens():
assert _sort_gens([]) == ()
assert _sort_gens([x]) == (x, )
assert _sort_gens([p]) == (p, )
assert _sort_gens([q]) == (q, )
assert _sort_gens([x, p]) == (x, p)
assert _sort_gens([p, x]) == (x, p)
assert _sort_gens([q, p]) == (p, q)
assert _sort_gens([q, p, x]) == (x, p, q)
assert _sort_gens([x, p, q], wrt=x) == (x, p, q)
assert _sort_gens([x, p, q], wrt=p) == (p, x, q)
assert _sort_gens([x, p, q], wrt=q) == (q, x, p)
assert _sort_gens([x, p, q], wrt="x") == (x, p, q)
assert _sort_gens([x, p, q], wrt="p") == (p, x, q)
assert _sort_gens([x, p, q], wrt="q") == (q, x, p)
assert _sort_gens([x, p, q], wrt="x,q") == (x, q, p)
assert _sort_gens([x, p, q], wrt="q,x") == (q, x, p)
assert _sort_gens([x, p, q], wrt="p,q") == (p, q, x)
assert _sort_gens([x, p, q], wrt="q,p") == (q, p, x)
assert _sort_gens([x, p, q], wrt="x, q") == (x, q, p)
assert _sort_gens([x, p, q], wrt="q, x") == (q, x, p)
assert _sort_gens([x, p, q], wrt="p, q") == (p, q, x)
assert _sort_gens([x, p, q], wrt="q, p") == (q, p, x)
assert _sort_gens([x, p, q], wrt=[x, "q"]) == (x, q, p)
assert _sort_gens([x, p, q], wrt=[q, "x"]) == (q, x, p)
assert _sort_gens([x, p, q], wrt=[p, "q"]) == (p, q, x)
assert _sort_gens([x, p, q], wrt=[q, "p"]) == (q, p, x)
assert _sort_gens([x, p, q], wrt=["x", "q"]) == (x, q, p)
assert _sort_gens([x, p, q], wrt=["q", "x"]) == (q, x, p)
assert _sort_gens([x, p, q], wrt=["p", "q"]) == (p, q, x)
assert _sort_gens([x, p, q], wrt=["q", "p"]) == (q, p, x)
assert _sort_gens([x, p, q], sort="x > p > q") == (x, p, q)
assert _sort_gens([x, p, q], sort="p > x > q") == (p, x, q)
assert _sort_gens([x, p, q], sort="p > q > x") == (p, q, x)
assert _sort_gens([x, p, q], wrt="x", sort="q > p") == (x, q, p)
assert _sort_gens([x, p, q], wrt="p", sort="q > x") == (p, q, x)
assert _sort_gens([x, p, q], wrt="q", sort="p > x") == (q, p, x)
X = symbols("x0,x1,x2,x10,x11,x12,x20,x21,x22")
assert _sort_gens(X) == X